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I have split off effective epimorphism in an (infinity,1)-category from effective epimorphism and polished and expanded slightly.
I have made at effective epimorphism in an (infinity,1)-category the characterization in an infinity-topos by “induces epi on connected components” more explicit.
This was in reaction to an MO question “What is the homotopy colimit of the Cech nerve as a bi-simplical set? “. However, when I was done compiling my reply, the question had been deleted, it seems.
The question seems to be on math.stackexchange still.
Ah, thanks!
You give an excellent reply there. I have just added a comment now on where to find this in Lurie’s book with a pointer to the above entry.
I wish I wrote that reply, but I'm afraid that was Akhil (\ne Adeel!) ;)
Oh, sorry. I should be paying more attention, that’s embarrassing. Sorry for the confusion.
But anyway, thanks for the pointer!
I added a remark, taken from an answer of David Carchedi on MO, about effective epimorphisms in sheaf toposes.
I recently had to be told that effective epimorphisms in the $(\infty, 1)$-category of spaces need not be epimorphisms. Perhaps a red herring principle warning is in order.
Yes, that’s actually already true in the (2,1)-category of groupoids. (Although I can’t remember whether I’ve ever heard someone use “epimorphism” to mean “monomorphism in the opposite category” for 2-categories or (∞,1)-categories.) Feel free to add.
I added some remarks to that effect.
I guess your example in 1-groupoids is $S^0 \to \Delta^0$?
Can we have some concrete statement other than ’it’s not true’? Or rather, what definition of ’epimorphism’ are you using?
I have moved Zhen Lin’s addition to a numbered example and added a hyperlink to epimorphism in an (infinity,1)-category in order to clarify what is meant. Also added more cross-links there.
This concept of epimorphism in an $\infty$-category is rarely used, isn’t it.
It does seem rarely used, though there are some nifty examples:
A map $A\to B$ of commutative ring spectra is an epimorphism iff $B$ is smashing over $A$, i.e., if $B\wedge_A B\approx B$.
A map $X\to Y$ between connected spaces is an epimorphism iff $Y$ is formed via a Quillen-plus construction from a perfect normal subgroup of $\pi_1 X$.
I sometimes try to find useful criteria for “epimorphism” in other settings. It’s usually pretty hard.
A map $A\to B$ of commutative ring spectra is an epimorphism iff $B$ is smashing over $A$, i.e., if $B\wedge_A B\approx B$.
Isn’t it true that in any category admitting fibred coproducts, $f : x \to y$ is an epimorphism iff the codiagonal morphism $x \coprod_y x \to x$ is an isomorphism (and dually for monomorphisms)? Is the same true for $(\infty,1)$-categories? (The above would then just be a special case of this.)
That’s right. That is why suspension shows up in the (counter)example: for a space $X$, $X \to \Delta^0$ is an epimorphism in the $(\infty, 1)$-category of spaces if and only if the (unreduced) suspension $\Delta^0 \amalg_X \Delta^0$ is contractible. More generally, it seems to me that $X \to Y$ is an epimorphism in the $(\infty, 1)$-category of spaces if and only if its homotopy fibres are spaces with contractible suspension.
Of course. The interesting feature of the commutative ring case is that the pushout is computed as a smash/tensor product, so whether $A\to B$ is an epimorphism of rings can be detected without appealing to the ring structures, and merely depends on $B$ as an $A$-module.
It goes the other way, of course: if $A$ is a commutative ring and $A\to B$ is a map of $A$-modules such that $B\approx B\wedge_A A\to B\wedge_A B$ is an equivalence, then $B$ is uniquely a commutative $A$-algebra, and $A\to B$ an epimorphism.
Another amusing fact: you can define a “Quillen plus-construction” of a commutative ring spectrum $A$, in complete analogy with the construction for spaces. Instead of killing a perfect subgroup of the fundamental group, the input data is a “perfect ideal” in the homotopy category of compact $A$-modules. All homotopy epimorphisms of commutative rings can be obtained this way.
People have discussed plus-constructions in other contexts (dg Lie algebras, for instance). These should probably give other examples of epimorphisms.
Is there a general $\infty$-categorical notion of “plus construction”?
I don’t know. By “plus construction”, I (approximately) mean a two step process where you (1) kill some stuff by introducing some “relations” $\{r_i\}_{i\in I}$, then (2) kill some more stuff by introducing some “higher relations” $\{s_i\}_{i\in I}$, where “relations” and “higher relations” are indexed by the same $I$. Furthermore, each “higher relation” should correspond to some kind of “redundancy” inherent in killing the “relations”.
For instance, given a space $X$ and a subgroup $P\subseteq \pi_1X$ generated by commutators $c_i=[x_i,y_i]$ of loops $x_i,y_i\in \Omega X$ representing elements of $P$, step (1) is: attach a 2-cell $d_i$ along each $c_i$, obtaining a space $Y$, while step (2) is: attach a 3-cell $e_i$ along the 2-sphere in $Y$ whose southern hemisphere is $d_i$, and whose northern hemisphere is $[H_i,K_i]$, built from choices of null-homotopies $H_i,K_i$ of the loops $x_i,y_i$ (which exist in $Y$ exactly because $P$ is generated by commutators).
The resulting map $X\to Z$ is an epimorphism.
Proof: The construction depends on the collection of choices $\alpha=\{(x_i,y_i,H_i,K_i)\}$ (assume fixed indexing set $I$), which themselves form a space $A$, and the plus-construction depends “continuously” on $\alpha\in A$. If $x_i$ and $y_i$ are themselves null-homotopic, then you can connect $\alpha$ to $\alpha_0=\{(*,*,*,*)\}$ (all constant maps) by a path in $A$, and it’s clear that the plus-construction built from $\alpha_0$ admits a deformation retraction, from which we conclude that $f$ is an equivalence when $[x_i],[y_i]$ are trivial in $\pi_1X$.
Next note that if $g\colon X\to X'$ is a map, then the pushout along $g$ of a plus construction $f\colon X\to Z$ built from an $\alpha$ is a map $g'\colon X'\to Z'$ which is itself a plus-construction built from $g(\alpha)$. It is clear that the plus construction map $f$ kills the elements $[x_i],[y_i]\in \pi_1X$, so the pushout of $f$ along itself must be an equivalence.
I don’t know too many other examples of this type of thing.
Very cool! I bet this has a nice formalization using HITs. However I don’t quite follow this bit:
If $x_i$ and $y_i$ are themselves null-homotopic, then you can connect $\alpha$ to $\alpha_0=\{(*,*,*,*)\}$ (all constant maps) by a path in $A$.
I see that you can connect $\alpha$ to something of the form $\{(\ast,\ast,H_i',K_i')\}$, but the constant loop can be nullhomotopic in a nontrivial way, so how do you know that $H_i'$ and $K_i'$ are also trivial?
Whoops. I don’t. The real argument is: if $x_i=*=y_i$, the map on the “northern hemisphere” factors through a map $[H_i,K_i]\colon S^2\to Y$, which is null homotopic because $\pi_2$ is abelian.
I wonder if comments 13-21 in this thread could be moved to a discussion thread for epimorphism in an (infinity,1)-category.
You can put a link.
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