nLab
weak equivalence of internal categories

Contents

Context

Internal categories

Equality and Equivalence

Contents

Idea

The naive 2-category Cat(S)Cat(S) of internal categories in an ambient category SS does in general not have enough equivalences of categories, due to the failure of the axiom of choice in SS. 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)Cat(S) at this class of 1-morphisms then serves as a more natural 2-category of internal categories.

Definition

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

X 1 f 1 Y 1 X 0×X 0 f 0×f 0 Y 0×Y 0 \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 SetSet. 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:XYf:X\to Y internal to a unary site (S,J)(S,J), ff is essentially JJ-surjective if the map tpr 2:X 0× f 0,Y 0,sY 1Y 0t\circ pr_2:X_0 \times_{f_0,Y_0,s}Y_1 \to Y_0 is a JJ-cover.

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

References

Last revised on August 29, 2021 at 09:03:10. See the history of this page for a list of all contributions to it.