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Stream: learning: questions

Topic: (Op)lax transformations as 2-cells in a tricategory

Amar Hadzihasanovic (Jan 08 2025 at 13:53):

It is “well known” at least among those who have wondered about this that there is no tricategory whose first three levels are bicategories, (strong) functors, and (op)lax natural transformations, since e.g. there is no way to whisker an (op)lax transformation with a functor.

However, as far as I can tell, this leaves open the possibility that all these components and their “valid” compositions may be faithfully represented in some wider tricategory, e.g. one in which the 1-cells are an appropriate notion of profunctors of bicategories; then, as some profunctors are representables associated with a functor, and these are closed under composition, so some 2-cells between representable profunctors would be representable as (op)lax transformations of functors, and these would be closed under vertical composition but not under horizontal composition and/or whiskering.

To your knowledge, are there any known positive or negative results about the existence of such tricategories?

Amar Hadzihasanovic (Jan 08 2025 at 13:53):

(A warning: I am interested specifically in n-categorical structures as opposed to multiple categories, so no matter if one thinks that “the correct structure that these form is a triple category” or something, that won't be a helpful answer for me, although you should feel free to share with others!)

Mike Shulman (Jan 08 2025 at 16:09):

Actually it's not whiskering that's the problem, but interchange. You can whisker (op)lax transformations with functors, but in the situation where you'd want to compose two of them horizontally, you get two composites from whiskering and vertical composition, and they aren't isomorphic (only related by a transformation).

Mike Shulman (Jan 08 2025 at 16:11):

At least at first, this seems like it would be an issue for your idea, since if you want the operations on (op)lax transformations that do exist to be respected in your tricategory of profunctors, that would include both vertical composition and whiskering, but then if interchange does hold in your tricategory of profunctors, the inclusion of (op)lax transformations into it couldn't be conservative, which also seems like something you'd want.

Amar Hadzihasanovic (Jan 09 2025 at 13:03):

Yes, that's right, I got confused for a moment with the issue of lax functors where iirc there is seemingly no way to even define whiskering.

Amar Hadzihasanovic (Jan 09 2025 at 13:20):

Still it may not be desperate if we don't require whiskering among the operations that we need to respect (at least not in the strongest possible sense).

Looking at the counterexample to horizontal composition of lax transformations on slide 7 of this talk by Bob Paré, I wonder if there might not be a generalisation of lax transformations where "the component at A" is allowed to be a pair of morphisms connected by a 2-morphism --- with "usual" lax transformations being represented as the case where the two morphisms are equal and the 2-morphism is an identity --- so that one is not actually forced to choose between the two "options" presented in that slide.

Amar Hadzihasanovic (Jan 09 2025 at 13:24):

I am vaguely thinking of something defined in "fibrational" form, so that instead of a "function from $n$-cells to $(n+1)$-cells” one has “an $(n+1)$-dimensional fibre over each $n$-cell” which can be relaxed to something higher-dimensional, if maybe “laxly” $(n+1)$-dimensional, e.g. the higher morphisms “converge” to some terminal $(n+1)$-morphism...
(I will need some time to think about whether there are obvious issues with this idea)

Mike Shulman (Jan 09 2025 at 16:09):

Amar Hadzihasanovic said:

I wonder if there might not be a generalisation of lax transformations where "the component at A" is allowed to be a pair of morphisms connected by a 2-morphism

Of course, then you have to say how to compose two of those horizontally...

Amar Hadzihasanovic (Jan 09 2025 at 20:17):

Let's work with strict 2-categories and strict functors for simplicity. Let's postulate that a 2-cell between functors $F, G: C \to D$ is a pair of lax natural transformations $\ell^-, \ell^+$ together with a modification $\ell: \ell^- \to \ell^+$.

It seems to me that there is a good notion of horizontal composition of these: given $F, G: C \to D$ and $H, K: D \to E$ and 2-cells $\ell: F \Rightarrow G$ and $m: H \Rightarrow K$, their horizontal composite $m \bullet \ell: HF \Rightarrow KG$ is the pair of lax natural transformations $K\ell^- \circ m^-F, m^+ G \circ H\ell^+$ together with the modification between the two whose component at $a$ is either side of this equation:
Screenshot from 2025-01-09 22-16-20.png

Amar Hadzihasanovic (Jan 09 2025 at 20:46):

(Haven't checked at all what equations this satisfies, just noticing that there does seem to be a "canonical" choice unlike in the case of lax natural transformations.)

Mike Shulman (Jan 09 2025 at 23:34):

Yes, you're right. But I think it won't be a tricategory, since in a tricategory the horizontal composite is still supposed to be isomorphic to both vertical-composites-of-whiskers.

Amar Hadzihasanovic (Jan 16 2025 at 13:13):

Here's another idea. Modulo some care in making pullbacks strictly functorial, for every $n$ there should be a “large strict $n$-category of strict $n$-categories” constructed as follows.
The $k$-cells are pairs of an $n$-category $C$ and a strict functor $p: C \to O^k$ where $k$ is the walking $k$-cell. The source/target operations as well as units are obtained by pullback along morphisms in the globe category represented as functors between walking cells. This gives the underlying reflexive $n$-graph ($n$-globular set).
When two $k$-cells are composable at the $j$-boundary, this determines a span of pullback squares of functors; taking its pushout in the arrow category produces an $n$-category over the “walking pasting diagram” $O^k \#_j O^k$. Pulling back along the “largest $k$-cell” functor $O^k \to O^k \#_j O^k$ produces a composite.
Functoriality of pullback does the rest -- the idea is that, with this “colimit then pullback” method, we can extend our $n$-graph to a presheaf on the cell category $\Theta_n$ that preserves certain limits by construction, hence defines a strict $n$-category.

Amar Hadzihasanovic (Jan 16 2025 at 13:16):

When $n = 1$, I claim that this produces something equivalent to the category of small categories and profunctors. Indeed, $O^0$ is the point i.e. terminal category so objects are just small categories, while $O^1$ is the walking arrow i.e. the poset (0 < 1), and a functor $p: E \to O^1$ is the “cograph” representation of a profunctor between the categories in the fibres over the endpoints of the walking arrow.

Amar Hadzihasanovic (Jan 16 2025 at 13:21):

Given a composable pair $p: E \to O^1$ and $q: F \to O^1$, the pushout in the arrow category “glues together” the cographs along the target of the first profunctor and the source of the second, producing a category together with a functor to the “walking composable pair of arrows”, which is equivalently the linear order $(0 < 1 < 2)$.
Pulling back along the canonical “endpoint-preserving” functor $(0 < 1) \to (0 < 1 < 2)$ gives the cograph of the composite profunctor.

Amar Hadzihasanovic (Jan 16 2025 at 13:27):

Anyway, when $n = 2$, this should produce a strict 2-category whose objects are small strict 2-categories.
Now, strict functors seem to be represented as morphisms in the same way as functors are represented as profunctors: namely, if $F: C \to D$ is a strict functor, we get a “cograph” 2-category whose objects are the disjoint union of the objects of $C$ and $D$, hom-categories in the two separate components are the same as in $C$ and $D$ respectively, and the hom-category $\mathrm{Hom}(c, d)$ for $c$ in $C$ and $d$ in $D$ is defined as $\mathrm{Hom}_D(Fc, D)$ (while $\mathrm{Hom}(d, c)$ is empty).

Amar Hadzihasanovic (Jan 16 2025 at 13:37):

If I'm not mistaken we can also represent at least oplax transformations between parallel strict functors as 2-cells between their representations: if $\alpha: F \Rightarrow G$ is an oplax natural transformation between $F, G: C \to D$, we define its “cograph” as the 2-category whose

• objects are the disjoint union of the objects of $C$ and $D$,
• hom-categories in the two components are the same as in $C$ and $D$ respectively,
• for $c \in C$ and $d \in D$, the objects of $\mathrm{Hom}(c, d)$ are the disjoint union of $\mathrm{Hom}_D(Fc, D)$ and $\mathrm{Hom}_D(Gc, D)$,
• hom-sets in the two separate components of $\mathrm{Hom}(c, d)$ are as they were,
• given $f: Fc \to D$ and $g: Gc \to D$, we let $\mathrm{Hom}(f, g) = \mathrm{Hom}_D(f, g\circ \alpha_c)$, while $\mathrm{Hom}(g, f)$ is empty.

Amar Hadzihasanovic (Jan 16 2025 at 13:46):

Oplax naturality should ensure that this is well-defined in that the cells and 2-cells of $C$ act naturally on $\mathrm{Hom}(f, g)$: for example, if $\gamma: f \Rightarrow g$ is a 2-cell between 1-cells $f, g: c' \to d$, and $h: c \to c'$ is a 1-cell in $C$, the whiskering of $\gamma$ with $h$ is the following 2-cell from $fh$ to $gh$:
Screenshot from 2025-01-16 18-44-31.png

Amar Hadzihasanovic (Jan 16 2025 at 14:09):

So this should give a bona fide 2-category (with strict interchange) where oplax transformations of 2-functors are represented as 2-cells... so either there's a mistake in my reasoning so far, or their horizontal composition or whiskering are genuinely different from the “naive” version; perhaps because of the role of 2-cells of $D$ in the “horizontally glued cographs”?

Mike Shulman (Jan 16 2025 at 19:05):

Well, to start with I don't think that "some care in making pullbacks strictly functorial" is going to be sufficient to get out a strict $n$-category. The construction involves not only pullbacks but also pushouts, and even in the two individual cases I don't know of a construction of an $n$-category of "higher (co)spans" that is strict in all dimensions, plus there is the interaction of the two to worry about.

Mike Shulman (Jan 16 2025 at 19:07):

It does seem likely that you'll get a non-strict $(n+1)$-category at least. But in that case I think your guess at the end is right: the inclusion of oplax transformations into this thing won't preserve their naive horizontal composition.

Amar Hadzihasanovic (Jan 16 2025 at 20:19):

Yes, the issue of “actual strictness” seems tricky, but then on the other hand, only pullbacks along morphisms in $\Theta$ and pushouts of $n$-categories that are “displayed over objects of $\Theta$” are involved, and maybe that helps; objects of $\Theta$ are particularly rigid. (It is only a vague sensation, I haven't given much thought to the issues involved.)

Amar Hadzihasanovic (Jan 16 2025 at 20:21):

When I have the time, I will try to explicitly compute the horizontal composite of two “2-cells represented by oplax transformations”.

Amar Hadzihasanovic (Jan 17 2025 at 15:16):

Ok, I have now computed the horizontal composite of two 2-cells representing oplax natural transformations, and the result is quite interesting!
I will just give the result for now, I can expand on how one gets to it if anyone's interested.

Amar Hadzihasanovic (Jan 17 2025 at 15:21):

So let $F, G: C \to D$ and $H, K: D \to E$ be 2-functors and let $\alpha: F \Rightarrow G$ and $\beta: H \Rightarrow K$ be oplax natural transformations. There are "cographs" $\mathcal{E}_\alpha$ and $\mathcal{E}_\beta$ associated to $\alpha$ and $\beta$ as I described before: to recall the interesting part,

• the 1-cells from $c \in C$ to $d \in D$ in $\mathcal{E}_\alpha$ are either $f: Fc \to d$ or $g: Gc \to d$ in $D$,
• the 2-cells from $f: Fc \to d$ to $g: Gc \to d$ are 2-cells $\gamma: f \Rightarrow g\alpha_c$ in $D$

and similarly

• the 1-cells from $d \in D$ to $e \in E$ in $\mathcal{E}_\beta$ are either $h: Hd \to e$ or $k: Kd \to e$ in $E$,
• the 2-cells from $h: Hd \to e$ to $k: Kd \to e$ are 2-cells $\delta: h \Rightarrow k\beta_d$ in $E$.

Amar Hadzihasanovic (Jan 17 2025 at 15:24):

Now, the horizontal composite of $\alpha$ and $\beta$ in my 2-category is described by a 2-category $\mathcal{E}_\beta \bullet \mathcal{E}_\alpha$ over the walking 2-cell $O^2$.
The 0-cells of this category are either objects of $C$ or objects of $E$. The interesting part is the following.

Amar Hadzihasanovic (Jan 17 2025 at 15:26):

The 1-cells from $c \in C$ to $e \in E$ are, as expected, either $h: HFc \to e$ or $k: KGc \to e$ in $E$.

Amar Hadzihasanovic (Jan 17 2025 at 15:33):

But now, a 2-cell from $h: HFc \to e$ to $k: KGc \to e$ is an equivalence class of 2-cells $\varphi: h \Rightarrow k(\beta_{Gc})(H\alpha_c)$ in $E$, under the equivalence relation where two 2-cells are identified if and only if their composites with $\beta(\alpha_c)$ as in the following diagram
Screenshot from 2025-01-17 17-32-09.png
are equal.

Amar Hadzihasanovic (Jan 17 2025 at 15:37):

This puts this horizontal composite exactly “in between” the cographs of the two, non-equal, “naive” horizontal composites obtained by whiskering + vertical composition of oplax natural transformations, that is, $(\beta G) (H\alpha)$ and $(K\alpha)(\beta F)$, in the sense that there are 2-functors

$$\mathcal{E}_{(\beta G) (H\alpha)} \to \mathcal{E}_\beta \bullet \mathcal{E}_\alpha \to \mathcal{E}_{(K \alpha) (\beta F)}$$

over $O^2$...

Amar Hadzihasanovic (Jan 17 2025 at 15:40):

The first sends $\varphi: h \Rightarrow k(\beta_{Gc})(H\alpha_c)$ to its equivalence class, whereas the second sends an equivalence class to the composite of each representative with $\beta(\alpha_c)$ as in the diagram above, which is by definition independent.

Amar Hadzihasanovic (Jan 17 2025 at 15:43):

(Presumably this is, in fact, just an epi-mono factorisation of the composite functor of cographs)

Amar Hadzihasanovic (Jan 17 2025 at 15:45):

Now I would be interested in characterising what is, precisely, the 2-subcategory spanned by oplax natural transformations via this horizontal and vertical composition.

Mike Shulman (Jan 17 2025 at 16:18):

Neat!

Mike Shulman (Jan 17 2025 at 16:19):

(I don't know what "non-actual strictness" would be... aren't things either strict or not?)

Amar Hadzihasanovic (Jan 17 2025 at 16:43):

Hmm, now I'm thinking that perhaps my description is missing some 2-cells... will have to give it some more thought. It seems to me that if this description were complete, it would imply that this representation is conservative also for "naive whiskerings", which ought to be incompatible with interchange.