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Stream: theory: category theory

Topic: Terminology: quotient of a 2-category

Chad Nester (Oct 25 2024 at 08:12):

Hi everyone. I have a terminological question.

Suppose I have a strict 2-category $\mathbb{C}$. Then there is a congruence $E$ on $\mathbb{C}$ in which $(f,g) \in E$ (for $f,g : X \to Y$) in case there exists a 2-cell $\alpha : f \to g$. What is the resulting quotient category $\mathbb{C}/E$ called? Is there a commonly agreed upon name for this?

Martti Karvonen (Oct 25 2024 at 08:30):

I don't know of a standard terminology, but if you don't require $\alpha$ to be invertible, you need to force this to be symmetric and take the transitive closure, so concretely $E$ is induced by zig-zags of 2-cells rather than just 2-cells. In other words you'd be taking the path components of the hom-categories.

Chad Nester (Oct 25 2024 at 08:37):

Ah yes, I did mean the transitive symmetric version. Thanks.

fosco (Oct 25 2024 at 10:13):

It seems you're considering the set of connected components $\pi_0(\mathbb C(X,Y))$ of each hom-category $\mathbb C(X,Y)$; this is part of a 2-functor $\pi_{0,*}$ from $2\textbf{-Cat}$ (Cat-enriched cats) to $\textbf{Cat}$ (Set-enriched cats)

Amar Hadzihasanovic (Oct 25 2024 at 13:53):

This is usually called the 1-truncation of a 2-category.

Amar Hadzihasanovic (Oct 25 2024 at 13:53):

It's the left adjoint to the inclusion of the category of 1-categories into the category of 2-categories.

Amar Hadzihasanovic (Oct 25 2024 at 13:55):

This has also been called the "intelligent truncation", where the "stupid truncation" is the right adjoint (I swear), but I do not think this is a good piece of terminology.

Amar Hadzihasanovic (Oct 25 2024 at 14:01):

You may want to say something like: “the 1-truncation (also known as “intelligent” 1-truncation)” when you first use it.

Todd Trimble (Oct 25 2024 at 14:35):

I don't think it's any more "intelligent" than the one that uses the change of base of enrichment $\pi_0 \mathsf{core}: \mathrm{Cat} \to \mathrm{Set}$ (connected components of underlying groupoid). Here the result is sometimes called the "homotopy category".

Chad Nester (Oct 25 2024 at 14:40):

Amar Hadzihasanovic said:

This is usually called the 1-truncation of a 2-category.

That seems pretty reasonable!

Mike Shulman (Oct 25 2024 at 21:10):

I'm not sure that "1-truncation" is well-established as a name for this when the 2-category has non-invertible 2-cells; I would be more likely to think that referred to the version that Todd described.

Chaitanya Leena Subramaniam (Oct 25 2024 at 22:58):

I'd be inclined to use the name "homotopy category" for the construction Amar described.

Kevin Carlson (Oct 25 2024 at 23:25):

That's surely got the same issue with quotienting by all paths versus only by isomorphisms.

Chaitanya Leena Subramaniam (Oct 25 2024 at 23:44):

My reasoning is that hom-sets you end up with are the connected components of the fundamental groupoids of the hom-categories, rather than their core groupoids.

Amar Hadzihasanovic (Oct 26 2024 at 06:59):

Mike Shulman said:

I'm not sure that "1-truncation" is well-established as a name for this when the 2-category has non-invertible 2-cells; I would be more likely to think that referred to the version that Todd described.

"Intelligent n-truncation" for what I have described is well-established in the context of strict n-categories. It dates back at least to Kapranov & Voevodsky and is used e.g. by Ara and Maltsiniotis, Gagna, Harpaz and Lanari, Guetta, Ozornova and Rovelli.

Amar Hadzihasanovic (Oct 26 2024 at 07:02):

And frankly I would not come here and tell others what is or isn't well-established terminology in their own field of which I am not an expert.

Amar Hadzihasanovic (Oct 26 2024 at 08:04):

(To be clear, I have no objection to criticising the terminology, and I think "intelligent"/"stupid" is, well, a stupid pair of adjectives to use in a mathematical context; but whether something is or is not "well-established" is not a value judgement.)

Amar Hadzihasanovic (Oct 26 2024 at 09:44):

Follow-up: it has been pointed out to me that I approached the question as a special case of a question about (strict) n-categories, but that it could also be approached as a question in particular about (not necessarily strict) 2-dimensional categories (of which Mike certainly is an expert, and whose community has its own specialised terminology); that's a fair point.

Mike Shulman (Oct 26 2024 at 15:09):

Thanks. (And thanks for the references in the field of strict $n$-categories!) And in addition, if generalizing a question about 2-categories to $n$-categories, I think the "default" generalization should be to non-strict $n$-categories, as those are generally more "correct" and useful.

Chad Nester (Oct 28 2024 at 08:18):

Thanks everyone. I think i'm just going to try and avoid calling it anything. If unavoidable "truncation" evokes appropriate imagery, so I think I'll go with that.