Definition Backwards and forwards morphisms [lcc-0027]

May 1, 2024
Eigil Fjeldgren Rischel

Let a morphism $\phi = (\phi ^+,\phi ^-)$ in $\mathsf {Set}^\Delta \times \mathsf {Set}^{\Delta ,\mathrm {op}}$ be called forwards if $\phi ^-$ is an isomorphism, and backwards if $\phi ^+$ is an isomorphism. Let $F$ denote the set of forwards morphisms, $B$ the set of backwards. Then clearly $(F,B)$ form an orthogonal factorization system - in fact, both $(F,B)$ and $(B,F)$ do.

Note that $F$ consists exactly of the local equivalences for the inclusion of $* \times \mathsf {Set}^{\Delta ,\mathrm {op}}$ , so that the localization of $(X,A)$ can be formed as the terminal forwards map from it, which is clearly $(X,A) \to (*,A)$ (of course, this is not surprising).

We will say a morphism in $\mathsf {Minmax}$ is forwards, respectively backwards, if it is so considered as a morphism in $\mathsf {Set}^\Delta \times \mathsf {Set}^{\Delta ,\mathrm {op}}$ , and reuse the notation $F,B$ for these subclasses of morphism.