nLab
weak equivalence of internal categories

Idea

The naive 2-category $\mathrm{Cat}(S)$ of internal categories in a category $S$ does not have enough equivalences in general, due to the failure of the axiom of choice in $S$. The functors which should be equivalences are called weak equivalences, and one often works with the localisation of $\mathrm{Cat}(S)$ at the weak equivalences. Roughly speaking, a weak equivalence is a functor which is ‘fully faithful’ and ‘essentially surjective’, but these terms need to be interpreted appropriately.

The concept of weak equivalences first arose in work of Bunge and Paré on stack completions of internal categories.

Definition

Let $f:X\to Y$ be a functor between categories internal to some category $S$. $f$ is fully faithful if the following diagram is a pullback

$$\begin{array}{ccc}{X}_1& \stackrel{f_1}{\to }& Y_1\\ \downarrow & & \downarrow \\ X_0\times X_0& \underset{f_0\times f_0}{\to }& Y_0\times Y_0\end{array}$$\begin{matrix} X_1& \stackrel{f_1}{\to} & Y_1 \\ \downarrow&& \downarrow \\ X_0\times X_0 &\underset{f_0\times f_0}{\to} & Y_0\times Y_0 \end{matrix}

To discuss the analogue of essential surjectivity, we need a notion of ‘surjectivity’, as this does not generalise cleanly from $\mathrm{Set}$. If we are working in a topos, a natural choice is to take epimorphisms, but weaker ambient categories are sometimes needed. A natural choice is to work in a unary site, where the covers are taken as the ‘surjective’ maps.

Given a functor $f:X\to Y$ internal to a unary site $(S,J)$, $f$ is essentially $J$-surjective if the map $t\circ {\mathrm{pr}}_2:X_0{\times }_{f_0,Y_0,s}{Y}_1\to Y_0$ is a $J$-cover.

We then define an internal functor to be a $J$-equivalence if it is fully faithful and essentially $J$-surjective.

Related concepts

• anafunctor

• equivalence of categories, weak equivalence

References

• M. Bunge, R. Paré, Stacks and equivalence of indexed categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no. 4 (1979), p. 373-399 NUMDAM
• T. Everaert, R. W. Kieboom and T. Van der Linden, Model structures for homotopy of internal categories, Theory Appl. Categ. 15 (2005), no. 3, 66-94. online
• D. Roberts, Internal categories, anafunctors and localisation, (will insert link to updated version soon)