Equivalence relations, Segal groupoids and groupoid objects in an infinity category

Question

There are three forms of "equivalence relations are effective" as part of Giraud's axioms in $1$-Grothendieck topoi, Model topoi and Infinity topoi. I am trying to understand how they relate to one another.

For $1$-Grothendieck topoi, an equivalence relation is a subobject $R \hookrightarrow X \times X$ satisfying some conditions. We say an equivalence relation is effective if when we take the coequalizer $R \rightrightarrows X \to X/R$, then $R = X \times_{X/R} X$. Intuitively I would have hoped an equivalence relation was just an internal groupoid, however the nlab says that an equivalence relation is only equivalent to an internal category with maps in both directions. Is there a good reason for this?

This is made more confusing because the way that equivalence relations are generalized to model topoi is using Segal groupoids. If an equivalence relation is not an internal groupoid, why should the generalization be groupoidal?

Segal groupoids are simplicial objects in a model category satisfying the usual Segal conditions and such that $X_2 \xrightarrow{(d_0, d_1)} X_1 \times_{X_0} X_1$ is a weak equivalence.

However a groupoid object in an infinity category requires much greater coherence, namely that $X_n$ must be a pullback of all possible partitions $S \cup S' = [n]$ with $|S \cap S'| = 1$.

For the nerve of a $1$-groupoid it seems that the two above notions are equivalent.

If I have a model topos, then a Segal groupoid object in it should be a groupoid object when I think of it as an infinity category. How do I show this? Equivalently, is the definition of groupoid object in an infinity category equivalent to just requiring the Segal condition and the condition on $X_2$?

Answer 1

Thanks to Marc Hoyois for affirming my assumptions. I will show that equivalence relations are examples of internal groupoids. I was having trouble proving this using the definition on the nlab, but will prove it for the definition of internal groupoid given in Section 7.1 in Borceux and Janelidze's Galois Theories which (I assume) is equivalent.

Let $R \overset{d_0}{\underset{d_1}{\rightrightarrows}} X$ be an equivalence relation (this definition is taken from the nlab), i.e. there are maps

1. $R \xrightarrow{d_0} X, R \xrightarrow{d_1} X$,
2. $X \xrightarrow{n} R$,
3. $R \xrightarrow{\tau} R$,
4. $R \times_X R \xrightarrow{m} R$,

such that $R \xrightarrow{(d_0, d_1)} X \times X$ is a monomorphism,

(a). $d_0 n = 1_X$, $d_1 n = 1_X$,

(b). $d_0 \tau = d_1$, $d_1 \tau = d_0$, and if

$\require{AMScd}$ \begin{CD} R\times_X R @>f_1>> R\\ @V f_0 V V @VV d_0 V\\ R @>>d_1> X \end{CD}

is a pullback, then

(c). $\require{AMScd}$ \begin{CD} R\times_X R @>m>> R\\ @V (d_0 f_0, d_1 f_1) V V @VV (d_0,d_1) V\\ X \times X @= X \times X \end{CD} commutes.

Now we check off the axioms for an internal category. (G1) is by definition, (G2) is equivalent to (a), (G3) is equivalent to (c), (G4) is proven by considering the diagram: $\require{AMScd}$ \begin{CD} R @>(nd_0, 1_R)>> R \times_X R @>(d_0 f_0, d_1 f_1)>> X \times X \\ @| @VmVV @A(d_0,d_1)AA\\ R @= R @= R \end{CD} and remembering that $(d_0, d_1): R \to X \times X$ is a monomorphism. Now $(d_0, d_1)m(nd_0, 1_R) = (d_0 f_0, d_1 f_1)(n d_0, 1_R) = (d_0, d_1) = (d_0, d_1) 1_R$, so the diagram commutes, and thus $m(nd_0, 1_R) = 1_R$. (G5) is more tedious but you use the fact that $(d_0, d_1)$ is a mono again.

Now to show it is an internal groupoid: (G6) is equivalent to (b), and (G7) follows similarly using $(d_0, d_1)$ is a mono, namely that $(d_0 f_0, d_1 f_1)(1_R,\tau) = (d_0, d_0)$ and $(d_0, d_1)n d_0 = (d_0 n d_0, d_1 n d_0) = (d_0, d_0)$ by (a).

Now I don't know if the following is an accepted definition of hom object of an internal category, but I'm going to give it a shot. Objects in an internal category $(R \rightrightarrows X)$ will be maps from the terminal object $* \to X$, and hom objects will be pullbacks $* \times_{X \times X} R$, where $* \xrightarrow{a,b} X \times X$ will be a pair of objects and $R \xrightarrow{(d_0, d_1)} X$ are the source and target maps. Since monomorphisms are stable under pullback, that means that $* \times_{X \times X} R$ is a subterminal object. In $\mathsf{Set}$ that means it is either empty or the terminal object itself. This is the closest I can get to interpreting that equivalence relations have no nontrivial automorphisms.

For the statement about groupoid objects in $\infty$-categories, this has actually been proven in detail by Severin Bunk in Principal $\infty$-Bundles and Smooth String Group Models, Lemma A.6.