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06 Feb 2022
ba3bbc6make on ref parenthetical -
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03 Feb 2022
5c0da58tweak terminology (opposite self-indexing => dual self-indexing) -
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02 Feb 2022
61d8e3bDiscuss codomain opfibration and domain fibration; clean up domain opfibration. -
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01 Feb 2022
8fb843bfeat: inheritable macrolib; prev/next navigation -
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30 Jan 2022
8597261first cut at definability à la Bénabou -
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- Foundations of Relative Category Theory
- Section 4. Properties of fibrations [000E]
- Section 4.4. Definable classes à la Bénabou [002L]
- Construction 4.4.1·a. Closure of a class of sets under base change [002M]
- Construction 4.4.1·b. From classes of families to classes of sets [002N]
- Remark 4.4.2·b [002O]
- Definition 4.4.1·c. Definable class of families of sets [002P]
- Section 4.4.1. Set-theoretic intuition for Bénabou's definability [002Q]
- Intuition 4.4.1·d [002R]
- Section 4.4.2. Bénabou's notion of definability [002S]
- Lemma 4.4.1·e. Characterization of definable classes of families [002T]
- Definition 4.4·a. Representable class of sets [002U]
- Definition 4.4·b. Definable class of sets [002V]
- Definition 4.4.2·a. Definable class in a fibration [002W]
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31 Jan 2022
79bafd1add nlab tag plugin -
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30 Jan 2022
b54be38topos: say “meta-bicategory” instead of “very large bicategory” -
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30 Jan 2022
624ca61topos: use diagrammatic order & flip an iso to make the pasting compatible -
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30 Jan 2022
20ef85dset a default global macrolib -
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- Foundations of Relative Category Theory
- Geometric universes and topoi (draft)
- Definition 2·a. Displayed category [0000]
- Definition 2·b. Cartesian morphisms [0001]
- Definition 2·c. Cartesian fibration [0002]
- Section 2.1. The canonical self-indexing [0003]
- Section 2.4. Displayed and fibered functors [0004]
- Section 2.5. Fiber categories and vertical maps [0005]
- Section 2.7. Example: the family fibration [0006]
- Section 2.8. Change of base [0007]
- Section 2. Displayed categories and fibrations [0008]
- Section 3. The Grothendieck construction [0009]
- Section 3.1. The total category and its projection [000A]
- Section 3.2. Displayed categories from functors [000B]
- Section 3.2.1. Relationship to Street's fibrations [000C]
- Section 3.3. Iteration and pushforward [000D]
- Section 4. Properties of fibrations [000E]
- Section 4.1. Locally small fibrations [000F]
- Section 4.1.1. Warmup: locally small family fibrations [000G]
- Section 4.1.2. A more abstract formulation [000H]
- Section 4.1.3. The definition of local smallness [000I]
- Section 4.2. Globally small fibrations [000J]
- Section 4.2.1. Generic objects [000K]
- Theorem 4.2.1·b. Globally small categories [000L]
- Section 5. Small fibrations and internal categories [000N]
- Section 5.1. Internal categories [000O]
- Definition 4.2·b. Globally small fibration [000P]
- Section 2.6. Opposite categories [000Q]
- Section 1. Foundational assumptions [000R]
- Section 2.6.3. Exegesis of opposite categories [000S]
- Section 2.6.1. Characterization of cartesian maps [000T]
- Section 2.6.2. Cartesian lifts in the opposite category [000U]
- Section 5.2. The externalization of an internal category [000V]
- Lemma 5.2·b [000W]
- Lemma 5.2·c [000X]
- Lemma 5.2·d [000Y]
- Section 5.4. The internalization of a small fibration [000Z]
- Construction 2.9·a. The full subfibration associated to a displayed object [0010]
- Section 5.3. The full internal subcategory associated to a displayed object [0011]
- Section 6. Other kinds of fibrations [0012]
- Section 6.2. Right fibrations [0013]
- Section 2.2. The generalized pullback lemma [0014]
- Section 6.1. Cocartesian fibrations [0015]
- Definition 6.1·a. Cocartesian morphism [0016]
- Definition 6.1·b. Cocartesian fibration [0017]
- Section 6.1.1. The total opposite of a displayed category [0018]
- Section 6.1.3. Example: The dual self-indexing [0019]
- Definition 5.1·a. Internal category [001A]
- Definition 4.1.3·b. Locally small fibration [001B]
- Definition 4.1.3·a. Hom candidates [001C]
- Definition 4.2·a [001D]
- Definition 4.2.1·a. Generic object [001E]
- Definition 1·a [001F]
- Definition 1·b [001G]
- Lemma 2.2·a. Generalized pullback lemma [001H]
- Construction 6.1.1·a [001I]
- Exercise 6.1.1·b [001J]
- Exercise 6.1.1·c [001K]
- Exercise 6.1.1·d [001L]
- Exercise 6.1.3·b [001M]
- Exercise 6.1.3·c [001N]
- Definition 6.2·a [001O]
- Theorem 6.2·b. Characterization of right fibrations [001P]
- Definition 5·a. Small fibration [001Q]
- Construction 5.2·a [001R]
- Theorem 5.3·a [001S]
- Exercise 3.1·a [001T]
- Exercise 3.2·a [001U]
- Exercise 3.2·b [001V]
- Exercise 3.2·c [001W]
- Construction 2.1·a. The canonical self-indexing [001X]
- Exercise 2.1·b [001Y]
- Construction 2.6·a [001Z]
- Lemma 2.6.1·a [0020]
- Corollary 2.6.2·a [0021]
- Definition 1·a. Geometric universe [0022]
- Section 1. Geometric universes [0023]
- Section 2. Topoi in a geometric universe [0024]
- Definition 2·a. Topos [0025]
- Notation 2·b [0026]
- Section 1.1. Relative and fibered geometric universes [0027]
- Section 2.3. An alternative definition of fibration [0029]
- Definition 2.3·a. Hypocartesian morphisms [002A]
- Lemma 2.3·c. Equivalence with Grothendieck's fibrations [002B]
- Lemma 2.3·b. Hypocartesian = cartesian in a cartesian fibration [002C]
- Remark 2.3·d. Two ways to generalize pullbacks [002D]
- Example 4.3·b. Well-pointedness of the category of sets [002F]
- Section 2.9. Full subfibrations and figure shapes [002J]
- Definition 2.9·b. Figures and figure shapes in the full subfibration [002K]
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29 Jan 2022
03e8f45fix missing macrolib -
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29 Jan 2022
09cbfd9tweak wording for clarity -
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29 Jan 2022
fa24378tweak a few refs -
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29 Jan 2022
3671a01cleanup in discussion of full subfibration -
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29 Jan 2022
25c493atidy up section 5 using new taxons -
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29 Jan 2022
a0cb966tweak a few references and node titles -
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29 Jan 2022
7358b63add support for authors -
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29 Jan 2022
9049427typo -
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29 Jan 2022
f88d9berefer to small separator in definition of topos -
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29 Jan 2022
846ea88missing word: small -
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29 Jan 2022
b8f66ecdisentangle a bunch of math, discussion of figure shapes -
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- Section 2. Displayed categories and fibrations [0008]
- Section 5. Small fibrations and internal categories [000N]
- Construction 2.9·a. The full subfibration associated to a displayed object [0010]
- Definition 4.3·c. Agreement on a class of figure shapes [002I]
- Section 2.9. Full subfibrations and figure shapes [002J]
- Definition 2.9·b. Figures and figure shapes in the full subfibration [002K]
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29 Jan 2022
082dcadfix confusing owrding -
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29 Jan 2022
1280abdDefine separators for fibrations (resolve #40) -
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- Section 4. Properties of fibrations [000E]
- Section 4.3. Separators for cartesian fibrations [002E]
- Example 4.3·b. Well-pointedness of the category of sets [002F]
- Definition 4.3·a. Separating family for a category [002G]
- Definition 4.3·d. Small separator for a fibration [002H]
- Definition 4.3·c. Agreement on a class of figure shapes [002I]
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29 Jan 2022
eb82d7dupdate several nodes to use cref and pref -
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- Definition 2·c. Cartesian fibration [0002]
- Section 3.3. Iteration and pushforward [000D]
- Section 4.1.1. Warmup: locally small family fibrations [000G]
- Section 4.1.2. A more abstract formulation [000H]
- Theorem 4.2.1·b. Globally small categories [000L]
- Section 5. Small fibrations and internal categories [000N]
- Section 2.6.3. Exegesis of opposite categories [000S]
- Section 2.6.2. Cartesian lifts in the opposite category [000U]
- Section 5.4. The internalization of a small fibration [000Z]
- Section 5.3. The full internal subcategory associated to a displayed object [0011]
- Section 6.2. Right fibrations [0013]
- Definition 6.1·a. Cocartesian morphism [0016]
- Section 6.1.3. Example: The dual self-indexing [0019]
- Exercise 6.1.1·b [001J]
- Exercise 6.1.3·c [001N]
- Exercise 3.1·a [001T]
- Section 2.3. An alternative definition of fibration [0029]
- Lemma 2.3·c. Equivalence with Grothendieck's fibrations [002B]
- Lemma 2.3·b. Hypocartesian = cartesian in a cartesian fibration [002C]
- Remark 2.3·d. Two ways to generalize pullbacks [002D]
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29 Jan 2022
3ed1724add a node title -
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29 Jan 2022
2a50e4fadd intuition about hypocartesian vs. cartesian re: pullbacks -
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29 Jan 2022
feb8243refactor and complete proof of equiv. between 2 defn’s of cart. fibration -
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29 Jan 2022
1108ee2title of generalized pullback lemma -
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29 Jan 2022
9832d71add \tick macro to get rid of \Sup{\prime} -
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29 Jan 2022
e6f37beresolve a todo -
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29 Jan 2022
622eba0hypocartesian morphisms -
28 Jan 2022
5f457b6start “sheafy-fying” topos lectures :) -
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- Geometric universes and topoi (draft)
- Definition 1·a. Geometric universe [0022]
- Section 1. Geometric universes [0023]
- Section 2. Topoi in a geometric universe [0024]
- Definition 2·a. Topos [0025]
- Notation 2·b [0026]
- Section 1.1. Relative and fibered geometric universes [0027]
- Definition 1.1·a. The gluing fibration [0028]
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28 Jan 2022
ad3f499add qed squares to proofs -
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- Section 4.1.2. A more abstract formulation [000H]
- Theorem 4.2.1·b. Globally small categories [000L]
- Section 2.6.3. Exegesis of opposite categories [000S]
- Section 2.6.2. Cartesian lifts in the opposite category [000U]
- Lemma 5.2·b [000W]
- Lemma 5.2·c [000X]
- Lemma 5.2·d [000Y]
- Lemma 2.2·a. Generalized pullback lemma [001H]
- Theorem 6.2·b. Characterization of right fibrations [001P]
- Theorem 5.3·a [001S]
- Lemma 2.6.1·a [0020]
- Corollary 2.6.2·a [0021]
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28 Jan 2022
a2d54a5add a warning about generic object terminology -
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28 Jan 2022
f1ea5fbfix a mistaken ref -
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28 Jan 2022
cc267e2split a lot more nodes -
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- Definition 2·a. Displayed category [0000]
- Definition 2·b. Cartesian morphisms [0001]
- Definition 2·c. Cartesian fibration [0002]
- Section 2.1. The canonical self-indexing [0003]
- Section 3.1. The total category and its projection [000A]
- Section 3.2. Displayed categories from functors [000B]
- Section 5. Small fibrations and internal categories [000N]
- Section 2.6. Opposite categories [000Q]
- Section 2.6.1. Characterization of cartesian maps [000T]
- Section 5.2. The externalization of an internal category [000V]
- Lemma 5.2·b [000W]
- Lemma 5.2·c [000X]
- Lemma 5.2·d [000Y]
- Construction 2.9·a. The full subfibration associated to a displayed object [0010]
- Section 5.3. The full internal subcategory associated to a displayed object [0011]
- Section 6.2. Right fibrations [0013]
- Section 6.1.1. The total opposite of a displayed category [0018]
- Section 6.1.3. Example: The dual self-indexing [0019]
- Construction 6.1.1·a [001I]
- Exercise 6.1.1·b [001J]
- Exercise 6.1.1·c [001K]
- Exercise 6.1.1·d [001L]
- Exercise 6.1.3·b [001M]
- Exercise 6.1.3·c [001N]
- Definition 6.2·a [001O]
- Theorem 6.2·b. Characterization of right fibrations [001P]
- Definition 5·a. Small fibration [001Q]
- Construction 5.2·a [001R]
- Theorem 5.3·a [001S]
- Exercise 3.1·a [001T]
- Exercise 3.2·a [001U]
- Exercise 3.2·b [001V]
- Exercise 3.2·c [001W]
- Construction 2.1·a. The canonical self-indexing [001X]
- Exercise 2.1·b [001Y]
- Construction 2.6·a [001Z]
- Lemma 2.6.1·a [0020]
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28 Jan 2022
63919fdsplit a bunch of nodes into result nodes -
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- Section 4.1.3. The definition of local smallness [000I]
- Section 4.2. Globally small fibrations [000J]
- Section 4.2.1. Generic objects [000K]
- Theorem 4.2.1·b. Globally small categories [000L]
- Definition 4.2·b. Globally small fibration [000P]
- Section 1. Foundational assumptions [000R]
- Section 2.2. The generalized pullback lemma [0014]
- Definition 4.1.3·b. Locally small fibration [001B]
- Definition 4.1.3·a. Hom candidates [001C]
- Definition 4.2·a [001D]
- Definition 4.2.1·a. Generic object [001E]
- Definition 1·a [001F]
- Definition 1·b [001G]
- Lemma 2.2·a. Generalized pullback lemma [001H]
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28 Jan 2022
cbb49bffeat: #6 sublayouts, taxa, clickers and better layouts -
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26 Jan 2022
358cf3ccocartesian fibrations (#37) -
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- Definition 2·b. Cartesian morphisms [0001]
- Section 6. Other kinds of fibrations [0012]
- Section 6.1. Cocartesian fibrations [0015]
- Definition 6.1·a. Cocartesian morphism [0016]
- Definition 6.1·b. Cocartesian fibration [0017]
- Section 6.1.1. The total opposite of a displayed category [0018]
- Section 6.1.3. Example: The dual self-indexing [0019]
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26 Jan 2022
3985d14fix typo (duplicate word) -
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26 Jan 2022
f8cc803fix missing arrows in diagrams -
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26 Jan 2022
1059902add a dotted pullback aura to a diagram -
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26 Jan 2022
af2b7adadd a back-reference -
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25 Jan 2022
8119533add back ref and fix typos -
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25 Jan 2022
5e20836the generalized pullback lemma -
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25 Jan 2022
412d604other kinds of fibrations (#29) -
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- Foundations of Relative Category Theory
- Definition 2·a. Displayed category [0000]
- Definition 2·c. Cartesian fibration [0002]
- Section 3.2.1. Relationship to Street's fibrations [000C]
- Section 4.1.2. A more abstract formulation [000H]
- Section 4.1.3. The definition of local smallness [000I]
- Section 4.2.1. Generic objects [000K]
- Section 5. Small fibrations and internal categories [000N]
- Definition 4.2·b. Globally small fibration [000P]
- Section 2.6.2. Cartesian lifts in the opposite category [000U]
- Lemma 5.2·b [000W]
- Section 5.4. The internalization of a small fibration [000Z]
- Section 5.3. The full internal subcategory associated to a displayed object [0011]
- Section 6. Other kinds of fibrations [0012]
- Section 6.2. Right fibrations [0013]
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24 Jan 2022
a956feetypos -
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19 Jan 2022
f788e92correct an internal link -
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19 Jan 2022
8d3e9bbinternalization of a small fibration -
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- Foundations of Relative Category Theory
- Section 4. Properties of fibrations [000E]
- Section 5. Small fibrations and internal categories [000N]
- Section 5.1. Internal categories [000O]
- Section 5.4. The internalization of a small fibration [000Z]
- Construction 2.9·a. The full subfibration associated to a displayed object [0010]
- Section 5.3. The full internal subcategory associated to a displayed object [0011]
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19 Jan 2022
f8d278atypos and a remark about the codomain functor -
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18 Jan 2022
72151a2typos -
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