defn: Pseudofunctor bicategories and the Grothendieck construction as a lax colimit#612
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aathn wants to merge 16 commits intothe1lab:mainfrom
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defn: Pseudofunctor bicategories and the Grothendieck construction as a lax colimit#612aathn wants to merge 16 commits intothe1lab:mainfrom
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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. Also adds the boilerplate to show modifications form a set, and sets up an extensionality characterization similar to that for natural 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. In addition, I added an explicit definition of the opposite product category duality which computes better.
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.
plt-amy
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Apr 13, 2026
| ×^op← .F-∘ f g = refl | ||
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| private | ||
| ×^op←-is-inverse : ×^op← ≡ is-equivalence.F⁻¹ ×^op-is-equiv |
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did this actually get used? paths of functors are kind of iffy, i would prefer if it was a natural isomorphism.
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So what happened is that before the bifunctor refactor I needed ×^op← when defining the co-bicategory, but the existing one was defined as is-equivalence.F⁻¹ ×^op-is-equiv (where ×^op-is-equiv was derived from a precat-iso) which gave a bunch of unreduced transports. So I changed it to a direct definition and thought I should at least give a proof that they are the same. But after the bifunctor refactor I'm not using it anymore.
Now I rewrote this to show the equivalence directly. I could also revert the changes entirely since they are not really related to the PR anymore.
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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.This is marked as a draft because I am still working on the prose, todo:Also note that this PR sits on top of the functor solver branch which is not yet merged (note that the first three commits are from that branch and are not part of the main content of this PR).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:.