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Stream: theory: category theory

Topic: colax Grothendieck

Christian Williams (Aug 19 2020 at 03:04):

The Grothendieck construction gives an equivalence between pseudofunctors $F:C\to \mathrm{Cat}$ and opfibrations $\int F\to C$. What if $F$ is not pseudo, but only colax? We have $\delta_{gf}:F(g\circ f)\to F(g)\circ F(f)$ satisfying "coassociativity", and $\varepsilon_c:F(1_c)\to 1_{F(c)}$ satisfying "counitality".

I believe that "$\int F$" is still a category.
Given $f:a\to b$, $g:b\to c$ and $w\in F(a)$, $x\in F(b)$, $y\in F(c)$, then
the composite of $(f:a\to b, \alpha:F(f)(w)\to b)$ with $(g:b\to c, \beta: F(g)(x)\to y)$ is
$(g\circ f:a\to c, \beta\circ F(g)(\alpha)\circ \delta_{gf}:F(g\circ f)(w)\to F(g)(F(f)(w))\to F(g)(x)\to y)$,
I've worked out that this composition is still associative, and similarly for the units.

This seems to be generalizing the usual construction, but of course we must be losing something. So then I would assume that $\int F\to C$ is no longer an opfibration. Does anyone have thoughts, or references? If I made a mistake in the calculations implied by the above paragraph, that is helpful as well.

Christian Williams (Aug 19 2020 at 03:27):

(If this is true, then we have the dual statement for lax functors $C^{op}\to \mathrm{Cat}$.)

Alexander Campbell (Aug 19 2020 at 03:57):

See Ross Street's note Powerful functors for an even more general Grothendieck construction (due to Bénabou), which applies to normal lax functors from a category to the bicategory (really the double category) of categories and profunctors. This includes the construction you describe as a special case.

Christian Williams (Aug 19 2020 at 03:59):

Great, thank you, I will. Do you know a bit about this "normal" condition?

Alexander Campbell (Aug 19 2020 at 04:05):

It means that the "unit" constraints $F(1_A) \to 1_{FA}$ are identities. So I suppose strictly speaking this isn't a generalisation of your construction. In the Bénabou construction, normality is used to define identities in the category $\int F$. It indeed looks like normality isn't necessary for your construction, but you can always replace an oplax functor by a normal one (by redefining it on identities), and I imagine the category $\int F$ you get will be the same (edit: this isn't true, see example below about comonads).

Christian Williams (Aug 19 2020 at 04:21):

Thanks, this is great. The connection to exponentiable functors is very interesting.

John Baez (Aug 19 2020 at 04:23):

This looks cool. Since a pseudofunctor $F : C \to \mathsf{Cat}$ gives a cloven fibration (not just a fibration), the functors $\int F \to C$ your construction gives must be some generalization of cloven fibrations. You could guess what this generalization is by looking at the definition of cloven fibration and seeing if there's someplace you could replace an isomorphism by a morphism... or trying to copy the usual proof that the Grothendieck construction gives a cloven fibration, and see what happens to it when you say $F$ is merely lax, not pseudo.

You could also ask Christina Vasilakopoulou this question. She's an expert on the Grothendieck construction. I don't think she reads this Zulip much.

John Baez (Aug 19 2020 at 04:26):

(Every fibration admits a cleaving given the axiom of choice, but a pseudofunctor $F: C \to \mathsf{Cat}$ gives you a fibration with a specific cleaving - a 'cloven' fibration. These nuances become important if you drop the axiom of choice, e.g. when all this math is happening internal to a topos. But I also expect they'll be relevant when you switch from 'pseudo' to 'lax'.)

Alexander Campbell (Aug 19 2020 at 04:30):

This generalised Grothendieck construction gives you an equivalence between normal oplax functors $C \to Cat$ and "locally cocartesian fibrations" over $C$. By the latter I mean functors $P \colon E \to C$ with the property that for every object $e \in E$ and every morphism $g \colon Pe \to c$ in $C$, there exists a "locally $P$-opcartesian" lift $f \colon e \to e'$ of $g$. Locally opcartesian morphisms are those satisfying the dual of the "weaker universal property" mentioned in https://ncatlab.org/nlab/show/Cartesian+morphism#traditional_definition.

Alexander Campbell (Aug 19 2020 at 04:37):

Another way of defining "locally cocartesian fibrations" (this is the ∞-categorical terminology; I forget if there is a classical name) is as those functors $P \colon E \to B$ whose pullback along any functor $\{0<1\} \to B$ is an opfibration in the usual sense.

John Baez (Aug 19 2020 at 04:38):

Neat! I assume that a normal oplax functor $F: C \to \mathsf{Cat}$ actually gives $P : E \to C$ such that every object $e \in E$ and $g : Pe \to e$ is equipped with a locally P-opcartesian lift. (A specified one, not just existence.)

Alexander Campbell (Aug 19 2020 at 04:40):

That's right, so without the axiom of choice, the equivalence is really between normal oplax functors $C \to \textbf{Cat}$ and cloven locally cocartesian fibrations over $C$.

Alexander Campbell (Aug 19 2020 at 04:45):

So I see that in SGA1, Grothendieck's name for "locally cartesian fibration" over $B$ is catégorie préfibrée sur $B$. There's even an nLab page for them.

Alexander Campbell (Aug 19 2020 at 06:11):

Returning to the non-normal case, things are a bit stranger than I said earlier. First, I should note that, for an oplax functor $F \colon C \to \mathbf{Cat}$, the category $\int F$ that Christian described is the oplax colimit of $F$. (I should have said that before anything else!) For example, if we take $C = \mathbf{1}$, then an oplax functor $F \colon \mathbf{1} \to \mathbf{Cat}$ amounts to a category $F(1)$ with a comonad on it, and the category $\int F$ is the coKleisli category of this comonad.

Christian Williams (Aug 19 2020 at 21:46):

Yes, that's very nice. It's strange that we usually think of laxness as deficient, when often it contains significant structure. I only just recently learned of the Kleisli construction as a colimit, and I had not yet realized that the general Grothendieck construction works the same way. Higher-dimensional co/limits are very expressive.

Ian Coley (Aug 19 2020 at 23:12):

Alexander Campbell said:

So I see that in SGA1, Grothendieck's name for "locally cartesian fibration" over $B$ is catégorie préfibrée sur $B$. There's even an nLab page for them.

I have no idea how I was having this same discussion in parallel over on MathOverflow without checking here

John Baez (Aug 20 2020 at 06:04):

Christian Williams said:

Yes, that's very nice. It's strange that we usually think of laxness as deficient, when often it contains significant structure.

I think most category theorists think of laxness as "weirdly productive of interesting structures" - like how a lax monoidal functor from 1 to a monoidal category is a monoid.

Nicolas Blanco (Aug 20 2020 at 13:54):

When considering a lax normal functor into distributors (aka profunctors) $F : \mathcal{B} \to \mathbf{Dist}$ as encoding a functor people sometimes use the term displayed category. It seems that it is possible to get rid of the normality condition by going into $\mathbf{Span}$ instead.
Then you can ask either for a pseudofunctor instead of a normal lax one which gets you the notion of Conduché functor or ask for factorisation through $\mathbf{Cat}$, $\mathbf{Cat}^{op}$ or $\mathbf{Adj}$ which gives preopfibration, prefibration and prebifibration (because the existence of the pullbacks and pushforwards correspond to representability condition on the distributors).
There are a lot of interesting things that happen when you generalise this to other structures (e.g. multicategories or polycategories) but that would be a little bit out of the subject and probably just me pushing my agenda/research.

Nicolas Blanco (Aug 20 2020 at 14:10):

Also if I recall correctly, the Bénabou construction can be derived as a (weak?) pullback (in the bicategorical sense not the fibrational one) by considering the strict 2-functor $\mathbf{Dist}_{\ast} \to \mathbf{Dist}$ from pointed distributors and pulling it along the lax normal functor considered.
I am trying to work out the details of something a little bit different : starting from a normal lax functor $F : \mathbf{B} \to \mathbf{C}$ from a category to a bicategory and considering a strict 2-opfibration (i.e. existence of opcartesian 2-morphisms) $p : \mathbf{D} \to \mathbf{C}$ to construct a category $\int_p F$ with a functor into $\mathbf{B}$ (and a lax one into $\mathbf{D}$ making the obvious diagram commute). The special case $p$ being the forgetful functor $\mathbf{Dist}_\ast \to \mathbf{Dist}$ will allow to get the Bénabou construction from the 2-fibrational properties of this forgetful functor.

dusko (Aug 27 2020 at 01:53):

John Baez said:

(Every fibration admits a cleaving given the axiom of choice, but a pseudofunctor $F: C \to \mathsf{Cat}$ gives you a fibration with a specific cleaving - a 'cloven' fibration. These nuances become important if you drop the axiom of choice, e.g. when all this math is happening internal to a topos. But I also expect they'll be relevant when you switch from 'pseudo' to 'lax'.)

are you sure that every fibration admits a cleavage? if you take a group epimorphism $e: G \to B$, it is clearly a fibration, even trivial. but as far as i can tell, to define a cleavage, you must choose a splitting of $e$. but you obviously cannot always do that, no matter how big is the axiom of choice. or did i misunderstand the claim?

[sorry, i login after weeks and the first thing i see immediately gets me in trouble...]

Alexander Campbell (Aug 27 2020 at 02:12):

A cleavage of the fibration you mention just amounts to a section of $e \colon G \to B$ as a function between sets; a splitting of this fibration is a section that is moreover a group homomorphism. So indeed, not every fibration admits a splitting, but every fibration does admit a cleavage (using the axiom of choice).

John Baez (Aug 27 2020 at 22:07):

Yeah, "cleaving" a fibration is much weaker than "splitting" it. This terminology goes back to Johnstone I think, along with "cloven" versus "split".

Alexander Campbell (Aug 27 2020 at 22:55):

In SGA1, Grothendieck uses the words "clivage" and "scindage" for "cleavage" and "splitting" of a fibration.

dusko (Sep 03 2020 at 08:48):

Alexander Campbell said:

A cleavage of the fibration you mention just amounts to a section of $e \colon G \to B$ as a function between sets; a splitting of this fibration is a section that is moreover a group homomorphism. So indeed, not every fibration admits a splitting, but every fibration does admit a cleavage (using the axiom of choice).

the idea that cleavage might be a function between sets (i presume you mean the underlying sets of groups) sounds a little misleading, since the sets in this case happen to be someone's hom-sets. since each inverse image functor in the cleavage in this case has to be a group automorphism, the cleavage in this case is a pseudofunctor $B \to Aut(G)$. a pseudofunctor is a little more than a set theoretic function. when it is a functor, we have a splitting. but a proper representation of an arbitrary group by an arbitrary group covering it may not exist. when it does not exist, then pseudofunctor chooses an automorphism $b^*$ of $G$ for every $b\in B$, and moreover a canonical isomorphism $(bd)^*\cong d^*\circ b^*$ for all $b,d\in B$. the canonical isomorphisms come with this silly requirement that they need to be coherent. the natural isomorphisms between automorphisms of $G$ are some elements of $G$. we can definitely select a family of canonical isomorphisms for the cleavage as a splitting of the group epimorphism from which we started. can we prove that we can select a coherent family of canonical isomorphisms for the cleavage that will not be a splitting of that epimorphism?

Alexander Campbell (Sep 03 2020 at 13:34):

Are we using different definitions of "cleavage"? For me, a cleavage of a fibration $P \colon E \to C$ is a choice, for each object $e$ in $E$ and each morphism $g \colon c \to P(e)$ in $C$, of a $P$-cartesian morphism $f$ in $E$ which has codomain $e$ and which satisfies $P(f) = g$.

dusko (Sep 04 2020 at 08:18):

Alexander Campbell said:

Are we using different definitions of "cleavage"? For me, a cleavage of a fibration $P \colon E \to C$ is a choice, for each object $e$ in $E$ and each morphism $g \colon c \to P(e)$ in $C$, of a $P$-cartesian morphism $f$ in $E$ which has codomain $e$ and which satisfies $P(f) = g$.

i thought you were using grothendieck's definition from SGA1, which you mentioned above. the liftings that you describe here would give just the object part of the inverse image functors.

dusko (Sep 04 2020 at 08:20):

(the fact that restricting the structure to the object parts would let us lift a group pretty much in the air, independently on the other group, seems like a nice example of why we need functors, and not just functions :)

Alexander Campbell (Sep 04 2020 at 08:46):

A cleavage in my sense canonically extends to a cleavage in the sense you indicate, using the universal property of (the chosen) $P$-cartesian morphisms.