David Roberts
anafunctor

This is my own notational preference for how to describe anafunctors. For others see the nLab.

Constructions

Let $X$ be a category internal to a site $(S,J)$ where $J$ is a subcanonical singleton pretopology. Let $U\to X_0$ be a cover, and $X[U]$ the category with objects $U$ and arrows $U\times U{\times }_{X_0\times X_0}{X}_1$. Call this the induced category. There is a canonical functor

$$X[U]\to X$$X[U] \to X

which is a $J$-equivalence.

Definition

An anafunctor in $(S,J)$ from $X$ to $Y$, both categories internal to $S$, is a cover $U\to X_0$ and a functor $f:X[U]\to Y$. Denote it $(U,f):X\to Y$.

Ordinary functors $g:X\to Y$ can be considered as anafunctors, with the identity map of $X_0$ as the cover. Denote these by $(X_0,g)$. There is a composition of anafunctors, which is composition of the spans that they define. This requires a little lemma to say that the pullback category so defined is (isomorphic to one) of the required form

There are also transformations between anafunctors, which are defined in a manner entirely analogous to coboundaries between Čech cocycles (which are, of course, examples of said transformations).

Definition

..of transformation goes here.

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Here is a result that helps to show that anafunctors compute the localisation of Gpd$(S)$ at the $J$-equivalences.

Let $(S,J)$ be a site with a subcanonical singleton pretopology, and $f:X\to Y$ a $J$-equivalence. Then there is an anafunctor $(U,\overline{f}):Y\to X$ and isotransformations

$$(U,\overline{f})\circ (X_0,f)\Rightarrow {\mathrm{id}}_X(X_0,f)\circ (U,\overline{f})\Rightarrow {\mathrm{id}}_Y.$$(U,\bar{f})\circ(X_0,f) \Rightarrow id_X \quad (X_0,f) \circ (U,\bar{f}) \Rightarrow id_Y.

That is, $f$ has an anafunctor pseudoinverse.