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Stream: learning: questions

Topic: quotient objects

Leopold Schlicht (Nov 15 2021 at 18:50):

Is there a categorical notion of "quotient of an object by something"? (In universal algebra there is the notion of quotients of algebraic structures by congruence relations, i.e., the something is a congruence relation, which generalizes normal subgroups, ideals in rings, and so on. So the categorical approach I am looking for should at least generalize this notion.)

Also I would be very interested in whether there is an established notion of "quotient of a simplicial set by something". If yes, is this a special case of the answer to my first question? Reid Barton mentions one special case of quotients of simplicial sets here. The something seems to be an edge there.

I picked up some category theorists mean by "quotient of $X$" any epimorphism with domain $X$. But this doesn't provide a way of quotienting by something.

I checked the page quotient object (nLab), but already the first sentence is pretty unclear:

The quotient object $Q$ of a congruence (an internal equivalence relation) $E$ on an object $X$ in a category $C$ is the coequalizer $Q$ of the induced pair of morphisms $E⇉X$.

What is an internal equivalence relation? (Does this notion reduce to congruence relations when considering categories of algebraic structures as studied in universal algebra?) What is the induced pair of morphisms?

What are the internal equivalence relations in the category of simplicial sets?

Mike Shulman (Nov 15 2021 at 19:00):

[[congruence]]

John Baez (Nov 15 2021 at 19:03):

It sounds like Leopold should put two brackets, [[ and ]], around the word "congruence" in the sentence he found unclear.

Nathanael Arkor (Nov 15 2021 at 19:03):

John Baez said:

It sounds like Leopold should put two brackets, [[ and ]], around the word "congruence" in the sentence he found unclear.

The nLab page already has a link there.

Leopold Schlicht (Nov 18 2021 at 17:52):

Ah, I see, thanks!

Reid Barton (Nov 18 2021 at 17:53):

All I meant by a "quotient" $X/A$ for a subobject $A \subseteq X$ is the pushout $1 \leftarrow A \to X$.

Reid Barton (Nov 18 2021 at 17:55):

Assuming that $A$ is nonempty it's also the quotient of $X$ by an equivalence relation ($x \sim y$ if $x = y$ or both are in $A$)

Leopold Schlicht (Nov 18 2021 at 18:13):

Ah, nice!

Is an internal equivalence relation $\sim$ on a simplicial set $S_\bullet$ an equivalence relation on $\bigsqcup_n S_n$ or a family of equivalence relations $\{\mathord{\sim}\subseteq S_n\times S_n\}_n$, i.e., can it identify simplices in different dimensions? (Reid Barton's comment suggests the first.) In both cases I guess the defining condition is that $\sigma\sim\tau\Rightarrow d_i\sigma\sim d_i\tau$ and $\sigma\sim\tau\Rightarrow s_i\sigma\sim s_i\tau$.

Is every quotient of a simplicial set $X$ by an internal equivalence relation of the form $X/A$ for some simplicial subset $A\subseteq X$?

Mike Shulman (Nov 18 2021 at 20:37):

I bet you could figure that out if you write down the definition of congruence in the category of simplicial sets.

Leopold Schlicht (Nov 20 2021 at 14:54):

Thanks! I came to the conclusion that it's a family of equivalence relations $\{\mathord{\sim}_n\subseteq S_n\times S_n\}_n$ with $\sigma\sim_n\tau\Rightarrow d_i\sigma\sim_{n-1} d_i\tau$ and $\sigma\sim_n\tau\Rightarrow s_i\sigma\sim_{n+1} s_i\tau$. Reid Barton's "$x\sim y$ if $x=y$ or both are in $A$" is probably just abuse of language for "for all $x,y\in S_n$, $x\sim_n y$ if and only if $x=y$ or both are in $A$".

Also, of course not every quotient is a "quotient by some subobject".

But here is a question that I can't answer at the moment: in which categories does each monomorphism $A\to X$ induce a congruence $R$ on $X$ such that $X/R$ is isomorphic to the pushout of $1 \leftarrow A \to X$? Is $R$ necessarily unique?

Why does the above first sentence of the nLab page on "quotient object" define quotients only for congruences and not for arbitary pairs of arrows (declaring their quotient to be their coequalizer)? Which kind of things one proves about quotients work only for congruences and not for arbitrary pairs of arrows?