defn: Pseudofunctor bicategories and the Grothendieck construction as a lax colimit#612
Open
aathn wants to merge 14 commits into
Open
defn: Pseudofunctor bicategories and the Grothendieck construction as a lax colimit#612aathn wants to merge 14 commits into
aathn wants to merge 14 commits into
Conversation
Member
New pages
Changed pages
|
aathn
force-pushed
the
cat-lax-colim
branch
16 times, most recently
from
e2f5d1a to
62bf56f
Compare
aathn
force-pushed
the
cat-lax-colim
branch
7 times, most recently
from
7c4cd8c to
f3ef5e2
Compare
aathn
force-pushed
the
cat-lax-colim
branch
5 times, most recently
from
8b4b980 to
b723bc9
Compare
aathn
force-pushed
the
cat-lax-colim
branch
2 times, most recently
from
6832902 to
d880749
Compare
plt-amy
requested changes
Apr 13, 2026
aathn
force-pushed
the
cat-lax-colim
branch
4 times, most recently
from
ed95d81 to
4f57700
Compare
aathn
force-pushed
the
cat-lax-colim
branch
2 times, most recently
from
c9758d6 to
0448384
Compare
plt-amy
requested changes
Jun 30, 2026
plt-amy
left a comment
plt-amy
left a comment
Member
There was a problem hiding this comment.
i just have some style quibbles, this is otherwise great work and very well-written!
Comment on lines
+80
to
+82
| lf .Lf.hexagon f g h = ext λ _ → bicat! C | ||
| lf .Lf.right-unit f = ext λ _ → bicat! C | ||
| lf .Lf.left-unit f = ext λ _ → bicat! C |
Adds the canonical inclusion functor from the fibre categories of a displayed category into the total category. Also adds lemmas relating this functor to the base change functor when the displayed category is a fibration.
Adds identity and composite constructions for lax functors and pseudofunctors, as well as reasoning modules with helper lemmas.
Adds identity and composition constructions for lax and pseudonatural transformations.
Adds identity, vertical, and horizontal composition constructions for modifications between lax transformations.
Defines opposite and conjugate bicategories, obtained by swapping the direction of 1-cells and 2-cells, respectively. Also defines an oplax functor as a lax functor on the conjugate categories, and defines the conjugate dual of a pseudofunctor.
Defines the category of lax transformations between two lax/pseudo-functors and modifications between them, and the same for oplax transformations. Defines the bicategories of lax/pseudo-functors between two bicategories and (op-)lax transformations between them.
Adds the covariant pseudofunctor which given a fixed object X maps an object Y to the hom-category from X to Y
Defines the constant pseudofunctor between bicategories B and C given a fixed object in C, as well as the curried pseudofunctor from C into the (oplax) pseudofunctor bicategory between B and C.
Defines adjunctions in a bicategory and adjoint equivalences, and proves that they correspond to the familiar notions in Cat. Also defines pseudonatural equivalences as a pseudonatural transformations which are componentwise equivalences.
Defines the canonical displayed category associated with a contravariant pseudofunctor from a locally discrete category into Cat, and proves it is a fibration. Also proves that the fibres of the displayed category coincide with the values of the pseudofunctor, and that the base change functors coincide with the pseudofunctor mapping. Defines the total category of this fibration using Cat.Displayed.Total and specializes the canonical inclusion functor using the equivalences mentioned above. Finally, the module defines some specialized lemmas for indexed categories and lax-transformations between them.
This defines the notion of a lax colimit, and shows that the contravariant Grothendieck construction of a pseudofunctor F corresponds to the lax colimit of F.
Co-authored-by: Amélia Liao <me@amelia.how>
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
3 participants
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
Description
Proves that the (contravariant) Grothendieck construction of a pseudofunctor
F : Pseudofunctor (Locally-discrete (I ^op)) (Cat _ _)is the lax colimit ofF(see nlab; note that the covariant construction is the oplax colimit and vice versa).This required adding a lot of bicategorical definitions, including composition of lax functors, lax transformations, and modifications. I defined the Grothendieck construction by way of the canonical displayed category associated with
F, and also proved some properties about the latter. I have tried to separate the PR into a series of meaningful commits with somewhat descriptive messages.Checklist
Before submitting a merge request, please check the items below:
support/sort-imports.hs(ornix run --experimental-features nix-command -f . sort-imports).If your change affects many files without adding substantial content, and
you don't want your name to appear on those pages (for example, treewide
refactorings or reformattings), start the commit message and PR title with
chore:.