The n-Category Café

I started thinking about this question during Maru Sarazola’s invited talk at Category Theory 2025 in Brno last month. She asked the question:

What is the right notion of equivalence between double categories?

and carefully went through the properties that the right notion of equivalence should have, some possible candidates, and different approaches one might take to deciding what “right” means.

The answer that Maru ultimately gave was that the right notion is “gregarious double equivalence”, proposed by Alexander Campbell in about 2020. (See these slides by Campbell.) And she gave a justification in terms of model categories, representing joint work between her, Lyne Moser and Paula Verdugo.

For the purposes of this post, it actually doesn’t matter what “gregarious double equivalence” means. What I want to talk about is the following principle, which popped into my head as Maru was speaking:

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 $A \to B$.

It occurred to me that this principle might give a rather different justification for why gregarious double equivalence is the right answer. And after some checking, I discovered that it does.

Let me explain.

A more concrete way to express the principle is that $A$ and $B$ are equivalent in the standard sense — whatever’s appropriate for the structures at hand — if and only if there exists a zigzag of strict surjective equivalences

$$A = A_0 \leftarrow A_1 \rightarrow \ \cdots \ \leftarrow A_n = B.$$

For any type of categorical structure I can think of, the pullback of a surjective equivalence is a surjective equivalence, so a simpler concrete condition is just that there exists a span of strict surjective equivalences

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

But hold on… what do I mean by “principle”?

What I mean is that for simple types of categorical structure, where “equivalence” and “strict surjective equivalence”, we have a theorem. Here are three examples.

• Categories. We certainly know what it means for two categories to be equivalent. A “surjective equivalence” is an equivalence that’s not just essentially surjective on objects, but literally surjective on objects.

  In this case, the theorem is that categories $A$ and $B$ are equivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of surjective equivalences between them.

  (The word “strict” does nothing in this case.)

• Monoidal categories. Again, we know what monoidal equivalence is, and it’s clear what a “strict surjective equivalence” is: a strict monoidal functor that’s a surjective equivalence of categories.

  The theorem is that monoidal categories $A$ and $B$ are monoidally equivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of strict surjective equivalences between them.

• Bicategories. The pattern is the same. The standard notion of equivalence for bicategories is biequivalence. A “strict surjective equivalence”, in this setting, is a strict $2$-functor that is literally surjective on objects and locally a surjective equivalence of categories. (Or put another way, surjective on $0$-cells, locally surjective on $1$-cells, and full and faithful on $2$-cells.)

  The theorem is that bicategories $A$ and $B$ are biequivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of strict surjective equivalences between them.

Probably all these theorems are known. I included them in my paper because I couldn’t find them anywhere in the literature, not even the first one. But if you know a reference, I’d be glad to hear it.

Since the principle holds for categories, monoidal categories and bicategories, it’s reasonable to suppose that it might hold for other types of structure. And if we’re investigating some type of structure where the full notion of equivalence isn’t clear, this principle might help guide us to it.

For example, here’s a theorem on double categories, the main result of my paper:

• Double categories. Again, it’s clear what “strict surjective equivalence” should mean: a strict double functor that’s surjective on $0$-cells, locally surjective on both horizontal and vertical $1$-cells, and full and faithful on $2$-cells.

  The theorem is that double categories $A$ and $B$ are gregariously double equivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of strict surjective equivalences between them.

Even without me telling you what “gregarious double equivalence” means, the four theorems I’ve stated suggest that it’s the right notion of equivalence for double categories, because it continues the pattern we’ve seen for simpler categorical structures.

So, I agree with the conclusion that Moser, Sarazola and Verdugo had already reached! But for different reasons.

Incidentally, this must be the fastest paper I’ve ever written: just under six weeks from sitting in Maru’s talk and hearing the mathematical term “gregarious” for the first time ever to putting the paper on the arXiv.

But the principle that all equivalences are generated by strict surjective equivalences was planted in my head in the late 1990s or early 2000s by Carlos Simpson. Back then, we were both working on higher category theory, and when he explained this principle, I found it very striking — so striking that I remembered it 20+ years later. There’s a bit more on that higher categorical context in the introduction to my paper.