Bicatégories mono?dales et extensions de $gr$-catégories

摘要

In this work, we study the 2-category ${f Ext}(underline{cal K},underline{cal G})$ 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 ${f Ext}(underline{cal K},underline{cal G})^{op}$ and the bicategory ${f Bimon}(underline{cal K}, underline{f Bieq}(underline{cal G}))$ of monoidal bicategory homomorphisms between $underline{cal K}$ and $underline{f Bieq}(underline{cal G})$, where $underline{f Bieq}(underline{cal G})$ is the monoidal bicategory of biequivalences of $underline{cal G}$ with itself.