Equivalences of Internal Categories

Question

I am trying to understand double categories and their relatives a little better. I do not understand much, so I apologize in advance if my question is too naive for this website.

A functor $F:X \rightarrow Y$ of ordinary $2$-categories is an equivalence if and only if:

(1) It is essentially surjective.

(2) It is fully faithful, in the sense that it induces equivalences of Hom categories.

Are there analogues of conditions (1) and (2) that detect equivalences of double categories?

To ask my question in a different way, section A.3.2 of Lurie's Higher Topos Theory constructs a model structure on $S$-enriched categories where $S$ is a monoidal model category satisfying some nice properties (an alternative set of properties were recently discovered by Berger and Moerdijk). In Lurie's model structure, an $S$-enriched functor $F:X \rightarrow Y$ is an equivalence if and only if

(1) $Map_X(x,x') \rightarrow Map_Y(F(x),F(x'))$ is always a weak equivalence in $S$

(2) $F$ induces an essentially surjective map of homotopy categories $Ho(X) \rightarrow Ho(Y)$. Here, arrows from $x$ to $x'$ in $Ho(X)$ are homotopy classes of morphisms from the unit of $S$ to $Map_X(x,x')$.

Suppose I have a nice model category $S$, for some definition of nice I am willing to determine. Can I expect there to be a model structure on category objects in $S$, analogous to the above model structure on $S$-enriched categories? What would be the analogues of conditions $(1)$ and $(2)$?

Answer 1

A quick note, since I can't write a full answer. See the 2005 paper by Everaert, Kieboom, van der Linden. Possibly also my paper on internal categories and anafunctors, which is on the arXiv (to appear in a journal soon). But I don't think this is quite what you want, since they deal with functors which are (analogues of) isomorphisms on hom-sets. I hope though that they can be a springboard or inspiration for you.

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That being said, it depends what you want to use double categories for. I'm guessing when you say 'internal to a model category' you mean the canonical model structure on $Cat$, where weak equivalences are categorical equivalences. In that case you propose for weak equivalences those double functors $F\colon C\to D$ which are essentially surjective on objects (which means that one has an epimorphism $C_0 \times_{D_0} D_1 \to D_1 \to D_0$ of categories, not a very good notion) and such that $C_1 \to C_0^2 \times_{D_0^2} D_1$ is an equivalence of categories in the slice $Cat/C_0^2$. This may or may not be what you want. I can think of other options, such as those double functors whose associated map of diagonals of bisimplicial sets is a weak equivalence in some sense (such as in the Quillen model structure), but this is a very homotopical notion, not an algebraic one. Alternatively, one could ask that the map on objects $C_{00} \to D_{00}$ is essentially surjective both horizontally and vertically, the functors of horizontal and vertical categories are fully faithful, and the map on squares $C_{11} \to D_{11}$ in fully faithful in the sense that for $S\in C_{11}$, $S \mapsto (s_v(S),t_v(S),s_h(S),t_h(S);F(S))$ is a bijection.

What you use also depends on whether the horizontal and vertical categories are afforded equal status, or one is privileged over the other.

Answer 2

As David says, the right notion of equivalence of double categories depends on what sort of double categories you're looking at, and what you want to use them for. You might also be interested in Theorem 7.7 of this paper.