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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2011

    I have split off effective epimorphism in an (infinity,1)-category from effective epimorphism and polished and expanded slightly.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2013

    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.

    • CommentRowNumber3.
    • CommentAuthoradeelkh
    • CommentTimeApr 25th 2013

    The question seems to be on math.stackexchange still.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2013

    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.

    • CommentRowNumber5.
    • CommentAuthoradeelkh
    • CommentTimeApr 25th 2013

    I wish I wrote that reply, but I'm afraid that was Akhil (\ne Adeel!) ;)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2013

    Oh, sorry. I should be paying more attention, that’s embarrassing. Sorry for the confusion.

    But anyway, thanks for the pointer!

    • CommentRowNumber7.
    • CommentAuthoradeelkh
    • CommentTimeNov 20th 2014
    • (edited Nov 20th 2014)

    I added a remark, taken from an answer of David Carchedi on MO, about effective epimorphisms in sheaf toposes.

    • CommentRowNumber8.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 24th 2015

    I recently had to be told that effective epimorphisms in the (,1)(\infty, 1)-category of spaces need not be epimorphisms. Perhaps a red herring principle warning is in order.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 24th 2015

    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.

    • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 25th 2015

    I added some remarks to that effect.

    I guess your example in 1-groupoids is S 0Δ 0S^0 \to \Delta^0?

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 25th 2015
    • (edited Feb 25th 2015)

    Can we have some concrete statement other than ’it’s not true’? Or rather, what definition of ’epimorphism’ are you using?

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2015

    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.

    • CommentRowNumber13.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 25th 2015

    It does seem rarely used, though there are some nifty examples:

    • A map ABA\to B of commutative ring spectra is an epimorphism iff BB is smashing over AA, i.e., if B ABBB\wedge_A B\approx B.

    • A map XYX\to Y between connected spaces is an epimorphism iff YY is formed via a Quillen-plus construction from a perfect normal subgroup of π 1X\pi_1 X.

    I sometimes try to find useful criteria for “epimorphism” in other settings. It’s usually pretty hard.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2015
    • CommentRowNumber15.
    • CommentAuthoradeelkh
    • CommentTimeFeb 25th 2015
    • (edited Feb 25th 2015)

    A map ABA\to B of commutative ring spectra is an epimorphism iff BB is smashing over AA, i.e., if B ABBB\wedge_A B\approx B.

    Isn’t it true that in any category admitting fibred coproducts, f:xyf : x \to y is an epimorphism iff the codiagonal morphism x yxxx \coprod_y x \to x is an isomorphism (and dually for monomorphisms)? Is the same true for (,1)(\infty,1)-categories? (The above would then just be a special case of this.)

    • CommentRowNumber16.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 25th 2015

    That’s right. That is why suspension shows up in the (counter)example: for a space XX, XΔ 0X \to \Delta^0 is an epimorphism in the (,1)(\infty, 1)-category of spaces if and only if the (unreduced) suspension Δ 0⨿ XΔ 0\Delta^0 \amalg_X \Delta^0 is contractible. More generally, it seems to me that XYX \to Y is an epimorphism in the (,1)(\infty, 1)-category of spaces if and only if its homotopy fibres are spaces with contractible suspension.

    • CommentRowNumber17.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 26th 2015

    Of course. The interesting feature of the commutative ring case is that the pushout is computed as a smash/tensor product, so whether ABA\to B is an epimorphism of rings can be detected without appealing to the ring structures, and merely depends on BB as an AA-module.

    It goes the other way, of course: if AA is a commutative ring and ABA\to B is a map of AA-modules such that BB AAB ABB\approx B\wedge_A A\to B\wedge_A B is an equivalence, then BB is uniquely a commutative AA-algebra, and ABA\to B an epimorphism.

    Another amusing fact: you can define a “Quillen plus-construction” of a commutative ring spectrum AA, 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 AA-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.

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 26th 2015

    Is there a general \infty-categorical notion of “plus construction”?

    • CommentRowNumber19.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 27th 2015
    • (edited Feb 27th 2015)

    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} iI\{r_i\}_{i\in I}, then (2) kill some more stuff by introducing some “higher relations” {s i} iI\{s_i\}_{i\in I}, where “relations” and “higher relations” are indexed by the same II. Furthermore, each “higher relation” should correspond to some kind of “redundancy” inherent in killing the “relations”.

    For instance, given a space XX and a subgroup Pπ 1XP\subseteq \pi_1X generated by commutators c i=[x i,y i]c_i=[x_i,y_i] of loops x i,y iΩXx_i,y_i\in \Omega X representing elements of P P, step (1) is: attach a 2-cell d id_i along each c ic_i, obtaining a space YY, while step (2) is: attach a 3-cell e ie_i along the 2-sphere in YY whose southern hemisphere is d id_i, and whose northern hemisphere is [H i,K i][H_i,K_i], built from choices of null-homotopies H i,K iH_i,K_i of the loops x i,y ix_i,y_i (which exist in YY exactly because PP is generated by commutators).

    The resulting map XZX\to Z is an epimorphism.

    Proof: The construction depends on the collection of choices α={(x i,y i,H i,K i)}\alpha=\{(x_i,y_i,H_i,K_i)\} (assume fixed indexing set II), which themselves form a space AA, and the plus-construction depends “continuously” on αA\alpha\in A. If x ix_i and y iy_i are themselves null-homotopic, then you can connect α\alpha to α 0={(*,*,*,*)}\alpha_0=\{(*,*,*,*)\} (all constant maps) by a path in AA, and it’s clear that the plus-construction built from α 0\alpha_0 admits a deformation retraction, from which we conclude that ff is an equivalence when [x i],[y i][x_i],[y_i] are trivial in π 1X\pi_1X.

    Next note that if g:XXg\colon X\to X' is a map, then the pushout along gg of a plus construction f:XZf\colon X\to Z built from an α\alpha is a map g:XZg'\colon X'\to Z' which is itself a plus-construction built from g(α)g(\alpha). It is clear that the plus construction map ff kills the elements [x i],[y i]π 1X[x_i],[y_i]\in \pi_1X, so the pushout of ff along itself must be an equivalence.

    I don’t know too many other examples of this type of thing.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 27th 2015

    Very cool! I bet this has a nice formalization using HITs. However I don’t quite follow this bit:

    If x ix_i and y iy_i are themselves null-homotopic, then you can connect α\alpha to α 0={(*,*,*,*)}\alpha_0=\{(*,*,*,*)\} (all constant maps) by a path in AA.

    I see that you can connect α\alpha to something of the form {(*,*,H i,K i)}\{(\ast,\ast,H_i',K_i')\}, but the constant loop can be nullhomotopic in a nontrivial way, so how do you know that H iH_i' and K iK_i' are also trivial?

    • CommentRowNumber21.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 28th 2015

    Whoops. I don’t. The real argument is: if x i=*=y ix_i=*=y_i, the map on the “northern hemisphere” factors through a map [H i,K i]:S 2Y[H_i,K_i]\colon S^2\to Y, which is null homotopic because π 2\pi_2 is abelian.

  1. added reference to HTT 6.2.3.10

    Shane

    diff, v25, current

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