11.2.9 Strict Epimorphisms

Let $\mathcal{C}$ be a bicategory.

A $1$-morphism $f\colon A\to B$ is a strict epimorphism in $\mathcal{C}$ if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor

\[ f^{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (B,X\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (A,X\webright ) \]

given by precomposition by $f$ is injective on objects, i.e. its action on objects

\[ f_{*}\colon \text{Obj}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (B,X\webright )\webright )\to \text{Obj}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (A,X\webright )\webright ) \]

is injective.

In detail, $f$ is a strict epimorphism if, for each diagram in $\mathcal{C}$ of the form

if $\phi \circ f=\psi \circ f$, then $\phi =\psi $.


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