6
$\begingroup$

Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions in the world of higher category theory (conceptual answers will be more useful to me than model specific answers).

My question are conceptually very simple:

  1. What are ($n$-?)monomorphisms and ($n$-?)epimorphisms in an $\infty$-catgory? What are the fundamental differences with $1$-categories?
  2. What are ($n$-?)coimages and ($n$-?)-images in an $\infty$-category? (it seems that on n-lab the definitions for both of them [1] [2] are exactly the same which is very confusing) What are the fundamental differences with $1$-categories? Which of these coincides with the Postnikov decomposition in $\infty$-groupoids?
  3. What is an $n$-ary factorization system for an $\infty$-category? Is this a direct analog of epi-mono factorization? If so In what sense?

I could elaborate but I tried and concluded that it will probably only serves to confuse me and the readers. I'm still confused by many of these notions.

Improve this question
$\endgroup$
3
  • $\begingroup$ Monomorphisms and epimorphisms tend to be quite boring: a monomorphism in spaces is just an inclusion of connected components and a map $f:x\to y$ in an ∞-category C is a monomorphism (resp. epimorphism) iff for every $z\in C$ the map $f_*:\mathrm{Map}_C(z,x)\to \mathrm{Map}_C(z,y)$ (resp. $f^*:\mathrm{Map}_C(y,z)\to \mathrm{Map}_C(x,z)$) is a monomorphism in spaces. $\endgroup$
    – Denis Nardin
    Commented Feb 2, 2017 at 15:04
  • $\begingroup$ @DenisNardin I claim that epimorphisms are not boring! For instance, epimorphisms of $\infty$-groupoids. $\endgroup$
    – Charles Rezk
    Commented Feb 2, 2017 at 16:48
  • $\begingroup$ @CharlesRezk Actually I have a very poor intuition for epimorphisms of ∞-groupoids. I guess what I meant with "boring" is that there are very few of them, so the notion of subobject for example is not that useful. $\endgroup$
    – Denis Nardin
    Commented Feb 2, 2017 at 16:52

2 Answers 2

Reset to default
12
$\begingroup$

An $n$-monomorphism is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers $n$-truncated, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all vanish in dimensions $>n$.

An $n$-epimorphism is an $n$-monomorphism in the opposite $\infty$-category.

When $n=-1$ you recover Denis's definition of monomorphism and epimorphism: note that a space is $(-1)$-truncated if and only if it is empty or contractible.

The phenomenon of "$n$-epi/mono factorization" in an $\infty$-category does not actually involve factorization as ($n$-mono)($n$-epi), but rather as ($n$-mono)($n$-connected map). The class of $n$-connected maps is usually defined as a kind of left complement to $n$-monomorphisms, so the existence of such factorizations is basically the assertion that your $\infty$-category admits a factorization system where the right-class is the class of $n$-monos. (These exist in all "reasonable" examples, e.g., presentable $\infty$-categories.)

This same issue shows up in 1-category theory: epi/mono factorization tends to be rare, and more commonly one considers factorizations of the form: (regular epi)(mono).

The factorization ($n$-mono)($n$-connected) can be thought of as factorization through an "$n$-image". It is clearly not a self-dual notion, hence there is a dual but distinct notion of factorization through an "$n$-coimage".

Improve this answer
$\endgroup$
5
  • $\begingroup$ So am i right to say that the notion of "image" only makes sense in a category with factorization whose right class are n-monos and "coimage" makes sense only for categories with factorization whose left class are n-epis? Both of these can be thought of as levels in a Postnikov tower correct? They both agree on $\infty$-groupoids right? If they both exist do they coincide in good circumstances? for instance (possibly hypercomplete) $\infty$-toposes? $\endgroup$
    – Saal Hardali
    Commented Feb 2, 2017 at 17:02
  • $\begingroup$ @SaalHardali I don't know about "only", but otherwise I agree with the first sentence. Image and coimage are not going to coincide often if ever: even in $\infty$-groupoids they are very different. I don't know much about existence of coimages in general. $\endgroup$
    – Charles Rezk
    Commented Feb 2, 2017 at 17:22
  • $\begingroup$ Thabks! So to sum up the analog of (mono)(regular epi) is n-mono n-connected and it can be rightfully called the generalization of postnikov towers. Coimages are weird. Interestingly what you say seems to contradict certain points in the nLab site. I'm far from an expert so I might be wrong about this. If I not i'm sure that editing what's there will help the rest of the community. $\endgroup$
    – Saal Hardali
    Commented Feb 2, 2017 at 17:29
  • 1
    $\begingroup$ @SaalHardali I just looked at those n-lab pages; it's very confusing. What they call "coimage" bears a relation to what I called image: their "coimage" computes the (mono)(-1-conn) factorization that I descibed in many cases. Presumably their "image" bears a relation to the the "dual" factorization construction, though I don't know if it actually computes it in any example. $\endgroup$
    – Charles Rezk
    Commented Feb 2, 2017 at 17:59
  • 1
    $\begingroup$ There's an unfortunate tradition in some parts of category theory whereby the "image" refers to the (regular mono, epi) factorization and the "coimage" to the (mono, regular epi) factorization, and I think the nLab is partly bowing to that. As you say, when they fail to coincide, often it is the latter that one wants to call the "image". $\endgroup$
    – Mike Shulman
    Commented Feb 2, 2017 at 23:14
6
$\begingroup$

I can answer to 4: https://arxiv.org/abs/1501.04658 here the definition is given for a generic poset $J$, provided it has a monotone $\mathbb{Z}$-action.

For 1,2,3 and the preamble, I think that the theory of factorization systems is pretty much unchanged from 1-category theory: see here; the same theorems hold, provided you translate them into quasicategorical terms (this is phrased in a specific model, and yet it is maybe possible to write it up model-independently -doing so in a few specific examples is part of my current plans :-) ).

What changes, when you upgrade to $(\infty,1)$-categories, is the fact that $(Epi, Mono)$ on $\bf Set$ is no longer the most natural example, replaced by $(n\text{-}Epi, n\text{-}Mono)$ on $\infty\text{-}\bf Gpd$.

Notice that all I'm saying is true for $(\infty\color{red}{,1})$-categories only, not $(\infty,n)$-categories, not $(\infty,\infty)$-categories.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .