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Stream: learning: questions

Topic: lax functors and connected components

Bruno Gavranović (Sep 06 2022 at 07:56):

I found an interesting way to transform lax functors into 1-functors.

Consider the monoidal adjunction $\pi_0 \dashv \textsf{discr}$ between $\mathbf{Set}$ and $\mathbf{Cat}$, where $\pi_0$ is the connected components functor. Via enriched change of base this induces a 2-adjunction $\textsf{discr}_* \dashv \pi_{0^*}$ between $\mathbf{Cat}$ (i.e. categories enriched in $\mathbf{Set}$) and $\mathbf{2}\textbf{-}\mathbf{Cat}$ (i.e. categories enriched in $\mathbf{Cat}$).

Now, I'm curious about $\pi_{0^*}$. As described in Section 4. of "Limits indexed by category-valued 2-functors", this 2-functor maps a 2-category $\mathcal{A}$ to a category $\pi_{0^*}(\mathcal{A})$ with the same objects, but whose hom-sets are connected components of hom-objects of $\mathcal{A}$. This can be extended to 2-functors between 2-categories in a straightforward way.

But I noticed something - this can be made even more general!
Namely, consider the category of 2-categories and lax functors between them. Then $\pi_{0^*}$ can be defined here too - and it sends a lax functor to a 1-functor. This happens because all the lax arrows $F(f) ; F(f) \Rightarrow F(f ; g)$ get identified together, strictifying the functor.
I find this really interesting, and it's the first time I see a way to transform lax functors into plain old 1-functors. Is anything more known about this?

I wondered whether the original adjunction works in this setting as well - but for that we'd need a 2-category of 2-categories, lax functors and some kinds of transformations. The only choice for this I know of is the 2-category $\mathbf{2}\textbf{-}\mathbf{Cat}^\textsf{ic}$ of 2-categories, lax functors and icons, but in that case it doesn't seem like the $\textsf{discr}_*$ functor can well defined on 2-cells.

I appreciate any pointers or thoughts about this.

fosco (Sep 06 2022 at 08:06):

The problem might be that the 1-category you obtain in this way says nothing about the 2-category you started with: $\pi_{0,*}({\cal H})$ is a 1-category that forgets a lot about $\cal H$: for example, if $\cal H$ is a 2-category where each hom-category is connected (for example, each ${\cal H}(X,Y)$ has a weakly initial or weakly terminal object, which is already a mild condition and not even the weakest one!), in the 1-category $\pi_{0,*}({\cal H})$ each hom-set is a singleton. A legitimate category, but not very interesting and most of all, knows nothing about $\cal H$

fosco (Sep 06 2022 at 08:08):

When you look at how $\pi_{0,*}$ acts on morphisms it's even worse: it is well-known that a monad $T$ on $Cat$ is a lax functor $T : 1\to Cat$ from the terminal 2-category; but then for every monad $T$ the map $\pi_{0,*}(T)$ is defined as the map choosing an object $X$ and the connected component of its identity functor

Not much information about $T$ is retained!

Bruno Gavranović (Sep 06 2022 at 08:09):

fosco said:

The problem might be that the 1-category you obtain in this way says nothing about the 2-category you started with: $\pi_{0,*}({\cal H})$ is a 1-category that forgets a lot about $\cal H$: for example, if $\cal H$ is a 2-category where each hom-category is connected (for example, each ${\cal H}(X,Y)$ has a weakly initial or weakly terminal object, which is already a mild condition and not even the weakest one!), in the 1-category $\pi_{0,*}({\cal H})$ each hom-set is a singleton. A legitimate category, but not very interesting and most of all, knows nothing about $\cal H$

Oh, of course. This even strengthens my motivation, as I'm starting from some category $\pi_{0^*}(\mathcal{H})$ that's already in the literature and trying to make a case it's all along been a shadow of the 2-category $\mathcal{H}$.

Mike Shulman (Sep 07 2022 at 17:33):

The analogue of the 2-category $\mathbf{2}\textbf{-}\mathbf{Cat}^\textsf{ic}$ whose objects are 1-categories is probably the 1-category $\mathbf{Cat}$, where the only 2-cells are identities.

Bruno Gavranović (Sep 07 2022 at 23:24):

I would've imagined it's the 2-category $\mathbf{Cat}$, as there's a 2-functor $\pi_{0^*} : \mathbf{2}\textbf{-}\mathbf{Cat}^\textsf{ic} \to \mathbf{Cat}$.

Mike Shulman (Sep 07 2022 at 23:29):

Every sufficiently good analogy may be yearning to become a functor, but not every functor is a sufficiently good analogy. (-:

An icon is, among other things, the identity on objects. Thus, I would say the analogue of an icon for 1-categories should also be the identity on objects. Indeed, an icon between 1-categories regarded as locally discrete 2-categories is just an identity natural transformation. Similarly, 2-categories naturally form a 3-category, but restricting the morphisms to icons allows them to be only a 2-category; thus an analogous restriction for 1-categories should reduce their collection from a 2-category to a 1-category. Finally, the absence of an adjoint to $\pi_{0*}$ is another clue that something is amiss.

Bruno Gavranović (Sep 07 2022 at 23:58):

That's a convincing argument. I never thought about the fact that an icon between 1-categories (regarded as locally discrete 2-cats) constrains it precisely to be the identity one as there are no choices left to be made: icon requires the components to be identity, and the oplax morphism has to be identity too since only available 2-cells are identities.

I suppose I was inclined to agree otherwise because I noticed $\pi_{0^*}$ sends adjunctions to equivalences, and keeping the 2-cells is important for stating that. But that seems like a different question.

John Baez (Sep 08 2022 at 07:17):

Every sufficiently good analogy may be yearning to become a functor...

This sounds like a hybrid of my remark "every analogy is yearning to become a functor" and Arthur C. Clarke's "any sufficiently advanced technology is indistinguishable from magic".

John Baez (Sep 08 2022 at 07:17):

Any sufficiently good analogy is indistinguishable from magic?

Beppe Metere (Sep 08 2022 at 08:08):

Hi @Bruno Gavranovic

Maybe this is related, and the context is not just 2-categories, but the more general one of bicategories.

Jean Bénabou, in his "Introduction to bicategories" (9 years before Street's paper) introduces the notion of the Poincaré category of a bicategory: given a bicategory $\mathbb B$, $\Pi(\mathbb B)$ is the category with the same objects and (guess!) $\Pi(\mathbb B)(a,b)$ is the set of connected components of the category $\mathbb B(a,b)$. This extends to morphisms of bicategories, so that you get the Poincaré functor $Bicat\to Cat$, which is left adjoint to the locally discrete functor (he calls it degeneracy).

Well, @fosco this is seems to be very interesting and meaningful, in some cases... for instance, one example in Bénabou paper is the following.

Let $A$ be an abelian category, and let $Ext_A$ be the bicategory of (finite) extensions in $A$: arrows are $n$-fold extensions $a\to\cdots\to b$, for any $n$, composition is Yoneda's composition (described in his 1960's paper on extensions), 2-cells are morphisms of $n$-fold extensions fixing $a$ and $b$ .

Well, in this case, the connected components of the hom-categories collect the cohomology groups :).

Ciao,

Beppe.

fosco (Sep 08 2022 at 08:58):

John Baez said:

Any sufficiently good analogy is indistinguishable from magic?

Assuming LEM, every functor is either a technology or a magic spell

Mike Shulman (Sep 08 2022 at 15:52):

John Baez said:

Every sufficiently good analogy may be yearning to become a functor...

This sounds like a hybrid of my remark "every analogy is yearning to become a functor" and Arthur C. Clarke's "any sufficiently advanced technology is indistinguishable from magic".

Thanks for the link! I looked briefly but didn't find a canonical citation for your quote; is that the best one? As for "sufficiently good", I think I just misremembered it -- although I see at that link that you did say "I didn't really mean 'any' analogy".

Mike Shulman (Sep 08 2022 at 15:56):

Bruno Gavranovic said:

I suppose I was inclined to agree otherwise because I noticed $\pi_{0^*}$ sends adjunctions to equivalences, and keeping the 2-cells is important for stating that.

I'm not sure what you mean by that. An internal adjunction in a 2-category $C$ becomes an internal isomorphism in $\pi_{0*}(C)$, since $\pi_{0*}(C)$ is a 1-category there's no room for anything else. An adjunction between 2-categories $C\rightleftarrows D$ becomes an adjunction between 1-categories $\pi_{0*}(C)\rightleftarrows \pi_{0*}(D)$, not an equivalence; $\pi_{0*}$ doesn't invert 2-natural transformations (although it makes lax natural transformations into strict ones).

fosco (Sep 08 2022 at 16:20):

@Beppe Metere I'm glad to be wrong when I discover such things! Can you write more precisely how the cohomology groups arise? Can you do the same for homology?

Some time ago I asked on MO what is the coend of the $Ext^n$ functor, I suspect there's a relation between the two ideas, what do you think?

Beppe Metere (Sep 08 2022 at 17:57):

@fosco , most of it is in Yoneda paper. Indeed, that paper is motivated by the the aim of doing cohomology in categories without (enough) injectives nor projectives.
The idea is to describe cohomology groups by means of a suitable operation on (the connected components of ) the $n$-extensions, for fixed $n$. The operation that results is called Baer product, or sometimes Baer sum. You can prove that this operation, is compatible with the equivalence relation generated by connectedness, and that gives an abelian group structure on the quotient sets $Ext^n(a,b)$. In concrete cases, (i.e. in the case of $R$-modules, when you have enough projectives) such groups are isomorphic to the classical $H^n(x,b)$, where $x$ is a projective resolution of $a$. Non abelian generalizations of this exists (Bourn, Janelidze, and others... something myself with Sandra Mantovani and Alan Cigoli). Concerning your second question, I have to think. However, as you know, that paper by Yoneda is where he (substantially) defines distributors and introduces your beloved integral notation... so, there should be a connection.

Bruno Gavranović (Sep 10 2022 at 08:35):

Mike Shulman said:

Bruno Gavranovic said:

I suppose I was inclined to agree otherwise because I noticed $\pi_{0^*}$ sends adjunctions to equivalences, and keeping the 2-cells is important for stating that.

I'm not sure what you mean by that. An internal adjunction in a 2-category $C$ becomes an internal isomorphism in $\pi_{0*}(C)$, since $\pi_{0*}(C)$ is a 1-category there's no room for anything else. An adjunction between 2-categories $C\rightleftarrows D$ becomes an adjunction between 1-categories $\pi_{0*}(C)\rightleftarrows \pi_{0*}(D)$, not an equivalence; $\pi_{0*}$ doesn't invert 2-natural transformations (although it makes lax natural transformations into strict ones).

Sorry, I was a bit vague. I was referring to the functor $\pi_{0^*}^\textsf{ic}: \mathbf{2}\textbf{-}\mathbf{Cat}^\textsf{ic} \to \mathbf{Cat}$ whose domain is the 2-category of 2-categories, lax functors and icons. I previously called that functor $\pi_{0^*}$, which gets confusing since I used that terminology for $\pi_{0^*} : \mathbf{2}\textbf{-}\mathbf{Cat} \to \mathbf{Cat}$ whose domain is the 2-category of 2-categories, 2-functors and lax/pseudo natural transformations.

I imagine here you were referring to the latter, with which I agree. But if I understand it correctly, for the former functor $\pi_{0^*}^\textsf{ic}: \mathbf{2}\textbf{-}\mathbf{Cat}^\textsf{ic} \to \mathbf{Cat}$ it is the case that icons are inverted -- every icon is sent to the identity natural transformation, turning adjunctions into equivalences. Even more - it turns adjunctions into isomorphisms, as we have identity natural transformations, not mere isomorphisms.

Bruno Gavranović (Sep 10 2022 at 08:38):

I suppose this allows us to think of the codomain of $\pi_{0^*}^\textsf{ic}: \mathbf{2}\textbf{-}\mathbf{Cat}^\textsf{ic} \to \mathbf{Cat}$ as a 1-category!

John Baez (Sep 11 2022 at 15:08):

Mike Shulman said:

As for "sufficiently good", I think I just misremembered it -- although I see at that link that you did say "I didn't really mean 'any' analogy".

In retrospect I think it depends on the precise definition of "yearning".

John Baez (Sep 11 2022 at 15:13):

Some hopes are really forlorn.