The Stacks project

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19.6 Abelian presheaves on a category

Let $\mathcal{C}$ be a category. Recall that this means that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is a set. On the one hand, consider abelian presheaves on $\mathcal{C}$, see Sites, Section 7.2. On the other hand, consider families of abelian groups indexed by elements of $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$; in other words presheaves on the discrete category with underlying set of objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let us denote this discrete category simply $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. There is a natural functor

\[ i : \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) \longrightarrow \mathcal{C} \]

and hence there is a natural restriction or forgetful functor

\[ v = i^ p : \textit{PAb}(\mathcal{C}) \longrightarrow \textit{PAb}(\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})) \]

compare Sites, Section 7.5. We will denote presheaves on $\mathcal{C}$ by $B$ and presheaves on $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ by $A$.

There are also two functors, namely $i_ p$ and ${}_ pi$ which assign an abelian presheaf on $\mathcal{C}$ to an abelian presheaf on $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, see Sites, Sections 7.5 and 7.19. Here we will use $u = {}_ pi$ which is defined (in the case at hand) as follows:

\[ uA(U) = \prod \nolimits _{U' \to U} A(U'). \]

So an element is a family $(a_\phi )_\phi $ with $\phi $ ranging through all morphisms in $\mathcal{C}$ with target $U$. The restriction map on $uA$ corresponding to $g : V \to U$ maps our element $(a_\phi )_\phi $ to the element $(a_{g \circ \psi })_\psi $.

There is a canonical surjective map $vuA \to A$ and a canonical injective map $B \to uvB$. We leave it to the reader to show that

\[ \mathop{Mor}\nolimits _{\textit{PAb}(\mathcal{C})}(B, uA) = \mathop{Mor}\nolimits _{\textit{PAb}(\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}))}(vB, A). \]

in this simple case; the general case is in Sites, Section 7.5. Thus the pair $(u, v)$ is an example of a pair of adjoint functors, see Categories, Section 4.24.

At this point we can list the following facts about the situation above.

  1. The functors $u$ and $v$ are exact. This follows from the explicit description of these functors given above.

  2. In particular the functor $v$ transforms injective maps into injective maps.

  3. The category $\textit{PAb}(\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}))$ has enough injectives.

  4. In fact there is a functorial injective embedding $A \mapsto \big (A \to J(A)\big )$ as in Homology, Definition 12.24.5. Namely, we can take $J(A)$ to be the presheaf $U\mapsto J(A(U))$, where $J(-)$ is the functor constructed in More on Algebra, Section 15.54 for the ring $\mathbf{Z}$.

Putting all of this together gives us the following procedure for embedding objects $B$ of $\textit{PAb}(\mathcal{C}))$ into an injective object: $B \to uJ(vB)$. See Homology, Lemma 12.26.5.

Proposition 19.6.1. For abelian presheaves on a category there is a functorial injective embedding.

Proof. See discussion above. $\square$


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