nLab
bicategory of fractions

Much has been said about inverting a class of morphisms in a category (see localization), and there are many different settings in which one wants to, and can, do this. Homotopical algebra is largely concerned with how to compute the homotopy category so it is locally small. One the other hand, we have simplicial localization which retains all the homotopy information and returns an $(\mathrm{\infty },1)$-category.

If we have a 2-category with a notion of weak equivalence, one could localize the underlying 1-category in a way hopefully compatible with the 2-arrows, or extend the result fully into the 2-dimensional setting. In general this will require bicategories, and is the subject of the paper Etendues and stacks as bicategories of fractions by Dorette Pronk.

Definition

Let $B$ be a bicategory with a class $W$ of 1-cells. $W$ is said to admit a right calculus of fractions if it satisfies the following conditions

• 2CF1. $W$ contains all equivalences

• 2CF2.

  • a) $W$ is closed under composition
  • b) If $a\in W$ and a iso-2-cell $a\stackrel{\sim }{\Rightarrow }b$ then $b\in W$

• 2CF3. For all $w:A\prime \to A$, $f:C\to A$ with $w\in W$ there exists a 2-commutative square

  $$\begin{array}{ccc}P& \stackrel{g}{\to }& A\prime \\ v\downarrow & \Rightarrow & \downarrow w\\ C& \underset{f}{\to }& A\end{array}$$\begin{matrix} P& \stackrel{g}{\to} & A' \\ v \downarrow&\Rightarrow &\, \downarrow w\\ C &\underset{f}{\to} & A \end{matrix}

  with $v\in W$.

• 2CF4. If $\alpha :w\circ f\Rightarrow w\circ g$ is a 2-cell and $w\in W$ there is a 1-cell $v\in W$ and a 2-cell $\beta :f\circ v\Rightarrow g\circ v$ such that $\alpha \circ v=w\circ \beta$. Moreover: when $\alpha$ is an iso-2-cell, we require $\beta$ to be an isomorphism too; when $v\prime$ and $\beta \prime$ form another such pair, there exist 1-cells $u,u\prime$ such that $v\circ u$ and $v\prime \circ u\prime$ are in $W$, and an iso-2-cell $ϵ:v\circ u\Rightarrow v\prime \circ u\prime$ such that the following diagram commutes:

  $$\begin{array}{ccc}f\circ v\circ u& \stackrel{\beta \circ u}{\Rightarrow }& g\circ v\circ u\\ f\circ ϵ\Downarrow \simeq & & \simeq \Downarrow g\circ ϵ\\ \\ f\circ v\prime \circ u\prime & \underset{\beta \prime \circ u\prime }{\Rightarrow }& g\circ v\prime \circ u\prime \end{array}$$\begin{matrix} f \circ v \circ u & \stackrel{\beta\circ u}{\Rightarrow} & g\circ v \circ u \\ f\circ \epsilon \Downarrow \simeq && \simeq \Downarrow g\circ \epsilon \\ \\ f\circ v' \circ u' &\underset{\beta'\circ u'}{\Rightarrow}& g\circ v' \circ u' \end{matrix}

If $B$ is a category, then these axioms reduce to the ones of Gabriel and Zisman for a calculus of fractions.

Given such a setup, Pronk constructs the localization of $B$ at $W$ and the universal functor sending elements of $W$ to equivalences.

Example

Let $S$ be a category with binary products and pullbacks together with a class of admissible maps $E$.

Theorem:

The 2-catgeories $\mathrm{Cat}(S)$ and $\mathrm{Gpd}(S)$ of categories and groupoids internal to $S$ admit bicategories of fractions for the class of $E$-equivalences.

The resulting localization is equivalent to the bicategory of anafunctors in $S$. For details, see the article

• David Roberts, Internal categories, anafunctors and localisations, arXiv:math/1101.2363