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Stream: learning: questions

Topic: coequalizers and quotient maps

Jonas Frey (Sep 05 2025 at 15:59):

Let $f,g:A \to B$ be a parallel pair in a category $ℂ$, assume that $e:B\to C$ is an equalizer of $f$ and $g$, and assume that $p,q:R\to B$ is a kernel pair of $e$. Then $e$ is a coequalizer of $p,q$.

This can be seen formally by looking at the Galois connection between sieves in the category $ℂ//B$ of parallel pairs into $A$, and co-sieves in the co-slice category $B/ℂ$ induced by the relation $hf=hg$ between parallel pairs $(f,g)$ into $B$ and arrows $h$ out of $B$. The Galois connection associates to every set $U$ of parallel arrows into $B$ the set $U^\bot$ of all arrows out of $B$ equalizing those two arrows, which is a co-sieve in $B/ℂ$, and dually in other direction. A coequalizer of $f,g$ is an initial object in the co-sieve $\{(f,g)\}^\bot$ (viewed as a full subcategory of $B/ℂ$), and a kernel pair of $h$ is a terminal object of the sieve ${}^\bot\{h\}$ in $ℂ//B$. The claim follows since we have
$\{(p,q)\}^\bot = (\downarrow\{(p,q)\})^\bot = ({}^\bot\{h\})^\bot = ({}^\bot(\{(f,g)\}^\bot))^\bot=\{(f,g)\}^\bot$
using idempotence of the Galois connection, where $\downarrow\{(p,q)\}$ is the sieve generated by $(p,q))$.

Jonas Frey (Sep 05 2025 at 16:04):

My question: is something analogous true for $\infty$-categories? Is every coequalizer or simplicial colimit in an $\infty$-category a quotient map in the sense of Lurie (kerodon), ie a colimit of its Cech nerve when the latter exists?

I don't think the above arguments transfers, since it relies on the fact that the relation $hf=hg$ between parallel pairs into and arrows out of $B$ is a property, whereas for the simplicial colimits in $\infty$-categories we need the additional structure of stitching together simplicial objects over $B$ and maps out of $B$ into augmented simplicial objects.

Jonas Frey (Sep 05 2025 at 16:33):

In the $\infty$-topos of spaces/types I think the statement is true since there the quotient maps are the surjections, and it seems obvious that all corqualizers and simplicial colims are surjective.

Mike Shulman (Sep 05 2025 at 16:59):

More generally I would expect it to be true in any $\infty$-topos for the same reason.

Adrian Clough (Sep 05 2025 at 18:08):

@Jonas Frey If I understand your question correctly, then it may be answered in the affirmative in any presentable $\infty$-category. For realisations this follows from combining Remark 6.1.4.4 + Lemma 6.1.4.6 in HTT, and for coequaliser from combining the previous two statements with Lemma 6.1.4.8.

Jonas Frey (Sep 05 2025 at 19:25):

Interesting, I'll have to look at that!

Jonas Frey (Sep 06 2025 at 18:16):

@Adrian Clough I had a look at 6.1.4.4&6.1.4.6 in HTT, but I don't think it answers my question for presentable $\infty$-categories. Lurie shows that the free groupoid on a simplicial object has the same colimit as the simplicial object, but my question was about the Cech nerve of the colimit, and w/o effectivity we don't know that the two coincide. Note that the free groupoid has an "initiality" universal property which is crucial in Lurie's reasoning, whereas the Cech nerve has a "terminality" property (being definable as a right adjoint to the restriction of augmented simplicial objects to the arrow category).

Jonas Frey (Sep 06 2025 at 18:17):

But in any case it settles the question for arbitrary $\infty$-toposes, confirming Mike's expectation above.

Adrian Clough (Sep 07 2025 at 06:24):

@Jonas Frey By Proposition 6.1.2.11, in any $\infty$-category, the underlying simplicial object $U_\bullet$ of an augmented simplicial object $U_\bullet^+$ is a groupoid iff it is a Čech nerve. Thus you get both the mapping-out and mapping-in properties in the generality of presentable $\infty$-categories.

Adrian Clough (Sep 07 2025 at 06:25):

(I'm just stating this for the record. You seem to be happy having the result for $\infty$-toposes :blush:)

Jonas Frey (Sep 07 2025 at 07:46):

Thanks for your reply. I'm not only interested in $\infty$-toposes, I want to understand the situation for arbitrary $\infty$-categories.

But I just looked at 6.1.2.11 in HTT, and to me it seems to state that an augmented simplicial object is a Cech nerve if the underlying sset is a groupoid and an additional pullback condition over the augmentation object holds.

Adrian Clough (Sep 07 2025 at 08:23):

Ah, yes, you're right.