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The Stacks project

7.3 Injective and surjective maps of presheaves

Definition 7.3.1. Let \mathcal{C} be a category, and let \varphi : \mathcal{F} \to \mathcal{G} be a map of presheaves of sets.

  1. We say that \varphi is injective if for every object U of \mathcal{C} the map \varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U) is injective.

  2. We say that \varphi is surjective if for every object U of \mathcal{C} the map \varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U) is surjective.

Lemma 7.3.2. The injective (resp. surjective) maps defined above are exactly the monomorphisms (resp. epimorphisms) of \textit{PSh}(\mathcal{C}). A map is an isomorphism if and only if it is both injective and surjective.

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

Definition 7.3.3. We say \mathcal{F} is a subpresheaf of \mathcal{G} if for every object U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) the set \mathcal{F}(U) is a subset of \mathcal{G}(U), compatibly with the restriction mappings.

In other words, the inclusion maps \mathcal{F}(U) \to \mathcal{G}(U) glue together to give an (injective) morphism of presheaves \mathcal{F} \to \mathcal{G}.

Lemma 7.3.4. Let \mathcal{C} be a category. Suppose that \varphi : \mathcal{F} \to \mathcal{G} is a morphism of presheaves of sets on \mathcal{C}. There exists a unique subpresheaf \mathcal{G}' \subset \mathcal{G} such that \varphi factors as \mathcal{F} \to \mathcal{G}' \to \mathcal{G} and such that the first map is surjective.

Proof. To prove existence, just set \mathcal{G}'(U) = \varphi _ U \left(\mathcal{F}(U)\right) for every U \in \mathop{\mathrm{Ob}}\nolimits (C) (and inherit the action on morphisms from \mathcal{G}), and prove that this defines a subpresheaf of \mathcal{G} and that \varphi factors as \mathcal{F} \to \mathcal{G}' \to \mathcal{G} with the first map being surjective. Uniqueness is straightforward. \square

Definition 7.3.5. Notation as in Lemma 7.3.4. We say that \mathcal{G}' is the image of \varphi .


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