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nLab > Latest Changes: effective epimorphism in an (infinity,1)-category

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJan 12th 2011
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Author: Urs
Format: MarkdownItexI have split off [[effective epimorphism in an (infinity,1)-category]] from [[effective epimorphism]] and polished and expanded slightly.

I have split off effective epimorphism in an (infinity,1)-category from effective epimorphism and polished and expanded slightly.
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeApr 25th 2013
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Author: Urs
Format: MarkdownItexI 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.

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
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Author: adeelkh
Format: MarkdownItexThe question seems to be on [math.stackexchange](http://math.stackexchange.com/questions/370941/what-is-the-homotopy-colimit-of-the-cech-nerve-as-a-bi-simplical-set) still.

The question seems to be on math.stackexchange still.
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeApr 25th 2013
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Author: Urs
Format: MarkdownItexAh, 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.

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
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Author: adeelkh
Format: MarkdownI wish I wrote that reply, but I'm afraid that was Akhil (\ne Adeel!) ;)

I wish I wrote that reply, but I'm afraid that was Akhil (\ne Adeel!) ;)
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeApr 25th 2013
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Author: Urs
Format: MarkdownItexOh, sorry. I should be paying more attention, that's embarrassing. Sorry for the confusion. But anyway, thanks for the pointer!

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)
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Author: adeelkh
Format: MarkdownItexI added a [remark](http://ncatlab.org/nlab/show/effective+epimorphism+in+an+%28infinity%2C1%29-category#in_a_sheaf_topos), taken from an [answer of David Carchedi](http://mathoverflow.net/a/177325/2503) on MO, about effective epimorphisms in sheaf toposes.

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
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Author: Zhen Lin
Format: MarkdownItexI 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.

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.
- CommentRowNumber9.
- CommentAuthorMike Shulman
- CommentTimeFeb 24th 2015
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Author: Mike Shulman
Format: MarkdownItexYes, 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.

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
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Author: Zhen Lin
Format: MarkdownItexI added some remarks to that effect. I guess your example in 1-groupoids is $S^0 \to \Delta^0$?

I added some remarks to that effect.

I guess your example in 1-groupoids is $S^0 \to \Delta^0$ ?
- CommentRowNumber11.
- CommentAuthorDavidRoberts
- CommentTimeFeb 25th 2015
- (edited Feb 25th 2015)
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Author: DavidRoberts
Format: MarkdownItexCan we have some concrete statement other than 'it's not true'? Or rather, what definition of 'epimorphism' are you using?

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
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Author: Urs
Format: MarkdownItexI have moved Zhen Lin's addition to a [numbered example](http://ncatlab.org/nlab/show/effective+epimorphism+in+an+%28infinity%2C1%29-category#AnEffectiveEpiNotBeingEpiInInfinityGroupoids) 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.

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
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Author: Charles Rezk
Format: MarkdownItexIt 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.
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.
- CommentRowNumber14.
- CommentAuthorUrs
- CommentTimeFeb 25th 2015
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Author: Urs
Format: MarkdownItex[thanks](http://ncatlab.org/nlab/show/epimorphism+in+an+%28infinity%2C1%29-category#Examples)

thanks
- CommentRowNumber15.
- CommentAuthoradeelkh
- CommentTimeFeb 25th 2015
- (edited Feb 25th 2015)
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Author: adeelkh
Format: MarkdownItex> 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.)

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.)
- CommentRowNumber16.
- CommentAuthorZhen Lin
- CommentTimeFeb 25th 2015
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Author: Zhen Lin
Format: MarkdownItexThat'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.

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.
- CommentRowNumber17.
- CommentAuthorCharles Rezk
- CommentTimeFeb 26th 2015
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Author: Charles Rezk
Format: MarkdownItexOf 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.

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.
- CommentRowNumber18.
- CommentAuthorMike Shulman
- CommentTimeFeb 26th 2015
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Author: Mike Shulman
Format: MarkdownItexIs there a general $\infty$-categorical notion of "plus construction"?

Is there a general $\infty$ -categorical notion of “plus construction”?
- CommentRowNumber19.
- CommentAuthorCharles Rezk
- CommentTimeFeb 27th 2015
- (edited Feb 27th 2015)
- PermaLink
Author: Charles Rezk
Format: MarkdownItexI 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.

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.
- CommentRowNumber20.
- CommentAuthorMike Shulman
- CommentTimeFeb 27th 2015
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Author: Mike Shulman
Format: MarkdownItexVery 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?

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?
- CommentRowNumber21.
- CommentAuthorCharles Rezk
- CommentTimeFeb 28th 2015
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Author: Charles Rezk
Format: MarkdownItexWhoops. 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.

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.
- CommentRowNumber22.
- CommentAuthornLab edit announcer
- CommentTimeApr 15th 2021
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Author: nLab edit announcer
Format: MarkdownItexadded reference to HTT 6.2.3.10 Shane <a href="https://ncatlab.org/nlab/revision/diff/effective+epimorphism+in+an+%28infinity%2C1%29-category/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/effective+epimorphism+in+an+%28infinity%2C1%29-category/25">v25</a>, <a href="https://ncatlab.org/nlab/show/effective+epimorphism+in+an+%28infinity%2C1%29-category">current</a>

added reference to HTT 6.2.3.10

Shane

diff, v25, current
- CommentRowNumber23.
- CommentAuthorMike Shulman
- CommentTimeJul 10th 2021
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Author: Mike Shulman
Format: MarkdownItexI wonder if comments 13-21 in this thread could be moved to a discussion thread for [[epimorphism in an (infinity,1)-category]].

I wonder if comments 13-21 in this thread could be moved to a discussion thread for epimorphism in an (infinity,1)-category.
- CommentRowNumber24.
- CommentAuthorUrs
- CommentTimeJul 10th 2021
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Author: Urs
Format: MarkdownItexYou can put a [link](https://nforum.ncatlab.org/discussion/13162/epimorphism-in-an-infinity1category/?Focus=93766#Comment_93766).

You can put a link.
- CommentRowNumber25.
- CommentAuthorDmitri Pavlov
- CommentTimeDec 14th 2023
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Author: Dmitri Pavlov
Format: MarkdownItexTerminology and redirects: quotient morphism. (Used in Lurie's Kerodon.) <a href="https://ncatlab.org/nlab/revision/diff/effective+epimorphism+in+an+%28infinity%2C1%29-category/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/effective+epimorphism+in+an+%28infinity%2C1%29-category/29">v29</a>, <a href="https://ncatlab.org/nlab/show/effective+epimorphism+in+an+%28infinity%2C1%29-category">current</a>

Terminology and redirects: quotient morphism. (Used in Lurie’s Kerodon.)

diff, v29, current

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nLab > Latest Changes: effective epimorphism in an (infinity,1)-category