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.
A meta-category
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
Assumption. We assume a meta-category
A category
Consequently there exists a meta-category
Assumption. At times we may assume that there exists a category
Let
When we have too many subscripts, we will write
Let
Above we have used the “pullback corner” to indicate
Let
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.
A displayed category
We will also refer to cartesian fibrations as simply fibrations or fibered categories.
There are other variations of fibration. For instance,
Let
Prove that the fundamental self-indexing
In light of our discussion of the fundamental self-indexing, the following result for displayed categories generalizes the ordinary “pullback lemma.”
Let
Suppose first that
Because
Conversely, suppose that
Because
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).
Let
Cartesian morphisms are clearly hypocartesian (setting
Let
Any cartesian map is clearly hypocartesian. To see that a hypocartesian map
As the cartesian lift
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.
Let
Suppose first that
Conversely, suppose that
Let
The composite
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.
Let
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.
Let
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.
Let
Given
such that
Let
Going forward, we will not be sensitive to the difference between the two constructions of opposite fibrations.
There is a simple characterization of cartesian maps in
A morphism
Suppose that
We must define the unique intervening map
The desired intervening map
We leave the converse to the reader.
The foregoing characterization of cartesian maps in
The displayed category
Fixing
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
A displayed morphism
Any ordinary category
The displayed category
This story for relative category theory reflects the way that ordinary categories are “based on”
The family fibration can be cloven, constructively. In particular, let
Suppose that
We visualize the change of base scenario as follows:
In a category
It often happens that a useful class of figure shapes can be arranged into a set-indexed family
Let
Let
Let
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.
Let
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.
Let
Let
In practice, we will frequently consider vertical displayed functors between fibrations over the same base category
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.
Let
Above, we have written
Note that any displayed category
The construction of the total category of a displayed category is called the Grothendieck construction.
Prove that the total category
In many cases, one starts with a functor
Suppose that
We have a functor
Explicitly construct the functorial action of
Verify that
In classical category theory, cartesian fibrations are defined by
Grothendieck to be certain functors
There is an alternative definition of cartesian fibration due to Street that avoids
equality of objects; here we require for each
By unrolling definitions, it is not difficult to see that the displayed
category
It also makes sense to speak of categories displayed over other displayed
categories; one way to formalize this notion is as follows. Let
Now let
Using the displayed category induced by a functor, we may define the pushforward of a displayed category along a functor. In particular, let
There are a number of (equivalent) variations on the definition of a locally small fibration. We attempt to provide some intuition for these definitions.
An ordinary category
Consider an index set
If
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
First we consider the restriction of
Explicitly the family
There is a canonical map
That each
To convince ourselves of this, we note that the family
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
For any
In the above,
Given
A cartesian fibration
In ordinary category theory, a category
An ordinary category is called globally small when it has a set of objects.
Up to equivalence of categories, we may detect global smallness of a category
Let
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.
An ordinary category
To see that this is the case, suppose that
Conversely assume that
A cartesian fibration
Let
Given an ordinary category
The intuition of separating families is that to compare two morphisms
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
We will now generalize the notion of separating family to the case of a cartesian fibration.
Let
Let
A class of sets
A better behaved notion of definability for sets than the formal one is
given model-theoretically, i.e. relative to a model
A class of sets
A class of sets
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
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.
Let
Conversely to the closure of a class of sets under change of base, we may take a class of families of sets
A class of families of sets
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.
Let
Suppose that
Conversely suppose that
We will now construe set-theoretic definability as the instantiation at the fundamental self-indexing
Let
Let
The purpose of this section is to develop the relationship between internal
categories (categories defined in the internal language of a category
A cartesian fibration is called small when it is both locally small and globally small.
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
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.
Let
For the details of these laws, we refer to any standard source.
Let
Given
The externalization is a cartesian fibration.
Given an object
The externalization is globally small
We may choose a generic object for
The externalization is locally small.
Fix
We define
We need to define a displayed evaluation map
Putting all this together, we assert that the terminal object of
Fixing another such candidate hom span
First we note that the evaluation map
The morphism
That
The full subfibration associated to a displayed object
Let
We will think of the fiber category
Because
The externalization
We will define a fibred equivalence
Fix
We must define
By composition with the “evaluation map” for our hom object, we have a map
Next we define
Let
By the characterization of the externalization we know that the externalization of
Cocartesian fibrations are a dual notion to cartesian fibrations, in which the variance of indexing is reversed.
Let
We use a “pushout corner” to indicate
A displayed category
Cocartesian fibrations are also known as opfibrations.
Let
Warning. Do not confuse this construction with the opposite fibered category, which produces a displayed category over
Let
Let
Prove that a displayed category
Recall that the fundamental self-indexing
Prove that the fundamental self-indexing
Dually to the fundamental self-indexing, every category
Let
Prove that
Prove that the total category of
Prove that
A cartesian fibration
Recall that for every
A cartesian fibration
Suppose that
Likewise, because
Conversely, suppose that
Because vertical maps are isomorphisms and
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.
Let
To understand the construction of the Day tensor, we will go through it step-by-step. Let
Under appropriate assumptions, we may also compute a “Day hom” by adjointness.
Let
First of all, we note that the pullback functor
Let
If
I believe, but did not check carefully, that when
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
Let
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
.
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.
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.
Atomicity in evergreen notes is enhanced by adhering to the following principles:
It can be a bit excessive to link every word: but the pertinent links could be added to a “related pages” section.
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.
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.
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.
We introduce and develop the notion of displayed categories. A displayed category over a category
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.
En hommage à Alexandre Grothendieck.
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.
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.