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 (,1)(\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 Pronk 96.


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

  • [2CF1.] WW contains all equivalences

  • [2CF2.]

    • a) WW is closed under composition
    • b) If aWa\in W and a iso-2-cell aba \stackrel{\sim}{\Rightarrow} b then bWb\in W
  • [2CF3.] For all w:AAw: A' \to A, f:CAf: C \to A with wWw\in W there exists a 2-commutative square

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

    with vWv\in W.

  • [2CF4.] If α:wfwg\alpha: w \circ f \Rightarrow w \circ g is a 2-cell and wWw\in W there is a 1-cell vWv \in W and a 2-cell β:fvgv\beta: f\circ v \Rightarrow g \circ v such that αv=wβ\alpha\circ v = w \circ \beta. Moreover: when α\alpha is an iso-2-cell, we require β\beta to be an isomorphism too; when vv' and β\beta' form another such pair, there exist 1-cells u,uu,\,u' such that vuv\circ u and vuv'\circ u' are in WW, and an iso-2-cell ϵ:vuvu\epsilon: v\circ u \Rightarrow v' \circ u' such that the following diagram commutes:

    fvu βu gvu fϵ gϵ fvu βu gvu \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 BB 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 BB at WW and the universal functor sending elements of WW to equivalences.


Let SS be a category with binary products and pullbacks together with a class of admissible maps EE.


The 2-categories Cat(S)Cat(S) and Gpd(S)Gpd(S) of categories and groupoids internal to SS admit bicategories of fractions for the class of EE-equivalences.

The resulting localization is equivalent to the bicategory of anafunctors in SS. For details, see Roberts (2012).


(Grothendieck toposes as a bicategory of fractions of localic groupoids)

The category of etale-complete localic groupoids (with open source and target maps) admits a bicategory of fractions at open essentially surjective fully faithful functors.

The resulting bicategory is equivalent to the bicategory of Grothendieck toposes, geometric morphisms, and natural isomorphisms.

See Theorem 7.7 in Moerdijk.


  • Ieke Moerdijk, The classifying topos of a continuous groupoid. I, Transactions of the American Mathematical Society, Volume 310, Number 2, December 1988.

  • O. Abbad, E. M. Vitale, Faithful calculus of fractions , Cah. Top. Géom. Diff. Catég. 54 No. 3 (2013) 221-239. (preprint)

  • E. Vitale, Bipullbacks and categories of fractions, pdf

  • Dorette A. Pronk, Etendues and stacks as bicategories of fractions, Comp. Math. 102 3 (1996) pp.243-303. (numdam:CM_1996__102_3_243_0)

  • David Roberts, Internal categories, anafunctors and localisations, TAC 26 (2012) pp.788-829. (pdf)

  • M. Tommasini, A bicategory of reduced orbifolds from the point of view of differential geometry , arXiv:1304.6959 (2013). (pdf)

  • M. Tommasini, Some insights on bicategories of fractions I , arXiv:1410.3990 (2014). (pdf)

  • M. Tommasini, Some insights on bicategories of fractions II , arXiv:1410.5075 (2014). (pdf)

  • M. Tommasini, Some insights on bicategories of fractions III , arXiv:1410.6395 (2014). (pdf)

See also:

Last revised on October 12, 2022 at 11:43:31. See the history of this page for a list of all contributions to it.