David Roberts weak equivalence of topological bigroupoids

Actually Andrew this page should have been called 'weak equivalence of topological bigroupoids', but I don’t have the renaming button. Can this be turned on? Ta

That’s odd. It’s there when I edit the page. Ah, but maybe this is the first edit - you don’t get the chance to rename a page when you’re first editing it. Then you should either submit then edit or cancel the creation and create the page you meant it to be.

Idea

The definition of equivalence of bigroupoids, in analogy with the definition of an equivalence of categories, comes in two flavours: the strict and the weak. The strict notion of equivalence is where the 2-functor (i.e. weak 2-functor) has a specified weak inverse. This may come with additional data, such as would give a biadjoint biequivalence (see Gurski’s work, for example). The weak notion is more along the lines of 'fully faithful and essentially surjective'. This is what we will look at here.

Quick and dirty definition of a 2-functor between bigroupoids

Recall that an internal bigroupoid (in some finitely complete category, say, such as $Top$) is basically an internal groupoid $\underline{B}_1$ over a space $B_0\times B_0$, together with composition, unit and inverse functors, satisfying some conditions. Really what we define here is a map between the underlying (truncated) globular set. All details are in the appendix to my thesis, available at HomePage.

Definition: Let $B,C$ be internal bigroupoids. A 2-functor $F:B \to C$ is given by a map $F_0:B_0\to C_0$ and a functor $\underline{F}_1:\underline{B}_1 \to \underline{C}_1$ over $F_0\times F_0$, such that there natural isomorphisms

$$\underline{F}_1(g)\cdot\underline{F}_1(k) \Rightarrow\underline{F}_1(g\cdot k)$$

$$\underline{F}_1(id_b)\Rightarrow id_{F_0(b)}$$

and

$$\underline{F}_1(\overline{g})\Rightarrow \overline{\underline{F}_1(g)}$$

in $\underline{C}_1$. These need to satisfy some coherence relations…

Weak equivalence

Now assume that our ambient category has a Grothendieck pretopology (such as the open cover pretopology on $Top$).

Firstly we say that a 2-functor is locally fully faithful if the functor $\underline{F}_1$ is fully faithful (it goes without saying this is in the internal sense).

Consider the functor $B_0\times_{F_0,C_0,S}\underline{C}_1 \to \underline{C}_1 \stackrel{T}{\to} C_0$. We would like this to be ‘surjective’ in the appropriate sense (this will give us essential surjectivity).

However, I would like a description in terms of internal weakly-enriched-in-groupoids groupoids. Will need truncated hypercover i.e. an identity-on-objects extension of a Cech groupoid where the arrow component is a cover. Also, take the most recent version of a weak equivalence from my anafunctors work…