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Stream: learning: questions

Topic: Epimorphism variants in infinity-category theory

Madeleine Birchfield (Dec 07 2024 at 17:39):

What is the relation between the notions of

• epimorphism
• extremal epimorphism
• strong epimorphism
• regular epimorphism
• strict epimorphism
• effective epimorphism
• split epimorphism
  in an $(\infty,1)$-category?

fosco (Dec 07 2024 at 18:17):

I'll start with a simple fact: you can define "$n$-epi" for every $n\ge 0$ (I think -1, in fact); the classes of $n$-epis are all incapsulated in one another, and taken together form a "$\omega$-ary factorization system"

fosco (Dec 07 2024 at 18:18):

factorization with respect to which is essentially the tower of killing homotopy groups (I would have to brush up this stuff which I knew back then, to remember if you kill homotopy above n or below, i.e. if you take a "Postnikov" or "Whitehead" tower)

fosco (Dec 07 2024 at 18:19):

This said, I suspect your question is about a fixed class of $P$ $n$-epi, where P is a property among {effective, regular, split, strict, extremal...}?

fosco (Dec 07 2024 at 18:20):

brush up, not off. That's a different thing

fosco (Dec 07 2024 at 18:22):

so if you fix $n$... For the nerve of a 1-category the implications you know for Cat must still hold.

It's an interesting exercise in seeing if one can make the 1-proof formal enough to adapt, mutatis mutandis, into an $\infty$-proof

fosco (Dec 07 2024 at 18:23):

Did you try doing that? (I myself would like to see the way one kickstarts this kind of reasoning, and I thought a lot about related questions!)

Madeleine Birchfield (Dec 07 2024 at 18:23):

The thing is that already we know that an epimorphism in an $(\infty,1)$-category is not the same as a 1-epimorphism in an $(\infty,1)$-category. The 1-epimorphisms in an $(\infty,1)$-category are the effective epimorphisms.

This was why I was asking about the other notions of epimorphisms.

fosco (Dec 07 2024 at 18:25):

Ah, I see. I didn't know or I don't remember this fact, can you tell me why?

Madeleine Birchfield (Dec 07 2024 at 18:27):

One example, in an $(\infty, 1)$-topos, the unique morphism from the coproduct $1 \coprod 1$ to the terminal object $1$ is an effective epimorphism but not an epimorphism. The former fact follows from the fact that the both $1 \coprod 1$ and $1$ is 0-truncated and the unique map is an effective epimorphism in the 1-topos of 0-truncated objects, and Buchholtz et al showed the latter fact using the internal language of the $(\infty, 1)$-topos in the introduction of this paper.

Jonas Frey (Dec 08 2024 at 14:39):

Madeleine Birchfield said:

The thing is that already we know that an epimorphism in an $(\infty,1)$-category is not the same as a 1-epimorphism in an $(\infty,1)$-category. The 1-epimorphisms in an $(\infty,1)$-category are the effective epimorphisms.

This was why I was asking about the other notions of epimorphisms.

I don't agree with this statement, but it's a matter of terminology.

It seems uncontroversial that the monos in an $\infty$-category are the maps $f:A\to B$ for which the map $\hom(X,A)\to\hom(X,B)$ between homotopy types "$0$-truncated", ie an "embedding".

For me the most reasonable definition of epi in an $\infty$-category is obtained by dualizing that: $f$ is an epi iff $\hom(f,X)$ is an embedding for all $X$.

It is true that this conflicts with Lurie's notion of effective epi in $\infty$-toposes : these are the maps that are the quotients of their Cech nerves, equivalently those that are right orthogonal to embeddings.

But it is not clear whether this notion is meaningful in arbitrary $\infty$-categories, and in my opinion the idea that epis are dual to monos is so fundamental that it should be carried over to $\infty$-categories, and Lurie's "effective epis" should be called "covers" , "surjections" (between h-types)), or "quotient maps" instead.

So in my opinion, $2\to 1$ is not an epi in eg in the $\infty$-category $\mathcal{S}$ of homotopy types, because $\delta_A:A\to A^2$ is not a mono in general, contrary to $\mathsf{Set}$.

Jonas Frey (Dec 08 2024 at 14:52):

Nvm, actually I do agree with your statement and the terminology in the Buchholtz/de Jong/Rijke paper !

It's just that their definition of $n$-epi is new to me. I expected it to be dual to $n$-truncated, but actually for them an $n$-epi is a map $f$ sth $\hom(f,X)$ is an embedding for all $k$-types! (they're using and "internalized" version using exponentials instead of homs since they're working in type theory)

I leave my above message since the issue with Lurie's notion of "effective epi" is still relevant to the discussion I think.

Jonas Frey (Dec 08 2024 at 14:58):

But @Madeleine Birchfield are you sure that the effective epis are the $1$-epis? To me it feels more like they should be the $0$-epis!

Tom de Jong (Dec 08 2024 at 15:03):

Indeed, effective epi = (-1)-connected = 0-epi (in the sense of our paper)