category object in an (∞,1)-category, groupoid object
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
identity type, equivalence of types, definitional isomorphism
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
The naive 2-category $Cat(S)$ of internal categories in an ambient category $S$ does in general not have enough equivalences of categories, due to the failure of the axiom of choice in $S$. Those internal functors which should but may not have inverses up to internal natural isomorphism, namely those which are suitably fully faithful and essentially surjective, may be regarded as weak equivalences of internal categories (Bunge & Paré 1979). The 2-category theoretic localisation of $Cat(S)$ at this class of 1-morphisms then serves as a more natural 2-category of internal categories.
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
To discuss the analogue of essential surjectivity, we need a notion of ‘surjectivity’, as this does not generalise cleanly from $Set$. 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 $t\circ pr_2:X_0 \times_{f_0,Y_0,s}Y_1 \to Y_0$ is a $J$-cover.
We then define an internal functor to be a $J$-equivalence if it is fully faithful and essentially $J$-surjective.
Marta Bunge, Robert Paré, Stacks and equivalence of indexed categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no. 4 (1979), p. 373-399 (numdam:CTGDC_1979__20_4_373_0)
Tomas Everaert, R.W.Kieboom and Tim Van der Linden, Model structures for homotopy of internal categories, Theory and Applications of Categories 15 (2005), no. 3, 66-94. (journal)
David Roberts, Internal categories, anafunctors and localisation, Theory and Applications of Categories, 26 (2012) No. 29, pp 788-829. (journal)
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