# Projective/injective object in functor category

Let $$\mathcal{C}$$ denote the functor category $$Fun(\textbf{Man} , \textbf{Ab})$$, where $$\textbf{Man}$$ and $$\textbf{Ab}$$ denote the category of smooth manifolds and abelian groups respectively. I want to know what are the projective objects or injective objects in this category $$\mathcal{C}$$? More precisely, I am interested in resolving objects of $$\mathcal{C}$$ by either projective or injective objects. So for that I need to know what are these objects? Any kind of partial help or comments would be great.

The approach which I thought of is the following: If we can find a category $$\mathcal{D}$$ with two functors from $$F : \mathcal{C} \rightarrow \mathcal{D}$$ and from $$G : \mathcal{D} \rightarrow \mathcal{C}$$ such that both are in adjunction. Then under some conditions using the unit or co-unit map, we can resolve objects.

• For any essentially small category $\mathcal{T}$, a generating class of projective objects of the functor category $Fun(\mathcal{T}, \mathbf{Ab})$ is given by the linearisation of reprensentable functors $\mathcal{T}(t,-)$ (thanks to the Yoneda lemma). So you get natural projective resolutions of functors by the multi-object bar construction. You can dualise it to get injective resolutions. I am afraid that it is not easy to say something more precise and useful in practice in your specific setting. Apr 3 '20 at 7:48
• You might find something useful in the following paper (data from MathSciNet): MR0204497 (34 #4336) 18.20 Watts, Charles E. A homology theory for small categories. 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 331–335 Springer, New York May 4 '20 at 3:33

One way to construct injectives on a presheaf category $$[\mathscr C^{\operatorname{op}},\mathbf{Ab}]$$ is to consider the forgetful functor $$i^* \colon \big[\mathscr C^{\operatorname{op}},\mathbf{Ab}\big] \to \big[\mathscr C^{\operatorname{disc,op}},\mathbf{Ab}\big]$$ induced by the inclusion $$i \colon \mathscr C^{\operatorname{disc}} \to \mathscr C$$ (where $$\mathscr C^{\operatorname{disc}}$$ is the subcategory with only identity morphisms). If $$\mathscr C$$ is small, then $$i^*$$ has left and right adjoints $$i_!$$ and $$i_*$$ given by \begin{align*} \big(i_! \mathscr F\big)(c) = \bigoplus_{c' \to c} \mathscr F(c'),\\ \big(i_* \mathscr F\big)(c) = \prod_{c \to c'} \mathscr F(c'). \end{align*} In particular, $$i^*$$ is an exact left adjoint to $$i_*$$, so $$i_*$$ takes injectives to injectives [Stacks, Tag 015N]. But in $$[\mathscr C^{\operatorname{disc,op}},\mathbf{Ab}]$$ injectives are computed pointwise, so this gives a recipe to construct injectives in $$[\mathscr C^{\operatorname{op}},\mathbf{Ab}]$$.

See for example [Stacks, Tag 01DJ] for a brief discussion, or [SGA IV$$_1$$, Exp. I, Prop. 5.1] for a more general discussion of adjoints (but without the mention of injectives).

In general the colimit for $$i_!$$ is taken over the opposite of the comma category $$(i \downarrow c)$$ (whose objects are $$(i(c') \to c)$$), which in this case is just a discrete category since $$\mathscr C^{\operatorname{disc}}$$ is, so we get a direct sum; similarly for $$i_*$$.

References.

[SGA IV$$_1$$] M. Artin, A. Grothendieck, J.-L. Verdier, Séminaire de géométrie algébrique du Bois-Marie 1963–1964. Théorie de topos et cohomologie étale des schémas (SGA 4), 1: Théorie des topos. Lecture Notes in Mathematics 269. Springer-Verlag (1972). ZBL0234.00007.

[Stacks] A.J de Jong et al, The stacks project.

For projective objects, see here: Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves.

As explained there for presheaves of sets (and the same argument works for abelian groups), projective presheaves are precisely coproducts of retracts of representables.

As for injective presheaves, the general consensus is that there is no general criterion to characterize them other than by their lifting properties. This question has been discussed before, see, for example, here: What are the fibrant objects in the injective model structure?

As for the last paragraph, this approach is known as the bar construction.

• It's seems very helpful at least projective case. Thanks. Apr 4 '20 at 5:47