11.1.9 Strict Monomorphisms

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

Definition 11.1.9.1.1 ▶ Strict Monomorphisms

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

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

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

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

is injective.

Remark 11.1.9.1.2 ▶ Unwinding Definition 11.1.9.1.1

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

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

Example 11.1.9.1.3 ▶ Examples of Strict Monomorphisms

Here are some examples of strict monomorphisms.

  1. Strict Monomorphisms in $\mathsf{Cats}_{\mathsf{2}}$. The strict monomorphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the functors which are injective on objects and injective on morphisms; see Chapter 9: Preorders, Item 1 of Proposition 9.7.2.1.2.
  2. Strict Monomorphisms in $\textbf{Rel}$. The strict monomorphisms in $\textbf{Rel}$ are characterised in Chapter 6: Relations, Proposition 6.3.7.1.1.

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