Bicatégories monoïdales et extensions de gr-catégories

Abstract

In this work, we study the 2-category ${\bf Ext}(\underline{\cal K},\underline{\cal G})$ of extensions of a $gr$-category $\underline{\cal K}$ by a $gr$-category $\underline{\cal G}$. Such an extension consists of a $gr$-category $\underline{\cal H}$, an essentially surjective homomorphism $p : \underline{\cal H} \longrightarrow \underline{\cal K}$ and a monoidal equivalence $q : \underline{\cal G} \longrightarrow N(p)$ where $N(p)$ is the {\it homotopy kernel} of the homomorphism $p$. The main result is a classification theorem which constructs a biequivalence between the 2-category ${\bf Ext}(\underline{\cal K},\underline{\cal G})^{op}$ and the bicategory ${\bf Bimon}(\underline{\cal K}, \underline{\bf Bieq}(\underline{\cal G}))$ of monoidal bicategory homomorphisms between $\underline{\cal K}$ and $\underline{\bf Bieq}(\underline{\cal G})$, where $\underline{\bf Bieq}(\underline{\cal G})$ is the monoidal bicategory of biequivalences of $\underline{\cal G}$ with itself.