# 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…

Revised on October 24, 2012 15:30:38 by David Roberts (121.216.167.70)