The n-Category Café

Ah, thanks, Mike. This is exactly why I tend to leave some time between posting about a preprint here and submitting it.

Unfortunately not. Maybe this is why I found the principle so striking when I first heard it from Carlos Simpson.

The best I can do is sketch the proof, which is essentially the same for all four structures in my paper (categories, monoidal categories, bicategories and double categories).

Start with an equivalence $F: A \to B$ between categorical structures of whatever type. Let $C$ be the structure whose objects consist of an object $a \in A$, an object $b \in B$, and an equivalence $F(a) \to b$ (and here we’re invoking the notion of equivalence in a structure). Then take the first and second projections

$$A \leftarrow C \rightarrow B.$$

These should be strict surjective equivalences.

To give a little more detail, it’s important to interpret “an equivalence $F(a) \to b$” correctly. For instance, in the bicategory case, you don’t just want a 1-cell $F(a) \to b$ that is an equivalence in the bicategory $B$. You want the full data of an adjoint equivalence between $F(a)$ and $b$, all specified, not just asserted to exist.

A more abstract way to look at the construction is that the span

$$A \stackrel{P}{\leftarrow} C \stackrel{Q}{\rightarrow} B$$

comes equipped with an isomorphism $F \circ P \to Q$ and is universal as such. I guess this is what’s called an “iso-comma object”, more or less.

I think it must be the case that if we start with equivalences $F$ and $G$ with common codomain, then the projections of the resulting iso-comma object are strict surjective equivalences. (In our case, $G = 1_B$.) But I don’t see a really direct explanation of why.

I personally think of Tom’s examples as instances of the fact that, in a model category, two fibrant objects are weakly equivalent if and only if they are connected by a span of trivial fibrations.

John’s provocative question

Do you have any argument for this principle, other than it seems to work in examples?

should, I think, have a good answer of an elementary kind (by which I partly mean “not involving model categories”).

I’ve been reflecting on this and don’t yet have an answer that I find completely satisfactory, but I think I see it more clearly than I did before.

Recall what the “principle” was:

For many types of categorical structure, the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ and $B$ when there exists a strict surjective equivalence $A \to B$.

I described how this works: given an equivalence $F: A \to B$ of categories with whatever kind of structure, one constructs the category-with-structure $C$ whose objects consist of an object $a \in A$, an object $b \in B$, and an equivalence between $F(a)$ and $b$. Higher morphisms are defined in a similar spirit. One can sort of see, in a hand-wavy way, that whatever kind of structure we’re considering, the projections $A \leftarrow C \rightarrow B$ ought to be surjective in every dimension and preserve the structure strictly.

But one could still ask whether this is just a trick, and I’d like to be able to argue that it isn’t.

Although I don’t see the whole story clearly, I believe the fundamental point here is that

Categorical structures are essentially algebraic.

To put it another way, many definitions in category theory are of the form “you’ve got some objects, and you’ve got some maps, and some equations hold”. Crucially, this is the case even if we’re talking about pseudo or lax things.

For example, a strong monoidal functor $A \to B$ between monoidal categories consists of an object $F(a)$ for each $a \in A$, a map $F(f): F(a) \to F(b)$ for each map $f: a \to b$ in $A$, an isomorphism $\phi_{a, b}: F(a) \otimes F(b) \to F(a \otimes b)$ for each pair $(a, b)$ of objects of $A$, and an isomorphism $\phi_\cdot: I_B \to F(I_A)$, such that some equations hold.

From this description, it follows that for each monoidal category $A$, we can build another monoidal category $A'$ such that the strong monoidal functors from $A$ to an arbitrary monoidal category $B$ correspond to the strict monoidal functors from $A'$ to $B$. Indeed, $A'$ is generated as a monoidal category as follows:

• For each object $a$ of $A$, an object of $A'$, which you could call just $a$ or $\langle a \rangle$ or whatever, but which I’d like to suggestively call $F(a)$, where now $F$ is simply a formal symbol.

• For each map $f: a \to b$ in $A$, a map $F(a) \to F(b)$ in $A'$ that I’ll call $F(f)$, where again the “$F$” of $F(f)$ is just a formal symbol.

• For each pair $(a, b)$ of objects of $A$, an isomorphism $\phi_{a, b} : F(a) \otimes F(b) \to F(a \otimes b)$ in $A'$, and also an isomorphism $\phi_\cdot: I_{A'} \to F(I_A)$ in $A'$.

And then, having generated a monoidal category freely from this data, you obtain $A'$ by quotienting out by all the obvious equations (e.g. $F(g \circ f) = F(g) \circ F(f)$ and equations looking like the coherence axioms for a strong monoidal functor).

The property I claimed then follows pretty much by definition: for any monoidal category $B$, a strict monoidal functor $A' \to B$ amounts to a strong monoidal functor $A \to B$.

There’s a canonical projection $P: A' \to A$: “remove the $F$s”. It’s a strict monoidal functor, and in fact a surjective equivalence. If we start with a strong monoidal functor $F: A \to B$, then we get the corresponding strict monoidal functor $F': A' \to B$, hence a span

$$A \stackrel{P}{\leftarrow} A' \stackrel{F'}{\rightarrow} B$$

of strict monoidal categories in which the left leg is a strict surjective equivalence and $F \circ P \cong F'$.

What I want to emphasize is that this is all a consequence of the very elementary point I mentioned: the definitions of monoidal category and strong monoidal functor are essentially algebraic.

The construction I just described, of $A'$ from $A$, is different from the construction in my paper of the monoidal (or whatever) category $C$ of equivalences. But it’s along the same lines, and I think it should be possible to understand the construction in my paper in a similar way.

by which I partly mean “not involving model categories”

I think that the story you wish to tell is essentially the same as the one underlying the model categorical point of view, if one unravels everything.

I think a good place to start is the question: what is the correct definition of (the analogue of) an ‘iso-fibration’ in this kind of situation? Roughly speaking the idea is clear: one wishes to lift ‘internal equivalences’ in the appropriate sense. For categories, one wishes to lift isomorphisms between objects. For 2-categories, one wishes to lift equivalences between objects. Etc. Suppose that we impose that our (analogues of) ‘iso-fibrations’ are strict ‘functors’ of the appropriate kind in each case, as you do.

Every external equivalence is not quite (the analogue of) an ‘iso-fibration’ in this sense: one lacks the necessary ‘strictness’ on objects. But given any ‘functor’, there is a universal way to construct a span of ‘iso-fibrations’ from it: the mapping co-cylinder construction (using the free-standing equivalence of the appropriate kind). This is basically the same kind of construction as you sketched. And applied to an equivalence, it has the effect of ‘strictifying’ in the manner you described, so that one obtains actual surjectivity on objects, not only essential surjectivity.

This kind of mapping co-cylinder construction is basically the same idea as is used in the model category setting.

If one does not impose that uses a strict 2-functor (or whatever), I’d imagine one can carry out roughly the same story, where the (strict) mapping co-cylinder is replaced by some 2-pullback (similar in the other cases), but I have not thought it through carefully.

Thanks, Yemon, that’s interesting and I think in the same direction.

The most basic example of the construction used repeatedly in my paper is as follows: given a bijection $f: A \to B$ between sets, one constructs the graph

$$C = \{(a, b) \in A \times B : f(a) = b\}$$

and the projections $A \leftarrow C \rightarrow B$, which are both in fact bijections. One can deduce from this the completely obvious and silly consequence that two sets are in bijection if and only if there is a span of bijections between them. Obvious and silly though this is, it’s the degenerate analogue of each of the not-so-obvious theorems in my paper.

I mention this because I think it’s also along the lines of your comment, and it has the universal algebra flavour that you mention. In fact, one could replace “set” by “group”, “ring”, “Boolean algebra”, etc., and “bijection” by “isomorphism”, in the previous paragraph and it would still be true.

Maru certainly spoke extensively about your work, including the gregarious model structure that you introduced in your 2020 talks. I may have done a poor job of conveying the relationship between your work then and her new work with Moser and Verdugo now (still to be published). To be honest, I haven’t really got into the model-categorical aspects. My focus has been purely on the question of equivalence.

It also occurs to me that there’s something a bit risky about putting talk slides on the web. When we prepare slides, we primarily envisage them as a complement to what we’re saying out loud. In other words, they’re only one of the two main components of a talk (visual and spoken). But when we upload slides afterwards, they’re missing their other half, and may sometimes give an inaccurate impression of the talk as a whole.

The Grandis–Paré notion of equivalence that you cite treats pseudo double categories as $1$-dimensional objects akin to categories, whereas the notion of gregarious double equivalence (which can be defined for pseudo double categories as well as for strict double categories) treats them as $2$-dimensional objects akin to bicategories. Recall that we can regard bicategories as pseudo double categories with only identity “tight” morphisms; the two induced notions of equivalence of bicategories are the bijective-on-objects biequivalences and the biequivalences respectively.

There are a few ways to make this precise. Let me mention just one before I go to sleep. Define the “dimension” of a relative category $(\mathcal{C},\mathcal{W})$ to be the smallest $-2 \leq n \leq \infty$ such that the $(\infty,1)$-categorical localisation $\mathcal{C}[\mathcal{W}^{-1}]$ is an $(n,1)$-category. The localisation of the category of strict/pseudo double categories and strict/pseudo double functors at the Grandis–Paré equivalences has dimension $2$, whereas the localisation of the category of strict/pseudo double categories and strict/pseudo double functors or double pseudofunctors (it doesn’t matter which choice of morphism you make, the localisations will be equivalent) at the gregarious double equivalences has dimension $3$.

So it all depends on how you want to think about double categories. I personally prefer the notion of gregarious equivalence even for pseudo double categories. For instance, in the pseudo double category of categories and profunctors, I want two objects, i.e. categories, to be seen as the same if they are equivalent, not isomorphic, as categories.

As to your question, that is indeed true. (Although I’m not sure what you mean by “compatible with source/target”; I’ve chosen to ignore it in my answer.) It’s another instance of the model category fact that I mentioned above, this time applied to the category of strict/pseudo double categories and strict double functors (this time we need strict morphisms in order to get a complete and cocomplete category on which to put a model structure, but the model structure also allows us to capture pseudo double functors via cofibrant replacement) with the “projective” model structure induced from the “levelwise” model structure on the category of graphs internal to $\mathbf{Cat}$, which is an instance of the model structures for categories of algebras of a $2$-monad from one of Steve’s papers.

Just picking up on one bit of Bryce’s comment:

one might define a pseudo functor $F:\mathbb{C}\to\mathbb{D}$ to be a surjective equivalence if…

The notion of surjective equivalence should be very easy (in fact, Carlos Simpson called them “easy equivalences” in another context). More exactly, it should only refer to the underlying graphs. So the definition should be exactly the same for pseudo double categories as for strict double categories: surjective on objects, horizontally and vertically full, and full and faithful on $2$-cells.

Is that the same as the definition you gave in your comment?