3511 frct-003I.rss.xml frct-003I frct-003I.xml 2022 1 1 Jon SterlingCarlo Angiuli Foundations of relative category theory

We assume knowledge of basic categorical concepts such as categories, functors, and natural transformations. The purpose of this lecture is to develop the notion of a category over another category.

We will draw on the work of Ahrens and Lumsdaine, Bénabou, Borceux, Jacobs, and Streicher.

1 3512 frct-000R.rss.xml frct-000R frct-000R.xml Jon Sterling Foundational assumptions 1 1 3513 frct-001F.rss.xml Definition frct-001F frct-001F.xml Jon Sterling Meta-category

A meta-category \mathfrak {C} is defined by explaining what an object of \mathfrak {E} is, and, given two objects x,y \in \mathfrak {E}, what a morphism from x to y is, together with the following operations:

1. for each object x \in \mathfrak {E}, an identity map 1 _{ x } : x \to x,
2. for any two maps f:x \to y and g:y \to z, a composite map f;g : x \to z,
3. such that the following equations hold: 1 _{ x } ;h = h \qquad h; 1 _{ y } = h \qquad f;(g;h) = (f;g);{h}
1 2 3514 frct-003J.rss.xml Remark frct-003J frct-003J.xml Collections

In the definition of meta-categories, we have not imposed any restrictions on what kinds of things the objects and morphisms are; our definition is pre-mathematical, so we do not assume beforehand that there is a such thing as a collection of “all” meta-categories.

We may define analogous notions of meta-functor, etc. But we do not assume that the notion of “all meta-functors \mathfrak {C} \to \mathfrak {D}” is well-defined; the notion is entirely schematic.

Assumption. We assume a meta-category \boldsymbol { \mathfrak {Coll} } whose objects we will refer to as “collections”. We assume that the meta-category of all collections satisfies the axioms of Lawvere’s ETCS.

1 3 3515 frct-001G.rss.xml Definition frct-001G frct-001G.xml Category

A category E is defined to be a meta-category whose objects are defined to be the elements of some collection, and for any two objects x,y \in E the morphisms x \to y are defined to be the elements of some collection.

1 4 3516 frct-003K.rss.xml Remark frct-003K frct-003K.xml Cartesian closure of the meta-category of all categories

Consequently there exists a meta-category \boldsymbol { \mathfrak {Cat} } of all categories. Following Lawvere (but deviating from some other authors that ground the notion of meta-categories in classes) we notice that \boldsymbol { \mathfrak {Cat} } is cartesian closed; in other words, all functor categories exist regardless of size.

Assumption. At times we may assume that there exists a category \mathbf {Set} \subseteq \boldsymbol { \mathfrak {Coll} } of collections that we will refer to as sets, such that \mathbf {Set} is closed under the axioms of ETCS. Rather than work with \mathbf {Set} at all times, our approach is to use the tools of relative category theory to objectify the notions of “small” and “locally small” category over any category B, generalizing the role of \mathbf {Set} from classical category theory.

2 3517 frct-0008.rss.xml frct-0008 frct-0008.xml Carlo AngiuliJon Sterling Displayed categories and fibrations 2 1 3518 frct-0000.rss.xml Definition frct-0000 frct-0000.xml 2022 Jon Sterling Displayed category

Let B be a category. A displayed category E over B is defined by the following data according to (Ahrens and Lumsdaine):

1. for each object x \in B, a collection of displayed objects E _{ x },
2. for each morphism { x } \xrightarrow {{ f }}{ y } \in B and displayed objects \bar {x} \in E _{ x } and \bar {y} \in E _{ y }, a family of collections of displayed morphisms \mathbf {hom} _{ E _{ f } } { \left ( \bar {x} , \bar {y} \right )}, an element of which shall denote by { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} },
3. for each x \in B and \bar {x} \in E _{ x }, a displayed morphism { \bar {x} } \xrightarrow [ 1 _{ x } ]{ 1 _{ \bar {x} } }{ \bar {x} },
4. for each { x } \xrightarrow {{ f }}{ y } and { y } \xrightarrow {{ g }}{ z } in B and objects \bar {x} \in E _{ x } , \bar {y} \in E _{ y } , \bar {z} \in E _{ z }, a function \mathbf {hom} _{ E _{ f } } { \left ( \bar {x} , \bar {y} \right )} \times \mathbf {hom} _{ E _{ g } } { \left ( \bar {y} , \bar {z} \right )} \to \mathbf {hom} _{ E _{ f;g } } { \left ( \bar {x} , \bar {z} \right )} that we will denote like ordinary (diagrammatic) function composition,
5. such that the following equations hold: 1 _{ \bar {x} } ; \bar {h} = \bar {h} \qquad \bar {h}; 1 _{ \bar {y} } = \bar {h} \qquad \bar {f};( \bar {g}; \bar {h}) = ( \bar {f}; \bar {g}); \bar {h} Note that these are well-defined because of the corresponding laws for the base category B.
2 1 1 3519 frct-003R.rss.xml Notation frct-003R frct-003R.xml 2022 12 27 Square brackets for subscripts

When we have too many subscripts, we will write E[x] instead of E _{ x }.

2 2 3520 frct-0001.rss.xml Definition frct-0001 frct-0001.xml Jon Sterling Cartesian morphism

Let E be displayed over B, and let { x } \xrightarrow {{ f }}{ y } \in B; a morphism { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } in E is called cartesian over f when for any { u } \xrightarrow {{ m }}{ x } and { \bar {u} } \xrightarrow [ m;f ]{ \bar {h} }{ \bar {y} } there exists a unique { \bar {u} } \xrightarrow [ m ]{ \bar {m} }{ \bar {x} } with \bar {m}; \bar {f} = \bar {h}. We visualize this unique factorization of \bar {h} through \bar {f} over m as follows: Above we have used the “pullback corner” to indicate \bar {x} \to \bar {y} as a cartesian map. We return to this in our discussion of the self-indexing of a category.

2 3 3521 frct-003T.rss.xml Definition frct-003T frct-003T.xml 2023 3 11 Jon Sterling Cleaving of a displayed category

Let E be a displayed category over C; following Ahrens and Lumsdaine, we define a cleaving of E to be a function assigning to each { x } \xrightarrow {{ f }}{ y } \in C and u \in C_y a cartesian morphism { f ^{ * } u } \xrightarrow [ f ]{ \bar {f} }{ u } in E called the chosen cartesian lift of u along f.

2 4 3522 frct-003U.rss.xml Remark frct-003U frct-003U.xml 2023 3 11 Jon Sterling Cleavings from the axiom of choice

Assuming the axiom of choice, any cartesian fibration may be equipped with a (non-canonical) cleaving. In the current version of these notes, we freely use this principle, but in the future we would like to follow Ahrens and Lumsdaine in distinguishing between weak and strong fibrations, in which the latter come equipped with cleavings.

2 5 3523 frct-0002.rss.xml Definition frct-0002 frct-0002.xml Cartesian fibration

A displayed category E over B is said to be a cartesian fibration, when for each morphism { x } \xrightarrow {{ f }}{ y } and displayed object \bar {y} \in E _{ y }, there exists a displayed object \bar {x} \in E _{ x } and a cartesian morphism { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} }. Note that the pair ( \bar {x}, \bar {f}) is unique up to unique isomorphism, so being a cartesian fibration is a property of a displayed category.

We will also refer to cartesian fibrations as simply fibrations or fibered categories.

There are other variations of fibration. For instance, E is said to be an isofibration when the condition above holds just for isomorphisms f : x \cong y in the base.

2 6 3524 frct-0003.rss.xml frct-0003 frct-0003.xml The fundamental self-indexing 2 6 1 3525 frct-001X.rss.xml Construction frct-001X frct-001X.xml The fundamental self-indexing

Let B be an ordinary category; there is an important displayed category \underline { B } over B given fiberwise by the slices of B.

1. For x \in B, we define \underline { B } _{ x } to be the collection { B } _{ / x } of pairs ( \bar {x} \in B, { \bar {x} } \xrightarrow {{ p _{ x } }}{ x } ).
2. For { x } \xrightarrow {{ f }}{ y } \in B, we define \underline { B } _{ f } to be the collection of commuting squares in the following configuration: 2 6 2 3526 frct-001Y.rss.xml Exercise frct-001Y frct-001Y.xml Pullbacks and cartesian maps

Prove that the fundamental self-indexing \underline { B } is a cartesian fibration if and only if B has pullbacks.

2 7 3527 frct-0014.rss.xml frct-0014 frct-0014.xml Carlo Angiuli The generalized pullback lemma

In light of our discussion of the fundamental self-indexing, the following result for displayed categories generalizes the ordinary “pullback lemma.”

2 7 1 3528 frct-001H.rss.xml Lemma frct-001H frct-001H.xml Carlo Angiuli Generalized pullback lemma

Let { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} }, and suppose that { \bar {y} } \xrightarrow [ g ]{ \bar {g} }{ \bar {z} } is cartesian over g. Then \bar {f}; \bar {g} is cartesian over f;g if and only if \bar {f} is cartesian over f. Proof.

Suppose first that \bar {f} is cartesian. To see that \bar {f}; \bar {g} is cartesian, we must construct a unique factorization as follows: Because \bar {g} is cartesian, we can factor \bar {h} = i; \bar {g} for a unique { \bar {u} } \xrightarrow [ m;f ]{ i }{ \bar {y} }. Then, because \bar {f} is cartesian, we can further factor i = j; \bar {f} for a unique { \bar {u} } \xrightarrow [ m ]{ j }{ \bar {x} }. We conclude that there is a unique { \bar {u} } \xrightarrow [ m ]{ j }{ \bar {x} } for which \bar {h} = j; \bar {f}; \bar {g}, as required.

Conversely, suppose that \bar {f}; \bar {g} is cartesian. To see that \bar {f} is cartesian, we must construct a unique factorization as follows: Because \bar {f}; \bar {g} is cartesian, we can factor \bar {h}; \bar {g} = i; \bar {f}; \bar {g} for a unique { \bar {u} } \xrightarrow [ m ]{ i }{ \bar {x} }. On the other hand, because \bar {g} is cartesian, there is a unique { \bar {u} } \xrightarrow [ m;f ]{ j }{ \bar {y} } for which \bar {h}; \bar {g} = j; \bar {g}; as both \bar {h} and i; \bar {f} satisfy this condition, we conclude \bar {h}=i; \bar {f}. Therefore, there is a unique { \bar {u} } \xrightarrow [ m ]{ i }{ \bar {x} } for which \bar {h} = i; \bar {f}, as required.

2 8 3529 frct-0029.rss.xml frct-0029 frct-0029.xml Carlo Angiuli Jon Sterling An alternative definition of fibration 2 8 1 3530 frct-003S.rss.xml Warning frct-003S frct-003S.xml 2023 3 2 Jon Sterling Competing terminologies for cartesian maps

Some authors including Grothendieck give an equivalent definition of cartesian fibration that factors through a nonequivalent definition of cartesian morphisms. Such authors refer to our notion of cartesian morphism as “hypercartesian” (see Streicher).

2 8 2 3531 frct-002A.rss.xml Definition frct-002A frct-002A.xml Carlo Angiuli Hypocartesian morphisms

Let E be displayed over B, and let f:x \to y \in B; a morphism { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } in E is called hypocartesian over f when for any \bar {u} \in E _{ x } and { \bar {u} } \xrightarrow [ f ]{ \bar {h} }{ \bar {y} } there exists a unique { \bar {u} } \xrightarrow [ 1 _{ x } ]{ i }{ \bar {x} } with i; \bar {f} = \bar {h} as follows: 2 8 3 3532 frct-003G.rss.xml Remark frct-003G frct-003G.xml Carlo Angiuli Hypocartesian morphisms need not be closed composable

Cartesian morphisms are clearly hypocartesian (setting u=x and m= 1 _{ x }), but the converse does not hold. The problem is that in an arbitrary displayed category, hypocartesian morphisms may not be closed under composition.

2 8 4 3533 frct-002C.rss.xml Lemma frct-002C frct-002C.xml Hypocartesian = cartesian in a cartesian fibration

Let E be a cartesian fibration, and let { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } be displayed over { x } \xrightarrow {{ f }}{ y }. The displayed morphism \bar {f} is cartesian if and only if it is hypocartesian.

Proof.

Any cartesian map is clearly hypocartesian. To see that a hypocartesian map { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } in a cartesian fibration is cartesian, we consider the cartesian lift of { x } \xrightarrow {{ f }}{ y } under \bar {y}: As the cartesian lift \bar {x} ' \to \bar {y} is also hypocartesian, it follows that there is a unique vertical isomorphism identifying \bar {x} with \bar {x} ' factoring { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } through { \bar {x} ' } \xrightarrow [ f ]{ \bar {f} ' }{ \bar {y} }. Being cartesian over f is clearly stable under isomorphism, hence we conclude that \bar {f} is cartesian from the fact that \bar {f} ' is cartesian.

Grothendieck defines a fibration in terms of (what we refer to as) hypocartesian morphisms rather than (what we refer to as) cartesian morphisms, and therefore imposes the additional constraint that the hypocartesian morphisms be closed under composition. Below, we verify that these two definitions of cartesian fibration coincide.

2 8 5 3534 frct-002B.rss.xml Lemma frct-002B frct-002B.xml Carlo Angiuli Equivalence with Grothendieck's fibrations

Let E be displayed over B. Then E is a cartesian fibration if and only if the following two conditions hold:

1. Hypocartesian lifts. For each f:x \to y \in B and \bar {y} \in E _{ y } there exists a displayed object \bar {x} \in E _{ x } and hypocartesian morphism \bar {f}: \bar {x} \xrightarrow [ f ]{} \bar {y}.
2. Closure under composition. If \bar {f}: \bar {x} \xrightarrow [ f ]{} \bar {y} and \bar {g}: \bar {y} \xrightarrow [ g ]{} \bar {z} are hypocartesian, then \bar {f}; \bar {g} is hypocartesian.
Proof.

Suppose first that E is a cartesian fibration in our sense. Then E has hypocartesian lifts because it has cartesian lifts. For closure under composition, fix hypocartesian \bar {f}, \bar {g}; because hypocartesian and cartesian maps coincide in a cartesian fibration we know that \bar {f}, \bar {g} are also cartesian and hence by the generalized pullback lemma so is the composite \bar {f}; \bar {g}; therefore it follows that \bar {f}; \bar {g} is also hypocartesian.

Conversely, suppose that E is a cartesian fibration in the sense of Grothendieck, and let \bar {f}: \bar {x} \xrightarrow [ f ]{} \bar {y} be the hypocartesian lift of f:x \to y at \bar {y} \in E _{ y }; we shall see that \bar {f} is also a cartesian lift of f at \bar {y} by constructing a unique factorization as follows: Let \bar {m}: \bar {u} ' \xrightarrow [ m ]{} \bar {x} be the hypocartesian lift of m at \bar {x}, where \bar {u} ' \in E _{ u }. By hypothesis, the composite \bar {m}; \bar {f} : \bar {u} ' \xrightarrow [ m;f ]{} \bar {y} is hypocartesian, so \bar {h} factors uniquely through \bar {m}; \bar {f} over 1 _{ u }: The composite i; \bar {m} : \bar {u} \xrightarrow [ m ]{} \bar {x} is the required (cartesian) factorization of \bar {h} through \bar {f} over m. To see that i; \bar {m} is the unique such map, we observe that all morphisms \bar {u} \xrightarrow [ m ]{} \bar {x} factor uniquely through \bar {m} over 1 _{ u } as a consequence of \bar {m} being hypocartesian.

2 8 6 3535 frct-002D.rss.xml Remark frct-002D frct-002D.xml Two ways to generalize pullbacks

Hypocartesian and cartesian morphisms can be thought of as two distinct ways to generalize the concept of a pullback, depending on what one considers the essential properties of pullbacks. Hypocartesian morphisms more directly generalize the “little picture” universal property of pullbacks as limiting cones, whereas cartesian morphisms generalize the “big picture” dynamics of the pullback pasting lemma. As we have seen, these two notions coincide in any cartesian fibration; the instance of this result for the fundamental self-indexing verifies that pullbacks can be equivalently presented in terms of cartesian morphisms.

2 9 3536 frct-0004.rss.xml frct-0004 frct-0004.xml Displayed and fibered functors

Let E be displayed over B and let F be displayed over C. If { B } \xrightarrow {{ U }}{ C } is an ordinary functor, than a displayed functor from E to F over U is given by the following data:

1. for each displayed object \bar {x} \in E _{ x }, a displayed object \bar {U} \bar {x} \in F _{ Ux },
2. for each displayed morphism { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} }, a displayed morphism { \bar {U} \bar {x} } \xrightarrow [ Uf ]{ \bar {U} \bar {f} }{ \bar {U} \bar {y} },
3. such that the assignment \bar {U}f preserves displayed identities and displayed composition.

From this notion, we can see that the variation of displayed categories over their base categories itself has a “displayed categorical” structure; up to size issues, we could speak of the displayed bicategory of displayed categories.

Note. The correct notion of morphism between cartesian fibrations is given by displayed functors that preserve cartesian maps. We will call these fibered functors.

2 10 3537 frct-0005.rss.xml frct-0005 frct-0005.xml Fiber categories and vertical maps

Let E be a category displayed over B. A vertical map in E is defined to be one that lies over the identity map in B. For every b \in B, there the collection E _{ b } of displayed objects has the structure of a category; in particular, we set \mathbf {hom} _{ E _{ b } } { \left ( u , v \right )} to be the collection of vertical maps u \xrightarrow [ 1 _{ b } ]{} v.

2 11 3538 frct-000Q.rss.xml frct-000Q frct-000Q.xml Jon Sterling Opposite categories

We adapt Bénabou’s construction as reported by Streicher. Our first construction works for an uncloven fibration, but it has the downside that it requires us to treat the collection of cartesian lifts as a set that can be quotiented, whereas our second construction avoids this by virtue of a cleaving. Up to equivalence, the two constructions coincide for a cloven fibration, which shows that our second construction is independent of the chosen cleaving.

2 11 1 3539 frct-001Z.rss.xml Construction frct-001Z frct-001Z.xml The opposite of an uncloven fibration

Let E be fibered over B; we may define the opposite fibered category E ^{ \mathsf {o} } over B like so:

1. An object of E ^{ \mathsf {o} } _{ x } is given by an object of E _{ x }.
2. Given { x } \xrightarrow {{ f }}{ y } \in B, a morphism \bar {x} \xrightarrow [ f ]{} \bar {y} in E ^{ \mathsf {o} } is given in terms of E by a cartesian map { \bar {y} _{ x } } \xrightarrow [ f ]{ \bar {y} _{ f } }{ \bar {y} } together with a vertical map { \bar {y} _{ x } } \xrightarrow [ 1 _{ x } ]{ h }{ \bar {x} } as depicted below: such that { \left \{ \bar {x} \leftarrow \bar {y} _{ x }^{ 1 } \to \bar {y} \right \} } is identified with { \left \{ \bar {x} \leftarrow \bar {y} _{ x }^{ 2 } \to \bar {y} \right \} } when they agree up to the unique vertical isomorphism \bar {y} _{ x }^{ 1 } \cong \bar {y} _{ x }^{ 2 } induced by the universal property of cartesian maps in the sense that the following diagram commutes: 2 11 2 3540 frct-003W.rss.xml Construction frct-003W frct-003W.xml 2023 3 11 Jon Sterling The opposite of a cloven fibration

Let E be a cloven fibration over B; in this case, we may use the cleaving of E to give a simple construction of the opposite fibration E ^{ \mathsf {o} }.

1. A displayed object of E ^{ \mathsf {o} } over x \in B is given by an object of E over x.
2. Given { x } \xrightarrow {{ f }}{ u } \in B, a displayed morphism \bar {x} \xrightarrow [ f ]{} \bar {y} in E ^{ \mathsf {o} } is given by a vertical map { f ^{ * } \bar {y} } \xrightarrow [ 1 _{ x } ]{ h }{ \bar {x} }.

Going forward, we will not be sensitive to the difference between the two constructions of opposite fibrations.

2 11 3 3541 frct-000T.rss.xml frct-000T frct-000T.xml Cartesian maps in the opposite category

There is a simple characterization of cartesian maps in E ^{ \mathsf {o} }.

2 11 3 1 3542 frct-0020.rss.xml Lemma frct-0020 frct-0020.xml Characterization of cartesian maps

A morphism { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } \in E ^{ \mathsf {o} } is cartesian over { x } \xrightarrow {{ f }}{ y } if and only if the vertical leg of f is an isomorphism.

Proof.

Suppose that { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } is represented by a span { \left \{ \bar {x} \leftarrow \bar {y} _{ x } \to \bar {y} \right \} } in E in which the vertical leg \bar {x} \leftarrow \bar {y} _{ x } is an isomorphism. We must show that { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } is cartesian in E ^{ \mathsf {o} }. We fix a morphism { \bar {w} } \xrightarrow [ m;f ]{ \bar {h} }{ \bar {y} } \in E ^{ \mathsf {o} } where { w } \xrightarrow {{ m }}{ x }, depicted below in terms of E ^{ \mathsf {o} }: We must define the unique intervening map { \bar {w} } \xrightarrow [ m ]{ }{ \bar {x} } in E ^{ \mathsf {o} }. We first translate the above into the language of E by unfolding definitions: The desired intervening map \bar {w} \xrightarrow [ m ]{} \bar {x} \in E ^{ \mathsf {o} } shakes out in the language of E to be a span { \left \{ \bar {w} \xleftarrow [ 1 _{ w } ]{} \bar {y} _{ w } \xrightarrow [ m ]{} \bar {x} \right \} } in which the left-hand leg is vertical and the right-hand leg is cartesian over { w } \xrightarrow {{ m }}{ x }. But the left-hand span { \left \{ \bar {w} \xleftarrow [ 1 _{ w } ]{} \bar {y} _{ w } \to \bar {y} _{ x } \cong \bar {x} \right \} } in the diagram above is exactly what we need.

We leave the converse to the reader.

2 11 4 3543 frct-000U.rss.xml frct-000U frct-000U.xml Cartesian lifts in the opposite category

The foregoing characterization of cartesian maps in E ^{ \mathsf {o} } immediately implies that E ^{ \mathsf {o} } is fibered over B.

2 11 4 1 3544 frct-0021.rss.xml Corollary frct-0021 frct-0021.xml

The displayed category E ^{ \mathsf {o} } is a cartesian fibration.

Proof.

Fixing \bar {y} \in E ^{ \mathsf {o} } _{ y } and f:x \to y \in B, we must exhibit a cartesian lift \bar {f} : \bar {x} \xrightarrow [ f ]{} \bar {y} \in E ^{ \mathsf {o} }. By the characterization it suffices to find any map over f whose vertical component is an isomorphism. Writing \bar {y} _{ x } \xrightarrow [ f ]{} \bar {y} for the cartesian lift of f in E, consider the map in E ^{ \mathsf {o} } presented by the following span in E: 2 11 5 3545 frct-000S.rss.xml frct-000S frct-000S.xml Exegesis of opposite categories

The construction of fibered opposite categories does appear quite involved, but it can be seen to be inevitable from the perspective of the fiber categories E ^{ \mathsf {o} } _{ x }. Indeed, let u,v \in E ^{ \mathsf {o} } _{ x } and fix a vertical map h : u \to v \in E ^{ \mathsf {o} } _{ x }; by unfolding definitions, we see that the vertical map h : u \to v is uniquely determined by a morphism v \to u \in E _{ x }.

Proof.

A displayed morphism u \xrightarrow [ 1 _{ x } ]{} v \in E ^{ \mathsf {o} } is determined by a span { \left \{ u \leftarrow v _{ x } \to v \right \} } \in E where the right-hand map is cartesian over 1 _{ x } : x \to x and the left-hand map is vertical, taken up to the identification of different cartesian lifts v _{ x } \to x. A displayed morphism that is cartesian over the identity is an isomorphism; hence, displayed morphisms u \xrightarrow [ 1 _{ x } ]{} v \in E ^{ \mathsf {o} } are equivalently determined by vertical maps v \to u \in E.

2 12 3546 frct-0006.rss.xml Example frct-0006 frct-0006.xml Jon Sterling The family fibration

Any ordinary category C can be viewed as a displayed category \boldsymbol { \mathcal {F}} _{ C } over \mathbf {Set}:

1. For S \in \mathbf {Set}, an object in \boldsymbol { \mathcal {F}} _{ C } [S] is specified by a functor C^S where S is regarded as a discrete category.
2. Given f : S \to T in \mathbf {Set} and x \in C^S and y \in C^T, a morphism x \xrightarrow [ f ]{} y is given by a morphism x \to f ^{ * } y in C^S where { C^T } \xrightarrow {{ f ^{ * } }}{ C^S } is precomposition with f.

The displayed category \boldsymbol { \mathcal {F}} _{ C } is in fact a cartesian fibration. This family fibration is the starting point for developing a relative form of category theory, the purpose of this lecture. By analogy with viewing an ordinary category C as a fibration \boldsymbol { \mathcal {F}} _{ C } over \mathbf {Set}, we may reasonably define a “relative category” over another base B to be a fibration over B.

This story for relative category theory reflects the way that ordinary categories are “based on” \mathbf {Set} in some sense in spite of the fact that they do not necessarily have sets of objects or even sets of morphisms between objects. Being small and locally small respectively will later be seen to be properties of a family fibration over an arbitrary base B, strictly generalizing the classical notions.

2 12 1 3547 frct-003V.rss.xml Construction frct-003V frct-003V.xml 2023 3 11 Jon Sterling Cleaving the family fibration

The family fibration can be cloven, constructively. In particular, let C be a category and consider the family fibration \boldsymbol { \mathcal {F}} _{ C } over \mathbf {Set}; let { T } \xrightarrow {{ f }}{ S } be a function between sets and let u \in C^S be an S-indexed family of objects of C. The family f ^{ * } u = f;u \in C^T is the object part of a cartesian lift for u along f. The cartesian morphism { f ^{ * } u } \xrightarrow [ f ]{ \bar {f} }{ u } \in \boldsymbol { \mathcal {F}} _{ C } is given by the identity map { f ^{ * } u } \xrightarrow {{ 1 _{ f ^{ * } u } }}{ f ^{ * } u } \in C^T.

2 13 3548 frct-0007.rss.xml Construction frct-0007 frct-0007.xml Change of base of displayed categories

Suppose that E is displayed over B and { X } \xrightarrow {{ F }}{ B } is a functor; then we may define a displayed category F ^{ * } E as over X follows:

1. An object of ( F ^{ * } E) _{ x } is an object of E _{ Fx }.
2. Given \bar {x} \in ( F ^{ * } E) _{ x }, \bar {y} \in ( F ^{ * } E) _{ y } and f : x \to y, a morphism \bar {x} \xrightarrow [ f ]{} \bar {y} in F ^{ * } E is given by a morphism \bar {x} \xrightarrow [ Ff ]{} \bar {y} in E.

We visualize the change of base scenario as follows: 2 14 3549 frct-002J.rss.xml frct-002J frct-002J.xml Full subfibrations and figure shapes

In a category E, a morphism { x } \xrightarrow {{ f }}{ y } can be thought of as a “figure” of shape x drawn in y. For instance, if x is the point (i.e. x= \mathbf {1} _{ E }) then a morphism x \to y is a “point” of the “space” y. We refer to x as the figure-shape in any such scenario. The perspective of morphisms as figures is developed in more detail by Lawvere and Schanuel.

It often happens that a useful class of figure shapes can be arranged into a set-indexed family { \left ( u _{ i } \right )} _{ i \in I }; viewed from the perspective of the family fibration \boldsymbol { \mathcal {F}} _{ E }, this family is just a displayed object \bar {u} over I and then a figure shape “in” this family is given by any cartesian morphism \bar {z} \to \bar {u}. We will generalize this situation to the case of an arbitrary fibration, by constructing the full subfibration spanned by displayed objects equipped with a cartesian morphism into \bar {u} in the associated full subfibration below.

2 14 1 3550 frct-0010.rss.xml Construction frct-0010 frct-0010.xml The full subfibration associated to a displayed object

Let E be a cartesian fibration over B; then any displayed object \bar {x} \in E _{ x } induces a full subfibration \mathbf {Full} { \left ( \bar {u} \right )} \subseteq E spanned by displayed objects that are classified by \bar {u}, i.e. arise from \bar {u} by cartesian lift.

1. An object of \mathbf {Full} { \left ( \bar {u} \right )} _{ x } is specified by an object \bar {x} \in E _{ x } together with a cartesian morphism \bar {x} \to \bar {u}.
2. Given { x } \xrightarrow {{ f }}{ y } \in B, a morphism from \bar {x} \to \bar {u} to \bar {y} \to \bar {u} \in over f is given by any displayed morphism \bar {x} \xrightarrow [ f ]{} \bar {y}.
2 14 2 3551 frct-002K.rss.xml Definition frct-002K frct-002K.xml Figures and figure shapes in the full subfibration

Let E be a cartesian fibration and let \mathbf {Full} { \left ( \bar {s} \right )} \subseteq E be the full subfibration determined by a displayed object \bar {s} \in E. We now develop the following vocabulary:

1. We will refer to each object of \mathbf {Full} { \left ( \bar {s} \right )} as a \bar {s}-figure shape.
2. A displayed morphism \bar {z} \to \bar {x} is called a \bar {s}-figure whenever \bar {z} \in \mathbf {Full} { \left ( \bar {s} \right )}.
2 14 3 3552 frct-002I.rss.xml Definition frct-002I frct-002I.xml Agreement on a class of figure shapes

Let \bar {s} be a displayed object in a cartesian fibration E over B. A pair of displayed morphisms { \bar {x} } \xrightarrow {{ f,g }}{ \bar {y} } \in E are said to agree on \bar {s}-figures when for any \bar {s}-figure { \bar {z} } \xrightarrow {{ h }}{ \bar {x} }, we have { \bar {z} } \xrightarrow {{ h;f = h;g }}{ \bar {y} }.

3 3553 jms-005H.rss.xml jms-005H jms-005H.xml 2023 5 31 Jon Sterling Displayed functors and natural transformations

We will develop the theory of functors between displayed categories or fibrations as well as natural transformations between these functors, building on Ahrens and Lumsdaine and Jacobs.

3 1 3554 jms-005I.rss.xml Definition jms-005I jms-005I.xml 2023 5 31 Jon Sterling Displayed functor

Let X be a displayed category over B and let Y be a displayed category over C, and let { B } \xrightarrow {{ F }}{ C } be a functor. A displayed functor from X to Y over F, written { X } \xrightarrow [ F ]{ \bar F }{ Y } is defined by the following data:

1. for each displayed object x \in X ^b, an assigned object \bar F _b x \in Y ^{ F b};
2. for each displayed morphism { x } \xrightarrow [ f ]{ \bar {f} }{ y } in X over { b } \xrightarrow {{ f }}{ c } in B, an assigned morphism { \bar F _b{x} } \xrightarrow [ F {f} ]{ \bar F _f \bar {f} }{ \bar F _c y };
3. such that \bar F _{ 1 _{ b } } 1 _{ x } = 1 _{ \bar F _b x } and \bar F _{ f;g } { \left ( \bar {f}; \bar {g} \right )} = \bar F _f{ \bar {f}}; \bar F _g{ \bar {g}}.

When it does not cause confusion, we may drop some of the subscripts in our notation.

The notion of fibered functor below makes sense for arbitrary displayed categories, but it is most useful when applied to fibrations.

3 2 3555 jms-005J.rss.xml Definition jms-005J jms-005J.xml 2023 5 31 Jon Sterling Fibered functor

Let X be a displayed category over B and let Y be a displayed category over C, and let { B } \xrightarrow {{ F }}{ C } be a functor. A displayed functor from X to Y over F is called fibered when it preserves cartesian morphisms.

3 3 3556 jms-005K.rss.xml Definition jms-005K jms-005K.xml 2023 5 31 Jon Sterling Displayed natural transformation

Let X be a displayed category over B and let Y be a displayed category over C, and let { X } \xrightarrow [ F ]{ \bar F }{ Y } and { X } \xrightarrow [ G ]{ \bar G }{ Y } be two displayed functors over { B } \xrightarrow {{ F , G }}{ C } respectively. Given a natural transformation { F } \xrightarrow {{ \alpha }}{ G }, a displayed dinatural transformation from \bar F to \bar G over \alpha, written { \bar F } \xrightarrow [ \alpha ]{ \bar \alpha }{ \bar G }, is defined by the following data:

1. for each displayed object x \in X ^b, an assigned displayed morphism { \bar F {x} } \xrightarrow [ \alpha _b ]{ \bar \alpha _x }{ \bar G {x} };
2. such that for each { x } \xrightarrow [ f ]{ \bar {f} }{ y } in X, we have \bar \alpha _x; \bar G \bar {f} = \bar F \bar {f}; \bar \alpha _y.
3 4 3557 jms-005N.rss.xml Remark jms-005N jms-005N.xml 2023 5 31 Jon Sterling Vertical functors and natural transformations

In practice, we will frequently consider vertical displayed functors between fibrations over the same base category B, i.e. functors displayed over the identity functor { B } \xrightarrow {{ 1 _{ B } }}{ B }. Likewise, we will often consider vertical natural transformations between vertical functors, i.e. natural transformations whose components are vertical. The vertical and non-vertical perspectives give rise to two different (very large) displayed categories of displayed categories, one with unfixed base, and the other with a fixed base. Both viewpoints are important, but it is important not to get them confused.

We wish to define displayed versions of dinatural transformations, but there is some question of how much verticality to assume in the definition: for instance, we could define a displayed dinatural transformation over a dinatural transformation, or we could define a totally vertical notion of dinatural transformation. It is unclear to me that the latter would be an instance of the former. We will focus on the latter, because it is most useful in practice when using displayed category theory to formalize reasoning over a fixed base category.

3 5 3558 jms-005L.rss.xml Definition jms-005L jms-005L.xml 2023 5 31 Jon Sterling Vertical dinatural transformation

Let X and Y be two fibrations over B, and let { X ^{ \mathsf {o} } \times _{ B } X } \xrightarrow [ 1 _{ B } ]{ F , G }{ Y } be two vertical fibered functors over B, recalling the theory of opposite fibrations. A vertical dinatural transformation \alpha from F to G is defined by the following data:

1. for each displayed object x \in X ^b, an assigned vertical morphism { F _b { \left ( x,x \right )} } \xrightarrow [ 1 _{ b } ]{ \alpha _x }{ G _b { \left ( x,x \right )} };
2. such that for each displayed morphism { x } \xrightarrow [ f ]{ \bar {f} }{ y } over { b } \xrightarrow {{ f }}{ c } \in B, the following hexagon commutes: Above, we have written { x } \xrightarrow [ 1 _{ b } ]{ \gamma _{ \bar {f} } }{ f ^{ * } {y} } \in X^b for the vertical gap map factoring { x } \xrightarrow [ f ]{ \bar {f} }{ y } through the cartesian map { f ^{ * } {y} } \xrightarrow [ f ]{ y ^{ * } f }{ x }. We have used the fact that a vertical map in X ^{ \mathsf {o} } is the same as a vertical map in X going the other way.

4 3559 frct-0009.rss.xml frct-0009 frct-0009.xml The Grothendieck construction 4 1 3560 frct-000A.rss.xml frct-000A frct-000A.xml 2022 The total category and its projection

Note that any displayed category E over B can be viewed as an undisplayed category \widetilde { E } equipped with a projection functor p _{ E } : \widetilde { E } \to B; in this case \widetilde { E } is called the total category of E.

1. An object of \widetilde { E } is given by a pair (x, \bar {x}) where x \in B and \bar {x} \in E _{ x }.
2. A morphism { (x, \bar {x}) } \xrightarrow {{ }}{ (y, \bar {y}) } in \widetilde { E } is given by a pair (f, \bar {f}) where { x } \xrightarrow {{ f }}{ y } and { \bar {x} } \xrightarrow [ f ]{ \bar {f} }{ \bar {y} } : \bar {x}.

The construction of the total category of a displayed category is called the Grothendieck construction.

4 1 1 3561 frct-001T.rss.xml Exercise frct-001T frct-001T.xml

Prove that the total category \widetilde { \underline { B } } of the fundamental self-indexing is the arrow category B^{ \to }, and the projection is the codomain functor.

4 2 3562 frct-000B.rss.xml frct-000B frct-000B.xml Displayed categories from functors

In many cases, one starts with a functor P:E \to B; if it were meaningful to speak of equality of objects in an arbitrary category then there would be an obvious construction of a displayed category P _{ \bullet } from P; we would simply set P _{ x } to be the collection of objects u \in E such that Pu=x. As it stands there is a more subtle version that will coincide up to categorical equivalence with the naïve one in all cases that the latter is meaningful.

1. We define an object of P _{ x } to be a pair (u, \phi _{ u } ) where u \in E and \phi _{ u } : Pu \cong x. It is good to visualize such a pair as a “crooked leg” like so: 2. A morphism (u, \phi _{ u } ) \xrightarrow [ f ]{} (v, \phi _{ v } ) over f : x \to y is given by a morphism h : u \to v that lies over f modulo the isomorphisms \phi _{ u } , \phi _{ v } in sense depicted below: 4 2 1 3563 frct-001U.rss.xml Exercise frct-001U frct-001U.xml

Suppose that B is an internal category in \mathbf {Set}, i.e. it has a set of objects. Exhibit an equivalence of displayed categories between P _{ \bullet } as described above, and the naïve definition which E _{ x } is the collection of objects u \in E such that Pu = x.

We have a functor \widetilde { P _{ \bullet } } \to E taking a pair (x,(u, \phi _{ u } )) to u.

4 2 2 3564 frct-001V.rss.xml Exercise frct-001V frct-001V.xml

Explicitly construct the functorial action of \widetilde { P _{ \bullet } } \to E.

4 2 3 3565 frct-001W.rss.xml Exercise frct-001W frct-001W.xml

Verify that \widetilde { P _{ \bullet } } \to E is a categorical equivalence.

4 2 4 3566 frct-000C.rss.xml frct-000C frct-000C.xml Relationship to Street's fibrations

In classical category theory, cartesian fibrations are defined by Grothendieck to be certain functors E \to B such that any morphism f:x \to Pv in B lies strictly underneath a cartesian morphism in E. As we have discussed, this condition cannot be formulated unless equality is meaningful for the collection of objects of B.

There is an alternative definition of cartesian fibration due to Street that avoids equality of objects; here we require for each f:x \to Pv a cartesian morphism h: f ^{ * } v \to v together with an isomorphism \phi : P( f ^{ * } v) \cong x such that \phi ^{-1};Ph = f.

By unrolling definitions, it is not difficult to see that the displayed category P _{ \bullet } is a cartesian fibration in our sense if and only if the functor P:E \to B was a fibration in Street’s sense. Moreover, it can be seen that the Grothendieck construction yields a Grothendieck fibration \widetilde { P _{ \bullet } } \to B; hence we have introduced by accident a convenient destription of the strictification of Street fibrations into equivalent Grothendieck fibrations.

4 3 3567 frct-000D.rss.xml frct-000D frct-000D.xml Iteration and pushforward

It also makes sense to speak of categories displayed over other displayed categories; one way to formalize this notion is as follows. Let E be displayed over B; we define a category displayed over E to be simply a category displayed over the total category \widetilde { E }.

Now let F be displayed over E over B. Then we may regard F as a displayed category B _{ ! } F over B as follows:

1. An object of (B _{ ! } F) _{ x } is a pair ( \bar {x},{ \ddot {x}}) with \bar {x} \in E _{ x } and { \ddot {x}} \in F _{ \bar {x} }.
2. A morphism ( \bar {x},{ \ddot {x}}) \xrightarrow [ f ]{} ( \bar {y},{ \ddot {y}}) is given by a pair ( \bar {f},{ \ddot {f}}) where \bar {f}: \bar {x} \xrightarrow [ f ]{} \bar {y} in E and { \ddot {f}}:{ \ddot {x}} \xrightarrow [ \bar {f} ]{} { \ddot {y}} in F.

Using the displayed category induced by a functor, we may define the pushforward of a displayed category along a functor. In particular, let E be displayed over B and let U:B \to C be an ordinary functor; then we may obtain a displayed category U _{ ! } E over C as follows:

1. First we construct the displayed category U _{ \bullet } corresponding to the functor U:B \to C.
2. We recall that there is a canonical equivalence of categories \widetilde { U _{ \bullet } } \to B.
3. Because E is displayed over B, we may regard it as displayed over the equivalent total category \widetilde { U _{ \bullet } } by change of base.
4. Hence we may define the pushforward U _{ ! } E to be the displayed category (U _{ \bullet } ) _{ ! } E as defined above.
5 3568 frct-000E.rss.xml frct-000E frct-000E.xml Jon Sterling Properties of fibrations 5 1 3569 frct-000F.rss.xml frct-000F frct-000F.xml Locally small fibrations

There are a number of (equivalent) variations on the definition of a locally small fibration. We attempt to provide some intuition for these definitions.

5 1 1 3570 frct-000G.rss.xml frct-000G frct-000G.xml Warmup: locally small family fibrations

An ordinary category E is called locally small when for any x,y \in E the collection of morphisms x \to y is a set. This property of E can be rephrased in terms of its family fibration \boldsymbol { \mathcal {F}} _{ E } over \mathbf {Set} as follows.

Consider an index set I \in \mathbf {Set} and two families u,v \in C^I. We may define an I-indexed collection { \left [ u,v \right ]} _{ i \in I } consisting (pointwise) of all the morphisms u _{ i } \to v _{ i } in C:

{ \left [ u,v \right ]} _{ i } = { \left \{ f \mid f: u _{ i } \to v _{ i } \right \} }

If C is locally small, { \left [ u,v \right ]} _{ i \in I } is in fact a family of sets for any I \in \mathbf {Set} as each { \left [ u,v \right ]} _{ i } is a set. Conversely, if { \left [ u,v \right ]} _{ i \in I } is a family of sets for any I \in \mathbf {Set}, then C is locally small as we may consider in particular the case that I= \mathbf {1}.

5 1 2 3571 frct-000H.rss.xml frct-000H frct-000H.xml A more abstract formulation of locally small family fibrations

We will reformulate the local smallness property of the family fibration in a way that uses only the language that makes sense for an arbitrary cartesian fibration, though for now we stick with \boldsymbol { \mathcal {F}} _{ C }. Given u,v \in \boldsymbol { \mathcal {F}} _{ C } [I], we have a “relative hom family” { \left [ u,v \right ]} \in { \mathbf {Set} } _{ / I }. The fact that each { \left [ u,v \right ]} _{ i } is the set of all morphisms u _{ i } \to v _{ i } can be rephrased more abstractly.

First we consider the restriction of u \in \boldsymbol { \mathcal {F}} _{ C } [I] to \boldsymbol { \mathcal {F}} _{ C } [ { \left [ u,v \right ]} ] as follows: Explicitly the family { \left [ u,v \right ]} ^{ * } u is indexed in a pair of an element i \in I and a morphism u _{ i } \to v _{ i }. We can think of { \left [ u,v \right ]} ^{ * } u as the object of elements of u _{ i } indexed in pairs (i,u _{ i } \to v _{ i } ).

There is a canonical map \epsilon _{ { \left [ u,v \right ]} } : { \left [ u,v \right ]} ^{ * } u \xrightarrow [ p _{ { \left [ u,v \right ]} } ]{} v that “evaluates” each indexing morphism u _{ i } \to v _{ i }.

That each { \left [ u,v \right ]} _{ i } is the set of all morphisms u _{ i } \to v _{ i } can be rephrased as a universal property: for any family h \in { \mathbf {Set} } _{ / I } and morphism \epsilon _{ h } : h ^{ * } u \xrightarrow [ h ]{} v in \boldsymbol { \mathcal {F}} _{ C }, there is a unique cartesian map h ^{ * } u \to { \left [ u,v \right ]} ^{ * } u factoring \epsilon _{ h } through \epsilon _{ { \left [ u,v \right ]} } in the sense depicted below: To convince ourselves of this, we note that the family H \coloneqq { \left \{ u _{ i } \to v _{ i } \right \} } _{ i \in I } itself satisfies the universal property above. Indeed, fix a candidate h \in { \mathbf {Set} } _{ / I } equipped with a map \epsilon _{ h } : h ^{ * } u \xrightarrow [ h ]{} v. Unfolding the meaning of this map in set theoretical notation, we see that it amounts to a family of maps \epsilon _{ h } [i] : \prod _{ x \in h_i } { \left \{ u_i \to v_i \right \} } for each i \in I; such a family immediately induces the desired map h \to H.

5 1 3 3572 frct-000I.rss.xml frct-000I frct-000I.xml The definition of local smallness

Based on our explorations above, we are now prepared to write down (and understand) the proper definition of local smallness for an arbitrary cartesian fibration E over B, which should be thought of as a (potentially large) category relative to B.

5 1 3 1 3573 frct-001C.rss.xml Definition frct-001C frct-001C.xml Hom candidates

For any x \in B and displayed objects u,v \in E _{ x }, we define a hom candidate for u,v to be a span u \leftarrow \bar {h} \rightarrow v in E in which the left-hand leg is cartesian: In the above, h should be thought of as a candidate for the “hom object” of u,v, and \epsilon _{ h } should be viewed as the structure of an “evaluation map” for h. This structure can be rephrased in terms of a displayed category \mathbf {H}_{ E _{ x } }( u , v ) over { B } _{ / x }:

1. Given h \in { B } _{ / x }, an object of \mathbf {H}_{ E _{ x } }( u , v ) _{ h } is given by a hom candidate whose apex in the base is h itself. We will write \bar {h} metonymically for the entire hom candidate over h.
2. Given \alpha :l \to h \in { B } _{ / x } and hom candidates \bar {l} \in \mathbf {H}_{ E _{ x } }( u , v ) _{ l } and \bar {h} \in \mathbf {H}_{ E _{ x } }( u , v ) _{ h }, a morphism \bar {h} \xrightarrow [ \alpha ]{} \bar {l} is given by a cartesian morphism \bar \alpha : \bar {l} \xrightarrow [ \alpha ]{} \bar {h} in E such that the following diagram commutes: 5 1 3 2 3574 frct-001B.rss.xml Definition frct-001B frct-001B.xml Locally small fibration

A cartesian fibration E over B is locally small if and only if for each x \in B and u,v \in E _{ x }, the total category \widetilde { \mathbf {H}_{ E _{ x } }( u , v ) } of hom candidates has a terminal object. Viewed as a displayed object in E, we shall write \overline { \mathbf {hom}} _{ E _{ x } } ( u , v ) lying over \mathbf {hom} _{ E _{ x } } { \left ( u , v \right )} for the terminal hom candidate.

5 2 3575 frct-000J.rss.xml frct-000J frct-000J.xml Jon Sterling Globally small fibrations

In ordinary category theory, a category C is called small when the objects of C can be arranged into a set, and so can for every x,y \in C the collection of morphisms x \to y. It is useful to separate these two conditions when we generalize them to fibrations. The latter is called local smallness; the former is called global smallness by Jacobs and factors through an important concept: the generic object.

5 2 1 3576 frct-001D.rss.xml Definition frct-001D frct-001D.xml Global smallness

An ordinary category is called globally small when it has a set of objects.

5 2 2 3577 frct-000K.rss.xml frct-000K frct-000K.xml Jon Sterling Generic objects

Up to equivalence of categories, we may detect global smallness of a category C from the perspective of the family fibration \boldsymbol { \mathcal {F}} _{ C }. In particular, a category is equivalent to a globally small category when its family fibration has a generic object in the following sense.

5 2 2 1 3578 frct-001E.rss.xml Definition frct-001E frct-001E.xml Jon Sterling Generic object

Let E be a cartesian fibration over B; a generic object for E is defined to be an object \bar {u} \in \widetilde { E } such that for any \bar {z} \in \widetilde { E } there exists a cartesian map \bar {z} \to \bar {u}.

Warning. Our terminology differs from that of Jacobs; what we refer to as a generic object here is Jacobs’ weak generic object. We prefer the unqualified terminology, as generic objects in the stronger sense are very rare.

5 2 2 2 3579 frct-000L.rss.xml Theorem frct-000L frct-000L.xml Globally small categories

An ordinary category C is equivalent to a globally small category if and only if the family fibration \boldsymbol { \mathcal {F}} _{ C } has a generic object.

Proof.

To see that this is the case, suppose that C has a set of objects. Then C \in \mathbf {Set} and we define \lfloor {C} \rfloor to be the displayed object { \left \{ x \right \} } _{ x \in C } \in \boldsymbol { \mathcal {F}} _{ C } [C]. Fixing I \in \mathbf {Set} and z \in C^I, we consider the cartesian map displayed over z : I \to C: Conversely assume that \boldsymbol { \mathcal {F}} _{ C } has a generic object \bar {u} \in \boldsymbol { \mathcal {F}} _{ C } [U] for some U \in \mathbf {Set}; then we may equip U with the structure of a globally small category such that U is equivalent to C, using the canonical cleaving of the family fibration. In particular, given { 1 } \xrightarrow {{ x,y }}{ U } we define a morphism from x to y to be given by a vertical map { x ^{ * } { \bar {u}} } \xrightarrow {{ h }}{ y ^{ * } { \bar {u}} } in \boldsymbol { \mathcal {F}} _{ C } { \left [ 1 \right ]} \simeq C.

5 2 3 3580 frct-000P.rss.xml Definition frct-000P frct-000P.xml Globally small fibration

A cartesian fibration E over B is called globally small when it has a generic object.

5 3 3581 frct-002E.rss.xml frct-002E frct-002E.xml Separators for cartesian fibrations

Let E be an ordinary category. In general, to compare two morphisms { x } \xrightarrow {{ f,g }}{ y } in E, it is not enough to see if they agree on global points { 1 } \xrightarrow {{ u }}{ x }, because the behavior of f,g may differ only on generalized elements. In some cases, however, there is a family of objects { \left ( s _{ i } \right )} _{ i \in I } \in E are together adequate for comparing morphisms of E in the sense of separating families below.

5 3 1 3582 frct-002G.rss.xml Definition frct-002G frct-002G.xml Separating family for a category

Given an ordinary category E, a set-indexed family { \left ( s _{ i } \right )} _{ i \in I } of E-objects is called a small separating family for E when, assuming that for all i \in I and all { s_i } \xrightarrow {{ u }}{ x } we have u;f=u;g, we then have f=g.

5 3 2 3583 frct-003L.rss.xml Intuition frct-003L frct-003L.xml

The intuition of separating families is that to compare two morphisms { x } \xrightarrow {{ f,g }}{ y } \in E, it suffices to check that they behave the same on all s _{ i }-shaped figures when { \left ( s _{ i } \right )} _{ i \in I } is a separating family for E.

5 3 3 3584 frct-002F.rss.xml Example frct-002F frct-002F.xml Well-pointedness of the category of sets

In the category of sets, to compare two morphisms it is enough to check that they agree on global points. This means that the unary family { \left \{ \mathbf {1} _{ \mathbf {Set} } \right \} } is a separator for \mathbf {Set}, a property referred to more generally as well-pointedness.

We will now generalize the notion of separating family to the case of a cartesian fibration.

5 3 4 3585 frct-002I.rss.xml Definition frct-002I frct-002I.xml Agreement on a class of figure shapes

Let \bar {s} be a displayed object in a cartesian fibration E over B. A pair of displayed morphisms { \bar {x} } \xrightarrow {{ f,g }}{ \bar {y} } \in E are said to agree on \bar {s}-figures when for any \bar {s}-figure { \bar {z} } \xrightarrow {{ h }}{ \bar {x} }, we have { \bar {z} } \xrightarrow {{ h;f = h;g }}{ \bar {y} }.

5 3 5 3586 frct-002H.rss.xml Definition frct-002H frct-002H.xml Small separator for a fibration

Let E be a cartesian fibration over B such that B has binary products. A displayed object \bar {s} \in E _{ s } is said to be a small separator for E when any two vertical maps f,g: \bar {u} \xrightarrow [ 1 _{ x } ]{} \bar {v} \in E _{ x } are equal when they are agree on \bar {s}-figures.

5 4 3587 frct-002L.rss.xml frct-002L frct-002L.xml Jon Sterling Definable classes à la <link href="jeanbénabou.xml" type="local" title="Jean Bénabou">Bénabou</link>

A class of sets \mathfrak {C} is sometimes said to be formally definable when there is a formula { \left ( x \mid \phi _{ \mathfrak {C} { \left ( x \right )} } \right )} in the language of set theory such that a set S lies in \mathfrak {C} if and only if \phi _{ \mathfrak {C} } { \left ( S \right )} holds. This concept is a bit sensitive, as it presupposes that we have a notion of “class” whose constituents are not all definable in this sense.

A better behaved notion of definability for sets than the formal one is given model-theoretically, i.e. relative to a model \mathscr {M} of set theory in some ambient set theory. We will refer to an element of \mathscr {M} as a set, and a subcollection of \mathscr {M} as a class.

5 4 1 3588 frct-002U.rss.xml Definition frct-002U frct-002U.xml Representable class of sets

A class of sets \mathfrak {C} \subseteq \mathscr {M} is called representable when there is a set C \in \mathscr {M} such that U \in C if and only if U \in \mathfrak {C}.

5 4 2 3589 frct-002V.rss.xml Definition frct-002V frct-002V.xml Definable class of sets

A class of sets \mathfrak {C} \subseteq \mathscr {M} is called definable when for any representable class \mathfrak {S}, the class \mathfrak {C} \cap \mathfrak {S} is representable.

Bénabou then generalizes these definitions to an arbitrary fibration, in such a way that the general fibered notion of definable class is equivalent in the fundamental self-indexing \underline { \mathscr {M} } to that of definable classes.

5 4 3 3590 frct-002Q.rss.xml frct-002Q frct-002Q.xml Jon Sterling Set-theoretic intuition for <link href="jeanbénabou.xml" type="local" title="Jean Bénabou">Bénabou</link>'s definability

To motivate Bénabou’s general notion of definability, we will first develop an alternative perspective on definability for classes of sets in terms of families of sets.

5 4 3 1 3591 frct-002M.rss.xml Construction frct-002M frct-002M.xml Closure of a class of sets under base change

Let \mathfrak {C} \subseteq \mathscr {M} be a class of sets; there exists a class \bar { \mathfrak {C}} of families of sets that contains { \left ( S _{ i } \right )} _{ i \in I } exactly when each S _{ i } lies in \mathfrak {C}. We will refer to \bar { \mathfrak {C}} as the closure under base change of \mathfrak {C}, a name motivated by the fact that when { \left ( S _{ i } \right )} _{ i \in I } lies in \bar { \mathfrak {C}}, then for any u:J \to I, the base change { \left ( S _{ uj } \right )} _{ j \in J } also lies in \bar { \mathfrak {C}}.

5 4 3 2 3592 frct-002N.rss.xml Construction frct-002N frct-002N.xml From classes of families to classes of sets

Conversely to the closure of a class of sets under change of base, we may take a class of families of sets \mathfrak {F} to the the class of sets \mathfrak {F} _{ 1 } \subseteq { \mathscr {M}} spanned by sets S such that the singleton family { \left \{ S \right \} } lies in \mathfrak {C}.

5 4 3 3 3593 frct-002P.rss.xml Definition frct-002P frct-002P.xml Definable class of families of sets

A class of families of sets \mathfrak {F} is said to be definable when it is stable and moreover, for any family of sets { \left ( S _{ i } \right )} _{ i \in I }, there exists a subset J \subseteq I such that the base change { \left ( S _{ j } \right )} _{ j \in J } lies in \mathfrak {F}, and moreover, such that u:K \to I factors through J \subseteq I whenever the base change { \left ( S _{ uk } \right )} _{ k \in K } lies in \mathfrak {F}.

5 4 3 4 3594 frct-002R.rss.xml Intuition frct-002R frct-002R.xml

A stable class of families of sets is definable when any family of sets can be restricted to a “biggest” subfamily that lies in the class.

5 4 3 5 3595 frct-002T.rss.xml Lemma frct-002T frct-002T.xml Characterization of definable classes of families

Let \mathfrak {C} \subseteq \mathscr {M} be a class of sets; then \bar { \mathfrak {C}} is a definable class of families of sets if and only if \mathfrak {C} is a definable class of sets.

Proof.

Suppose that \mathfrak {C} is a definable class of sets. To check that \bar { \mathfrak {C}} is a definable class of families of sets, we fix a family { \left ( S _{ i } \right )} _{ i \in I } not necessarily lying in \bar { \mathfrak {C}}. Because \mathfrak {C} is definable, the intersection \mathfrak {C} \cap \bigcup _{ i \in I } S _{ i } is represented by a set U. We therefore take the subset J = { \left \{ i \in I \mid S_i \in U \right \} } \subseteq I, and verify that the base change { \left ( S _{ j } \right )} _{ j \in J } is the largest approximation of { \left ( S _{ i } \right )} _{ i \in I } by a family lying in \bar { \mathfrak {C}}.

Conversely suppose that \bar { \mathfrak {C}} is a definable class of families of sets. To see that \mathfrak {C} is definable, we fix a class \mathfrak {U} represented by a set U \in \mathscr {M} to check that \mathfrak {C} \cap \mathfrak {U} is representable. We consider the family of sets { \left ( u \right )} _{ u \in U }; because \bar { \mathfrak {C}} is definable, there is a largest subset V \subseteq U such that the change of base { \left ( v \right )} _{ v \in V } lies in \bar { \mathfrak {C}}, i.e. such that each v \in V lies in \mathfrak {C}. Therefore \mathfrak {C} \cap \mathfrak {U} is represented by the set V.

5 4 4 3596 frct-002S.rss.xml frct-002S frct-002S.xml Jon Sterling Bénabou’s notion of definability

We will now construe set-theoretic definability as the instantiation at the fundamental self-indexing \underline { \mathscr {M} } of a more general notion of definability for classes of objects in a cartesian fibration, defined forthwith. The following definition of definability in a cartesian fibration is due to Bénabou, but is discussed in print by Borceux, Jacobs, and Streicher.

5 4 4 1 3597 frct-002W.rss.xml Definition frct-002W frct-002W.xml Jon Sterling Definable class in a fibration

Let E be a cartesian fibration over B. A definable class \mathfrak {F} in E is a stable subcollection of the displayed objects of E such that for any \bar {u} \in E _{ u }, there exists a cartesian map \bar {v} \to \bar {u} lying over a monomorphism v \rightarrowtail u such that \bar {v} \in \mathfrak {F} and, moreover, any cartesian morphism \bar {w} \to \bar {u} such that \bar {w} \in \mathfrak {F} factors through \bar {v} \to \bar {u}. 5 4 4 2 3598 frct-002O.rss.xml Remark frct-002O frct-002O.xml Jon Sterling

Let \mathscr {M} be a model of ETCS; and let \mathfrak {C} be a class of families of sets in \mathscr {M}. Then \mathfrak {C} is a class of objects in the fundamental self-indexing \underline { \mathscr {M} } over \mathscr {M}. Furthermore, \mathscr {M} a definable class in \underline { \mathscr {M} } if and only if it is a definable class of families of sets.

6 3599 frct-000N.rss.xml frct-000N frct-000N.xml Jon Sterling Small fibrations and internal categories

The purpose of this section is to develop the relationship between internal categories (categories defined in the internal language of a category B) and cartesian fibrations over B, generalizing the relationship between categories internal to \mathbf {Set} (i.e. small categories) and their family fibrations over \mathbf {Set}.

6 1 3600 frct-001Q.rss.xml Definition frct-001Q frct-001Q.xml Small fibration

A cartesian fibration is called small when it is both locally small and globally small.

6 2 3601 frct-003O.rss.xml frct-003O frct-003O.xml Jon Sterling Internal categories

We have already seen in our discussion of locally small and globally small categories that smallness in the fibered sense appropriately generalizes the ordinary notion of smallness for categories over \mathbf {Set}. Another perspective on smallness is given by the internal language, in which a category is viewed as an algebra for the “theory of categories” computed in another category with enough structure. The notion of internal categories is credited (independently) to Alexander Grothendieck and Charles Ehresmann.

The notion of a (meta-)category is an essentially algebraic theory. As such it is possible to compute models of this theory in any category that has finite limits.

6 2 1 3602 frct-001A.rss.xml Definition frct-001A frct-001A.xml Internal category

Let E be a category with finite limits; then an internal category in E is defined by the following data:

1. an object of objects C _{ 0 } \in E,
2. an object of morphisms C _{ 1 } \in E,
3. source and target morphisms s,t:C _{ 1 } \to C _{ 0 },
4. a generic identity morphism C _{ 0 } \to C _{ 1 },
5. a generic composition morphism C _{ 1 } \times _{ C _{ 0 } } C _{ 1 } \to C _{ 1 },
6. satisfying a number of laws corresponding to those of a category.

For the details of these laws, we refer to any standard source.

6 2 2 3603 frct-000V.rss.xml frct-000V frct-000V.xml Jon Sterling The externalization of an internal category 6 2 2 1 3604 frct-001R.rss.xml Construction frct-001R frct-001R.xml Externalization

Let C be an internal category in E. We may define a fibered category { \left [ C \right ]} over E called the externalization of C.

1. Given x \in E, an object of { \left [ C \right ]} _{ x } is defined to be a morphism x \to C _{ 0 } in E.
2. Given x,y \in E and f:x \to y and u \in { \left [ C \right ]} _{ x } and v \in { \left [ C \right ]} _{ y }, a morphism u \xrightarrow [ f ]{} v is defined to be a morphism h : x \to C _{ 1 } in E such that the following diagram commutes: 6 2 2 2 3605 frct-000W.rss.xml Lemma frct-000W frct-000W.xml Cartesian lifts in the externalization

The externalization is a cartesian fibration.

Proof.

Given an object v \in { \left [ C \right ]} _{ y } and a morphism f:x \to y in E, we may define a cartesian lift f ^{ * } v \xrightarrow [ f ]{} v by setting f ^{ * } v = v \circ f : x \to C _{ 0 }.

6 2 2 3 3606 frct-000X.rss.xml Lemma frct-000X frct-000X.xml Globally small externalization

The externalization is globally small

Proof.

We may choose a generic object for { \left [ C \right ]}, namely the identity element (C _{ 0 } , 1 _{ C _{ 0 } } ) \in \widetilde { { \left [ C \right ]} }. Given any object (x,u) \in \widetilde { { \left [ C \right ]} } the cartesian map (x,u) \to (C _{ 0 } , 1 _{ C _{ 0 } } ) is given as follows: 6 2 2 4 3607 frct-000Y.rss.xml Lemma frct-000Y frct-000Y.xml Jon Sterling Locally small externalization

The externalization is locally small.

Proof.

Fix x \in E and u,v \in { \left [ C \right ]} _{ x }, we must exhibit a terminal object to the (total) category \widetilde { \mathbf {H}_{ { \left [ C \right ]} _{ x } }( u , v ) } of “hom candidates”. First we define { \left [ u,v \right ]} to be the following pullback in E: We define \overline {p}: { \left [ u,v \right ]} ^{ * } {u} \xrightarrow [ p ]{} u \in { \left [ C \right ]} _{ { \left [ u,v \right ]} } to be the cartesian lift of u \in { \left [ C \right ]} _{ x } along p: { \left [ u,v \right ]} \to x: We need to define a displayed evaluation map \epsilon : { \left [ u,v \right ]} ^{ * } u \xrightarrow [ p ]{} v; unraveling the definition of a displayed morphism in the externalization of C, we choose the following diagram: Putting all this together, we assert that the terminal object of \widetilde { \mathbf {H}_{ { \left [ C \right ]} _{ x } }( u , v ) } is the following span in { \left [ C \right ]}: Fixing another such candidate hom span { \left \{ u \leftarrow \bar {h} \rightarrow v \right \} } \in \widetilde { \mathbf {H}_{ { \left [ C \right ]} _{ x } }( u , v ) }, we must exhibit a unique cartesian morphism \bar \alpha : \bar {h} \to { \left [ u,v \right ]} ^{ * } {u} making the following diagram commute: First we note that the evaluation map \epsilon _{ h } : \bar {h} \to v amounts to an internal morphism h \to C _{ 1 } satisfying the appropriate compatibility conditions. Therefore we may define the base \alpha :h \to { \left [ u,v \right ]} of the universal map using the universal property of the pullback that defines { \left [ u,v \right ]}: The morphism \alpha :h \to { \left [ u,v \right ]} defined above is the unique map in E satisfying the conditions required of the base for \bar \alpha; therefore, it suffices to show that there exists a cartesian morphism \bar \alpha : \bar {h} \xrightarrow [ \alpha ]{} { \left [ u,v \right ]} ^{ * } u since it will be unique if it exists. We define \bar \alpha using the universal property of the cartesian lift: That \bar { \alpha }: \bar {h} \xrightarrow [ \alpha ]{} { \left [ u,v \right ]} ^{ * } u is cartesian follows from the generalized pullback lemma for cartesian morphisms: it suffices to observe that both \bar {p} _{ h } : \bar {h} \to u and its second factor \bar {p}: { \left [ u,v \right ]} ^{ * } u \to u are cartesian.

6 2 3 3608 frct-0011.rss.xml frct-0011 frct-0011.xml The full internal subcategory associated to a displayed object

The full subfibration associated to a displayed object \bar {u} of a locally small cartesian fibration E over B can be seen to be equivalent to the externalization of an internal category { \left \langle \bar {u} \right \rangle } in B.

6 2 3 1 3609 frct-003Q.rss.xml Construction frct-003Q frct-003Q.xml The internal category associated to a displayed object

Let \bar {u} be a displayed object in a locally small fibration E over B. We will define the internal category { \left \langle \bar {u} \right \rangle } in B associated to \bar {u}. In particular, we let the object of objects { \left \langle \bar {u} \right \rangle } _{ 0 } be u itself; defining the object of arrows { \left \langle \bar {u} \right \rangle } _{ 1 } is more complex, making critical use of the local smallness of E over B.

We will think of the fiber category E _{ u \times u } as the category of objects indexed in the boundary (source and target) of a morphism. Restricting \bar {u} along the source and target projections, we obtain the displayed objects of “points of the source” and “points of the target” respectively: Because E is locally small, there is an object \mathbf {hom} _{ E _{ u \times u } } { \left ( \bar \partial _{ 1 } , \bar \partial _{ 2 } \right )} \in { B } _{ / u \times u } that behaves as the “generic hom set”. We define { \left \langle \bar {u} \right \rangle } _{ 1 } \in B and its source and target projections to be this very object.

6 2 3 2 3610 frct-001S.rss.xml Theorem frct-001S frct-001S.xml Characterization of the externalization

The externalization { \left [ { \left \langle \bar {u} \right \rangle } \right ]} of the internal category { \left \langle \bar {u} \right \rangle } associated to a displayed object \bar {u} in a locally small fibered category E over B is equivalent to the full internal subfibration \mathbf {Full} { \left ( \bar {u} \right )}.

Proof.

We will define a fibred equivalence F : { \left [ { \left \langle \bar {u} \right \rangle } \right ]} \to \mathbf {Full} { \left ( \bar {u} \right )} over B.

1. Fix x \in B and \chi _{ x } \in { \left [ { \left \langle \bar {u} \right \rangle } \right ]} _{ x }, i.e. \chi _{ x } : x \to u; we define F { \left ( \chi _{ x } \right )} to be an arbitrary cartesian map \phi _{ x } : \bar {x} \xrightarrow [ \chi _{ x } ]{} \bar {u}. (Here we have used the axiom of choice for collections.)
2. Fix f : x \to y \in B and \chi _{ x } :x \to u and \chi _{ y } :y \to u and a diagram representing a displayed morphism h from \chi _{ x } to \chi _{ y } over f as below: We must define F { \left ( h \right )} : \bar {x} \xrightarrow [ f ]{} \bar {y}, fixing arbitrary cartesian maps \bar \chi _{ x } : \bar {x} \xrightarrow [ \chi _{ x } ]{} \bar {u} and \bar \chi _{ y } : \bar {y} \xrightarrow [ \chi _{ y } ]{} \bar {u}. First we lift h:x \to { \left [ { \left \langle \bar {u} \right \rangle } \right ]} _{ 1 } into E using the universal property of the cartesian lift: By composition with the “evaluation map” for our hom object, we have a map \bar {x} \xrightarrow [ f; \chi _{ y } ]{} \bar {u}: Next we define F { \left ( h \right )} : \bar {x} \xrightarrow [ f ]{} \bar {y} using the universal property of (another) cartesian lift: 6 2 4 3611 frct-000Z.rss.xml frct-000Z frct-000Z.xml The internalization of a small fibration

Let E be a small fibration over B a category with finite limits, i.e. a cartesian fibration that is both locally small and globally small. We will show that E is equivalent to the externalization { \left [ C \right ]} of an internal category C in B, namely the full internal subcategory associated to the generic object \bar {u} \in E.

Proof.

By the characterization of the externalization we know that the externalization of C so-defined is equivalent to the full subfibration \mathbf {Full} { \left ( \bar {u} \right )} of E spanned by objects that are “classified” by \bar {u}. Because \bar {u} is generic, we know that every object of E is classified by \bar {u}, so we are done.

7 3612 frct-0012.rss.xml frct-0012 frct-0012.xml Carlo AngiuliJon Sterling Other kinds of fibrations 7 1 3613 frct-0015.rss.xml frct-0015 frct-0015.xml Carlo AngiuliJon Sterling Cocartesian fibrations

Cocartesian fibrations are a dual notion to cartesian fibrations, in which the variance of indexing is reversed.

7 1 1 3614 frct-0016.rss.xml Definition frct-0016 frct-0016.xml Carlo Angiuli Cocartesian morphism

Let E be displayed over B, and let f:x \to y \in B; a morphism \bar {f}: \bar {x} \xrightarrow [ f ]{} \bar {y} in E is called cocartesian over f when for any m:y \to u and \bar {h}: \bar {x} \xrightarrow [ f;m ]{} \bar {u} there exists a unique \bar {m} : \bar {y} \xrightarrow [ m ]{} \bar {u} with \bar {f}; \bar {m} = \bar {h}: We use a “pushout corner” to indicate \bar {x} \to \bar {y} as a cocartesian morphism, a notation justified by our discussion of the dual self-indexing.

7 1 2 3615 frct-0017.rss.xml Definition frct-0017 frct-0017.xml Carlo Angiuli Cocartesian fibration

A displayed category E over B is a cocartesian fibration when for each f : x \to y \in B and \bar {x} \in E _{ x }, there exists a displayed object \bar {y} \in E _{ y } and a cocartesian morphism \bar {f} : \bar {x} \xrightarrow [ f ]{} \bar {y}.

Cocartesian fibrations are also known as opfibrations.

7 1 3 3616 frct-0018.rss.xml frct-0018 frct-0018.xml Carlo AngiuliJon Sterling The total opposite of a displayed category 7 1 3 1 3617 frct-001I.rss.xml Construction frct-001I frct-001I.xml Total opposite category

Let E be displayed over B; we define its total opposite E ^{ \tilde { \mathsf {o}} } displayed over B ^{ \mathsf {o} } as follows:

1. An object of E ^{ \tilde { \mathsf {o}} } _{ x } is given by an object of E _{ x }.
2. Given f : x \to y \in B ^{ \mathsf {o} }, a displayed morphism \bar {x} \xrightarrow [ f ]{} \bar {y} in E ^{ \tilde { \mathsf {o}} } is given by a displayed morphism \bar {y} \xrightarrow [ f ]{} \bar {x} in E.

Warning. Do not confuse this construction with the opposite fibered category, which produces a displayed category over B and not B ^{ \mathsf {o} }.

7 1 3 2 3618 frct-001J.rss.xml Exercise frct-001J frct-001J.xml

Let E be displayed over B. Prove that the total category \widetilde { E ^{ \tilde { \mathsf {o}} } } is { \left ( \widetilde { E } \right )} ^{ \mathsf {o} }, and its projection functor is { \left ( p _{ E } \right )} ^{ \mathsf {o} } : \widetilde { E } ^{ \mathsf {o} } \to B ^{ \mathsf {o} }.

7 1 3 3 3619 frct-001K.rss.xml Exercise frct-001K frct-001K.xml

Let E be displayed over B, and let f:x \to y \in B. Prove that a morphism \bar {f}: \bar {x} \xrightarrow [ f ]{} \bar {y} is cartesian over f in E if and only if \bar {f}: \bar {y} \xrightarrow [ f ]{} \bar {x} is cocartesian over f in E ^{ \tilde { \mathsf {o}} }.

7 1 3 4 3620 frct-001L.rss.xml Exercise frct-001L frct-001L.xml

Prove that a displayed category E is a cartesian fibration over B if and only if E ^{ \tilde { \mathsf {o}} } is a cocartesian fibration over B ^{ \mathsf {o} }.

7 1 4 3621 frct-002X.rss.xml Example frct-002X frct-002X.xml Carlo Angiuli Revisiting the fundamental self-indexing

Recall that the fundamental self-indexing \underline { B } of a category B is a displayed category with \underline { B } _{ x } = { B } _{ / x }. Recall that \underline { B } is a cartesian fibration over B if and only if B has pullbacks. However, \underline { B } is unconditionally a cocartesian fibration.

7 1 4 1 3622 frct-002Y.rss.xml Exercise frct-002Y frct-002Y.xml Carlo Angiuli

Prove that the fundamental self-indexing \underline { B } is a cocartesian fibration for any category B.

7 1 5 3623 frct-0019.rss.xml Example frct-0019 frct-0019.xml Carlo AngiuliJon Sterling The dual self-indexing

Dually to the fundamental self-indexing, every category B can also be displayed over itself via its coslices B ^{ \setminus x }.

7 1 5 1 3624 frct-002Z.rss.xml Construction frct-002Z frct-002Z.xml Carlo Angiuli The dual self-indexing

Let B be a category. Define the displayed category \overline { B } over B as follows:

1. For x \in B, define \overline { B } _{ x } as the collection of pairs ( \bar {x} \in B,p _{ x } :x \to \bar {x}).
2. For f : x \to y \in B, define \overline { B } _{ f } to be the collection of commuting squares in the following configuration: 7 1 5 2 3625 frct-001M.rss.xml Exercise frct-001M frct-001M.xml

Prove that \overline { B } is a cocartesian fibration if and only if B has pushouts.

7 1 5 3 3626 frct-001N.rss.xml Exercise frct-001N frct-001N.xml

Prove that the total category of \overline { B } is the arrow category B ^{ \to }, and the projection is the domain functor.

7 1 5 4 3627 frct-0030.rss.xml Exercise frct-0030 frct-0030.xml Carlo Angiuli

Prove that \overline { B } is a cartesian fibration for any category B.

7 2 3628 frct-0013.rss.xml frct-0013 frct-0013.xml Jon Sterling Right fibrations 7 2 1 3629 frct-001O.rss.xml Definition frct-001O frct-001O.xml Jon Sterling Right fibration

A cartesian fibration E over B is said to be a right fibration when all displayed morphisms in E are cartesian.

7 2 2 3630 frct-003P.rss.xml Remark frct-003P frct-003P.xml

Recall that for every b \in B, the collection of displayed objects E _{ b } and vertical maps E _{ 1 _{ b } } forms a category. When E is a right fibration over B, this category is in fact a groupoid.

7 2 3 3631 frct-001P.rss.xml Theorem frct-001P frct-001P.xml Jon Sterling Characterization of right fibrations

A cartesian fibration E over B is a right fibration if and only if all its vertical maps are isomorphisms.

Proof.

Suppose that E is a right fibration over B, and fix b \in B, \bar {b} \in E _{ b }, and a vertical map f: \bar {b} \xrightarrow [ 1 _{ b } ]{} \bar {b}. Using the hypothesis that f is cartesian, it has a unique section g: \bar {b} \xrightarrow [ 1 _{ b } ]{} \bar {b} as follows: Likewise, because g is cartesian, f is the unique section of g; thus f is an isomorphism in E _{ b }.

Conversely, suppose that E is a cartesian fibration whose vertical maps are isomorphisms. Fix f:x \to y \in B and an arbitrary displayed morphism \bar {g}: \bar {x} \xrightarrow [ f ]{} \bar {y}. Then \bar {g} is the precomposition of a cartesian lift \bar {f}: \bar {x} ' \xrightarrow [ f ]{} \bar {y} with a vertical map: Because vertical maps are isomorphisms and \bar {f} is cartesian, we can observe that \bar {g} is cartesian as follows, writing \bar {m} : \bar {u} \xrightarrow [ m ]{} \bar {x} ' for the unique factorization of \bar {h} through \bar {f} over m: 3632 jms-009F.rss.xml jms-009F jms-009F.xml 2023 9 20 Jon Sterling Day tensors of displayed categories

I have been thinking about monoidal closed structures induced by slicing over a monoid, which has been considered by Combette and Munch-Maccagnoni as a potential denotational semantics of destructors à la C++. It occurred to me that this construction is, in fact, almost a degenerate case of Day convolution on an internal monoidal category — and this made me realize that there might be a nice way to understand Day convolution in the language of fibered categories. In fact, many of these results (in particular the relativization to internal monoidal categories) are probably a special case of Theorem 11.22 of Shulman’s paper on enriched indexed categories.

8 3633 jms-009A.rss.xml Definition jms-009A jms-009A.xml 2023 9 20 Jon Sterling The “Day tensor” of displayed categories

Let E and F be two displayed categories over a semimonoidal category { \left ( B , \otimes , \alpha \right )}. We may define the “Day tensor” product E \otimes _ B F of E and F to be the following displayed category over B:

E \otimes _ B F : \equiv \otimes _! { \left ( \pi _1^* E \times _{ B \times B } \pi _2^* F \right )}
9 3634 jms-009C.rss.xml Exegesis jms-009C jms-009C.xml 2023 9 20 Jon Sterling Stepping through the <link href="jms-009A.xml" type="local" title="The “Day tensor” of displayed categories">Day tensor of displayed categories</link>

To understand the construction of the Day tensor, we will go through it step-by-step. Let E and F be two displayed categories over a semimonoidal category { \left ( B , \otimes , \alpha \right )}.

1. Given that E and F are displayed over B, we may restrict them to lie the two disjoint regions of B \times B by base change along the two projections: The above can be seen as cartesian lift in the 2-bifibration of displayed categories over \mathbf {Cat}.
2. Next, we took the fiber product of displayed categories, giving a vertical span over B: \pi _1^* E \leftarrow \pi _1^* E \times _{ B \times B } \pi _2^* F \rightarrow \pi _2^* F Of course, this corresponds to pullback in \mathbf {Cat} or cartesian product in { \mathbf {Cat} } _{ / B }.
3. Finally, we take a cocartesian lift along the tensor functor { B \times B } \xrightarrow {{ \otimes }}{ B } to obtain the Day tensor: The cocartesian lift above corresponds to composition in \mathbf {Cat}.

Under appropriate assumptions, we may also compute a “Day hom” by adjointness.

10 3635 jms-009G.rss.xml Definition jms-009G jms-009G.xml 2023 9 20 Jon Sterling The “Day hom” of displayed categories

Let E and F be two displayed categories over a semimonoidal category { \left ( B , \otimes , \alpha \right )} such that E is a cartesian fibration. We may define the “Day hom” E \multimap _ B F of E and F to be the following displayed category over B:

E \multimap _ B F : \equiv { \left ( \pi _1 \right )} _* { \left ( \pi _2^* E \Rightarrow _{ B \times B } \otimes ^* F \right )} Proof.

First of all, we note that the pullback functor { { \mathbf {Cat} } _{ / B } } \xrightarrow {{ \pi _1^* }}{ { \mathbf {Cat} } _{ / B \times B } } has a right adjoint \pi _1^* \dashv { \left ( \pi _1 \right )} _*, as { B \times B } \xrightarrow {{ \pi _1 }}{ B } is (as any cartesian fibration) a Conduché functor, i.e. an exponentiable arrow in \mathbf {Cat}. We have already assumed that E is a Cartesian fibrations, and thus so is its restriction \pi _1^*E; it therefore follows that \pi _1^*E is exponentiable. With that out of the way, we may compute the hom by adjoint calisthenics:

\begin {aligned} & \mathbf {hom} _{ { \mathbf {Cat} } _{ / B } } { \left ( X \otimes _B E , F \right )} \\ & \quad \equiv \mathbf {hom} _{ { \mathbf {Cat} } _{ / B } } { \left ( \otimes _! { \left ( \pi _1^*X \times _{B \times B} \pi _2^* E \right )} , F \right )} \\ & \quad \simeq \mathbf {hom} _{ { \mathbf {Cat} } _{ / B \times B } } { \left ( \pi _1^*X \times _{B \times B} \pi _2^* E , \otimes ^* F \right )} \\ & \quad \simeq \mathbf {hom} _{ { \mathbf {Cat} } _{ / B \times B } } { \left ( \pi _1^*X , \pi _2^* E \Rightarrow \otimes ^* F \right )} \\ & \quad \simeq \mathbf {hom} _{ { \mathbf {Cat} } _{ / B } } { \left ( X , { \left ( \pi _1 \right )} _* { \left ( \pi _2^* E \Rightarrow \otimes ^* F \right )} \right )} \end {aligned}
11 3636 jms-009B.rss.xml Definition jms-009B jms-009B.xml 2023 9 20 Jon Sterling The “Day unit” of displayed categories

Let { \left ( B , \otimes ,I, \alpha , \lambda , \rho \right )} be a monoidal category. We may define a “Day unit” for displayed categories over B to be given by the discrete fibration { B } _{ / I } \to B, which corresponds under the Grothendieck construction to the presheaf represented by I.

12 3637 jms-009D.rss.xml Conjecture jms-009D jms-009D.xml 2023 9 20 Jon Sterling The <link href="jms-009C.xml" type="local" title="Stepping through the Day tensor of displayed categories">Day tensor</link> preserves cartesian fibrations

If E and F are cartesian fibrations over a semimonoidal category { \left ( B , \otimes , \alpha \right )}, then the Day tensor E \otimes _ B F is also a cartesian fibration.

I believe, but did not check carefully, that when E and F are discrete fibrations over a semimonoidal category { \left ( B , \otimes , \alpha \right )} then the Day tensor is precisely the discrete fibration corresponding to the (contravariant) Day convolution of the presheaves corresponding to E and F. Likewise when { \left ( B , \otimes ,I, \alpha , \lambda , \rho \right )} is monoidal, it appears that the Day unit corresponds precisely to the traditional one.

There remain some interesting directions to explore. First of all, the claims above would obviously lead to a new construction of the Day convolution monoidal structure on the 1-category of discrete fibrations on B that coincides with the traditional one up to the Grothendieck construction. But in general, we should expect to exhibit both { \mathbf {Cat} } _{ / B } and \mathbf {Fib}_{ B } as monoidal bicategories, a result that I have not seen before.

13 3638 jms-009E.rss.xml Conjecture jms-009E jms-009E.xml 2023 9 20 A monoidal bicategory of displayed categories

Let { \left ( B , \otimes ,I, \alpha , \lambda , \rho \right )} be a monoidal category. Then the Day tensor and unit extend to a monoidal structure on the bicategory of displayed categories over B.

The conjecture above is highly non-trivial, as monoidal bicategories are extremely difficult to construct explicitly. I am hoping that Mike Shulman’s ideas involving monoidal double categories could potentially help.

This website is a “forest” created using the Forester tool. I organize my thoughts here on a variety of topics at a granular level; sometimes these thoughts are self-contained, and at times I may organize them into larger notebooks or lecture notes. My nascent ideas about the design of tools for scientific thought are here. I welcome collaboration on any of the topics represented in my forest. To navigate my forest, press Ctrl-K.

3640 tfmt-0007.rss.xml tfmt-0007 tfmt-0007.xml 2022 12 27 Jon Sterling Atomicity of scientific notes

One of the design principles for evergreen notes described by Matuschak is atomicity (Evergreen notes should be atomic): a note should capture just one thing, and if possible, all of that thing. A related point is that it should be possible to understand a note by (1) reading it, and (2) traversing the notes that it links to and recursively understanding those notes.

Traditional mathematical writing does not achieve this kind of atomicity: understanding the meaning of a particular node (e.g. a theorem or definition) usually requires understanding everything that came (textually) before it. In the context of the hierarchical organization of evergreen notes, this would translate to needing to go upward in the hierarchy in order to understand the meaning of a leaf node. I regard this property of traditional notes as a defect: we should prefer explicit context over implicit context.

High-quality scientific notes should make sense with minimal context; hierarchical context is imposed in order to tell a story, but consumers of scientific notes should not be forced into a particular narrative. Indeed, as many different hierarchical structures can be imposed, many different narratives can be explored.

My first exploration of hypertext science was the lecture notes on relative category theory; in hindsight, these lecture notes are very much traditional lecture notes, not written with the atomicity principle in mind. As a result, it is often difficult to understand a given node without ascending upward in the hierarchy.

3641 tfmt-000A.rss.xml tfmt-000A tfmt-000A.xml 2022 12 27 Jon Sterling Lessons from <link href="frct-003I.xml" type="local" title="Foundations of relative category theory">“Foundations of Relative Category Theory”</link>

Here I compile some of the lessons that I have synthesized from the experience of writing the foundations of relative category theory in hypertext, and my subsequent investigations into the concept of evergreen notes. As it was my first attempt to develop transformative hypertext media, I believe I made some mistakes.

14 3642 tfmt-0008.rss.xml tfmt-0008 tfmt-0008.xml 2022 12 27 Jon Sterling Achieving atomicity

Atomicity in evergreen notes is enhanced by adhering to the following principles:

1. no free variables: do not rely on one-off objects that are defined incidentally upwards in the hierarchy; turn them into atomic nodes that can be linked;
2. favor explicit dependency: whenever using a terminology or construction that has been defined elsewhere, link it;
3. notation should be decodable: all notations (except the most very basic) should be recalled via a link.

It can be a bit excessive to link every word: but the pertinent links could be added to a “related pages” section.

15 3643 tfmt-0009.rss.xml tfmt-0009 tfmt-0009.xml 2022 12 27 Jon Sterling The best structure to impose is relatively flat

It is easy to make the mistake of prematurely imposing a complex hierarchical structure on a network of notes, which leads to excessive refactoring. Hierarchy should be used sparingly, and its strength is for the large-scale organization of ideas. The best structure to impose on a network of many small related ideas is a relatively flat one. I believe that this is one of the mistakes made in the writing of the foundations of relative category theory, whose hierarchical nesting was too complex and quite beholden to my experience with pre-hypertext media.

3644 tfmt-0009.rss.xml tfmt-0009 tfmt-0009.xml 2022 12 27 Jon Sterling The best structure to impose is relatively flat

It is easy to make the mistake of prematurely imposing a complex hierarchical structure on a network of notes, which leads to excessive refactoring. Hierarchy should be used sparingly, and its strength is for the large-scale organization of ideas. The best structure to impose on a network of many small related ideas is a relatively flat one. I believe that this is one of the mistakes made in the writing of the foundations of relative category theory, whose hierarchical nesting was too complex and quite beholden to my experience with pre-hypertext media.

3645 streicher-fcjb.rss.xml Reference streicher-fcjb streicher-fcjb.xml 2022 12 16 Thomas Streicher Fibred categories à la <link href="jeanbénabou.xml" type="local" title="Jean Bénabou">Jean Bénabou</link> @unpublished{streicher-fcjb, author = {Streicher, Thomas}, year = {2021}, eprint = {1801.02927}, eprintclass = {math.CT}, eprinttype = {arXiv}, title = {Fibered Categories \`{a} la {Jean B\'{e}nabou}}, } 10.48550/arXiv.1801.02927

These are notes about the theory of Fibred Categories as I have learned it from Jean Bénabou. I also have used results from the Thesis of Jean-Luc Moens from 1982 in those sections where I discuss the fibered view of geometric morphisms. Thus, almost all of the contents is not due to me but most of it cannot be found in the literature since Bénabou has given many talks on it but most of his work on fibered categories is unpublished. But I am solely responsible for the mistakes and for misrepresentations of his views. And certainly these notes do not cover all the work he has done on fibered categories. I just try to explain the most important notions he has come up with in a way trying to be as close as possible to his intentions and intuitions. I started these notes in 1999 when I gave a course on some of the material at a workshop in Munich. They have developed quite a lot over the years and I have tried to include most of the things I want to remember.

3646 ahrens-lumsdaine-2019.rss.xml Reference ahrens-lumsdaine-2019 ahrens-lumsdaine-2019.xml 2019 3 5 Benedikt AhrensPeter LeFanu Lumsdaine Displayed categories @article{ahrens-lumsdaine-2019, author = {Ahrens, Benedikt and Lumsdaine, Peter LeFanu}, url = {https://lmcs.episciences.org/5252}, year = {2019}, month = mar, doi = {10.23638/LMCS-15(1:20)2019}, eprint = {1705.04296}, eprinttype = {arXiv}, issue = {1}, journal = {Logical Methods in Computer Science}, title = {Displayed Categories}, volume = {15}, }10.23638/LMCS-15(1:20)2019

We introduce and develop the notion of displayed categories. A displayed category over a category C is equivalent to “a category D and functor F : D \to C, but instead of having a single collection of “objects of D” with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.

3647 jacobs-1999.rss.xml Reference jacobs-1999 jacobs-1999.xml 1999 1 14 Bart Jacobs Categorical logic and type theory @book{jacobs-1999, author = {Jacobs, Bart}, address = {Amsterdam}, publisher = {North Holland}, year = {1999}, number = {141}, series = {Studies in Logic and the Foundations of Mathematics}, title = {Categorical Logic and Type Theory}, }

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

3648 benabou-1985.rss.xml Reference benabou-1985 benabou-1985.xml 1985 Jean Bénabou Fibered categories and the foundations of naïve category theory 10.2307/2273784The Journal of Symbolic Logic@article{benabou-1985, author = {B\'{e}nabou, Jean}, publisher = {Cambridge University Press}, year = {1985}, doi = {10.2307/2273784}, journal = {The Journal of Symbolic Logic}, number = {1}, pages = {10--37}, title = {Fibered categories and the foundations of naive category theory}, volume = {50}, }

En hommage à Alexandre Grothendieck.

3649 lawvere-1964-etcs.rss.xml Reference lawvere-1964-etcs lawvere-1964-etcs.xml 1964 10 26 F. William Lawvere An elementary theory of the category of sets 10.1073/pnas.52.6.1506@article{lawvere-1964-etcs, author = {F. William Lawvere}, doi = {10.1073/pnas.52.6.1506}, journal = {Proceedings of the National Academy of Sciences}, number = {6}, pages = {1506-1511}, title = {An Elementary Theory of the Category of Sets}, volume = {52}, year = {1964}, } 3650 street-1980.rss.xml Reference street-1980 street-1980.xml Ross Street Fibrations in bicategories 3651 sga-1.rss.xml Reference sga-1 sga-1.xml Alexander GrothendieckMichele Raynaud Revêtements étales et groupe fondamental (SGA 1) 10.48550/ARXIV.MATH/0206203

Le texte présente les fondements d’une théorie du groupe fondamental en Géométrie Algébrique, dans le point de vue “kroneckerien” permettant de traiter sur le même pied le cas d’une variété algébrique au sens habituel, et celui d’un anneau des entiers d’un corps de nombres, par exemple.

The text presents the foundations of a theory of the fundamental group in Algebraic Geometry from the Kronecker point of view, allowing one to treat on an equal footing the case of an algebraic variety in the usual sense, and that of the ring of integers in a number field, for instance.

3652 borceux-hca-2.rss.xml Reference borceux-hca-2 borceux-hca-2.xml Francis Borceux Handbook of categorical algebra 2: categories and structures @book{borceux-hca-2, author = {Borceux, Francis}, publisher = {Cambridge University Press}, year = {1994}, isbn = {978-0-521-44179-7}, series = {Encyclopedia of Mathematics and its Applications}, title = {Handbook of Categorical Algebra 2 -- Categories and Structures}, volume = {2}, }

A Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence, with the first being essentially self-contained, and are accessible to graduate students with a good background in mathematics. Volume 1, which is devoted to general concepts, can be used for advanced undergraduate courses on category theory. After introducing the terminology and proving the fundamental results concerning limits, adjoint functors and Kan extensions, the categories of fractions are studied in detail; special consideration is paid to the case of localizations. The remainder of the first volume studies various ‘refinements’ of the fundamental concepts of category and functor.