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Stream: theory: category theory

Topic: Anti-Grothendieck Construction

Joshua Meyers (Feb 10 2021 at 03:44):

Given a functor $F:C^\text{op}\to\textbf{Cat}$, the Grothendieck construction has objects $(c:C,x:Fc)$ and a morphism $(c,x)\to (c',x')$ is a morphism $f:c\to c'$ and a morphism $f^\sharp:Ff(x')\to x$.

But what happens if instead we define a morphism $(c,x)\to (c',x')$ to be a morphism $f:c\to c'$ and a morphism $f^\sharp:x\to Ff(x')$?

Turns out it still works, it still gives a category. So why has nobody bothered to give it a name? Is it boring for some reason? Does it always naturally reduce to some simple thing (it does in the case $F:\textbf{Set}\to\textbf{Cat},S\mapsto \textbf{Set}/S$, where it just gives the arrow category)?

Joe Moeller (Feb 10 2021 at 03:46):

This is the Grothendieck construction for $C^{op}$-opindexed categories.

Joshua Meyers (Feb 10 2021 at 03:47):

What do you mean?

Joe Moeller (Feb 10 2021 at 03:50):

A functor $F \colon C^{op} \to \mathbf{Cat}$ is a $C$-indexed category, and then you describe the Grothendieck consruction for that. But $F$ is also a $C^{op}$-opindexed category. Opindexed categories have a different Grothendieck construction, which is given by what you described.

Joshua Meyers (Feb 10 2021 at 03:51):

I see

Joshua Meyers (Feb 10 2021 at 04:00):

Ah you're right! I see the two Grothendieck constructions here: https://ncatlab.org/nlab/show/Grothendieck+construction

Now, both Grothendieck constructions have the same objects --- is there any nice way that the morphisms play with each other?

Joshua Meyers (Feb 10 2021 at 05:07):

I just found something cool! The two types of morphisms are vertical and horizontal morphisms in a double category! 2-cells are commuting squares, which actually can be defined.

Joe Moeller (Feb 10 2021 at 05:09):

This sounds familiar. Do you have a link?

Joshua Meyers (Feb 10 2021 at 05:09):

No I just figured it out, I haven't looked it up yet

Joshua Meyers (Feb 10 2021 at 05:10):

This sounds similar, in the nLab article on "double category":

There is a double category MonCat whose objects are monoidal categories, whose horizontal arrows are lax monoidal functors, whose vertical arrows are colax monoidal functors, and whose 2-cells are generalized monoidal natural transformations. An analogous double category can be constructed involving the algebras for any 2-monad; see double category of algebras.

Joe Moeller (Feb 10 2021 at 05:10):

Oh ok. Have you read Shulman's Framed bicategories and monoidal fibrations? It's relevant.

Joshua Meyers (Feb 10 2021 at 05:11):

No not yet

Joshua Meyers (Feb 10 2021 at 05:12):

Perhaps lax/colax stuff forms a double category in general?

Joshua Meyers (Feb 10 2021 at 05:13):

Also, this double category has a special property: the source and target maps are jointly faithful. In other words, a square either commutes or doesn't, there aren't ever multiple 2-cells you can put in a square while the edges remain constant. This means that it can be thought of as a double category in 2 ways: the vertical and horizontal 1-morphisms can be interchanged, so the standard definition is a bit biased in this regard.

Mike Shulman (Feb 10 2021 at 16:02):

Joshua Meyers said:

Perhaps lax/colax stuff forms a double category in general?

I think this is what is indicated by the second sentence of the nLab quote:

An analogous double category can be constructed involving the algebras for any 2-monad; see double category of algebras.

Mike Shulman (Feb 10 2021 at 16:05):

Joshua Meyers said:

The two types of morphisms are vertical and horizontal morphisms in a double category! 2-cells are commuting squares, which actually can be defined.

I don't remember offhand seeing this before, but it is probably an instance of a general Grothendieck construction for double categories, where the domain is an opposite of the double category of commuting squares in $C$.

Joshua Meyers (Feb 10 2021 at 16:36):

Thanks @Mike Shulman! Can you elaborate a bit on the second conjecture? What is the functor that this may be a Grothendieck construction on?

Joshua Meyers (Feb 10 2021 at 16:36):

And the double category of algebras seems very similar, I'll read about it

Mike Shulman (Feb 10 2021 at 16:44):

Let $QC$ be the double category of commutative squares in $C$, and $Q\rm Cat$ the double category of pseudo-commutative squares in $\rm Cat$. Then a pseudofunctor $F:C^{\rm op} \to \rm Cat$ should induce a double pseudofunctor $QF : QC^{\rm h\cdot op} \to Q\rm Cat^{\rm v\cdot op}$, and there should be a Grothendieck construction for such functors that behaves like the two 1-categorical Grothendieck constructions on the vertical and horizontal pieces. More generally, $Q\rm Cat^{\rm v\cdot op}$ should embed in $Q \rm Prof$, which has profunctors as both kinds of morphisms, and there should be a Grothendieck construction for functors landing therein.

Mike Shulman (Feb 10 2021 at 16:45):

Since the most general Grothendieck construction for a 1-category acts on a lax functor to $\rm Span$ (any category over $C$ can be obtained from such a Grothendieck construction), one might suspect that for double categories this would be true for $Q\rm Span$. However, I'm not sure whether one can make sense of a double functor that is "lax in both directions".

Joshua Meyers (Feb 10 2021 at 17:22):

That last fact is so cool! I see how it extends the pair of facts that a monad in $C$ is a lax functor $1\to C$ and a monad in $\text{Span}$ is a category.