Proof.
We shall show that \varphi : \mathcal{F} \to \mathcal{G} is injective if and only if it is a monomorphism of \textit{PSh}(\mathcal{C}). Indeed, the “only if” direction is straightforward, so let us show the “if” direction. Assume that \varphi is a monomorphism. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}); we need to show that \varphi _ U is injective. So let a, b \in \mathcal{F}(U) be such that \varphi _ U (a) = \varphi _ U (b); we need to check that a = b. Under the isomorphism (7.2.1.1), the elements a and b of \mathcal{F}(U) correspond to two natural transformations a', b' \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}). Similarly, under the analogous isomorphism \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{G}) = \mathcal{G}(U), the two equal elements \varphi _ U (a) and \varphi _ U (b) of \mathcal{G}(U) correspond to the two natural transformations \varphi \circ a', \varphi \circ b' \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{G}), which therefore must also be equal. So \varphi \circ a' = \varphi \circ b', and thus a' = b' (since \varphi is monic), whence a = b. This finishes (1).
We shall show that \varphi : \mathcal{F} \to \mathcal{G} is surjective if and only if it is an epimorphism of \textit{PSh}(\mathcal{C}). Indeed, the “only if” direction is straightforward, so let us show the “if” direction. Assume that \varphi is an epimorphism.
For any two morphisms f : A \to B and g : A \to C in the category \textit{Sets}, we let \text{inl}_{f,g} and \text{inr}_{f,g} denote the two canonical maps from B and C to B \coprod _ A C. (Here, the pushout is evaluated in \textit{Sets}.)
Now, we define a presheaf \mathcal{H} of sets on \mathcal{C} by setting \mathcal{H}(U) = \mathcal{G}(U) \coprod _{\mathcal{F}(U)} \mathcal{G}(U) (where the pushout is evaluated in \textit{Sets} and induced by the map \varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U)) for every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}); its action on morphisms is defined in the obvious way (by the functoriality of pushout). Then, there are two natural transformations i_1 : \mathcal{G} \to \mathcal{H} and i_2 : \mathcal{G} \to \mathcal{H} whose components at an object U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) are given by the maps \text{inl}_{\varphi _ U, \varphi _ U} and \text{inr}_{\varphi _ U, \varphi _ U}, respectively. The definition of a pushout shows that i_1 \circ \varphi = i_2 \circ \varphi , whence i_1 = i_2 (since \varphi is an epimorphism). Thus, for every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), we have \text{inl}_{\varphi _ U, \varphi _ U} = \text{inr}_{\varphi _ U, \varphi _ U}. Thus, \varphi _ U must be surjective (since a simple combinatorial argument shows that if f : A \to B is a morphism in \textit{Sets}, then \text{inl}_{f,f} = \text{inr}_{f,f} if and only if f is surjective). In other words, \varphi is surjective, and (2) is proven.
We shall show that \varphi : \mathcal{F} \to \mathcal{G} is both injective and surjective if and only if it is an isomorphism of \textit{PSh}(\mathcal{C}). This time, the “if” direction is straightforward. To prove the “only if” direction, it suffices to observe that if \varphi is both injective and surjective, then \varphi _ U is an invertible map for every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), and the inverses of these maps for all U can be combined to a natural transformation \mathcal{G} \to \mathcal{F} which is an inverse to \varphi .
\square
Comments (0)