Abstract Nonsense

Crushing one theorem at a time

Functor Categories (Pt. II)


Point of Post: This is a continuation of this post.

Properties of Functor Categories

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Let’s now look at some of the properties inherited by functor categories from their source and target categories. The first thing we’d like to note is that much of the “work” in functor categories can be thought of being done “componentwise” i.e. by examining individual objects in the source category. As a first homage to this idea we have the following:

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Theorem: Let \mathcal{S} be a small category and \mathcal{C} a category. Then, if \eta:F\implies G is a morphism of functors F,G:\mathcal{S}\to\mathcal{C} in \mathcal{C}^\mathcal{S} then if \eta_x enjoys any of the following listed properties as a morphism in \mathcal{C} for all x\in\text{Obj}(\mathcal{S}) then \eta enjoys that property as a morphism in \mathcal{C}^\mathcal{S}:

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\begin{aligned}&\mathbf{(1)}\quad\textit{is monic}\\ &\mathbf{(2)}\quad\textit{is epic}\\ &\mathbf{(3)}\quad\textit{is a section}\\ &\mathbf{(4)}\quad\textit{is a retraction}\\ &\mathbf{(5)}\quad\textit{is an isomorphism}\end{aligned}

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Proof: Suppose that \eta_x is monic for each object x of \mathcal{S} and suppose that \nu,\nu':H\to F are natural transformations, where H:\mathcal{S}\to\mathcal{C} is a functor, such that \eta\circ\nu=\eta\circ\nu'. Then, by definition, for each object x of \mathcal{C} we have that \eta_x\circ\nu_x=\eta_x\circ\nu'_x, now since \eta_x is monic for each x we can conclude that \nu_x=\nu'_x for all x and so \nu=\nu'. Since \nu,\nu' were arbitrary we may conclude that \eta is monic as desired.

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Suppose now that \eta_x is an isomorphsim for all x. For each x\in X let \eta^{-1}_x be the morphism G(x)\to F(x) such that \eta_x\circ\eta^{-1}_x=1_{G(x)} and \eta_x^{-1}\circ\eta_x=1_{F(x)}, we claim that \eta^{-1}:G\implies F is a natural transformation. In other words, we have to just check that for each morphism x\xrightarrow{f}y in \mathcal{S} we have that F(f)\circ\eta_x^{-1}=\eta_y^{-1}\circ G(f), but this follows by hitting both sides of the known identity \eta_y\circ F(f)=G(f)\circ\eta_x by \eta_y^{-1} on the left and \eta_x^{-1} on the right. Thus, \eta^{-1} really is a natural transformation, and since \eta\circ \eta ^{-1}=1_G and \eta\circ\eta^{-1}=1_F we see that \eta is an isomorphism F\implies G, and so the conclusion follows.

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The other parts of the theorem follow similarly. \blacksquare

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We see that, just as functor categories “preserve” special types of morphsims they also preserve special types of objects.

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Theorem: Let \mathcal{S} be a small category and \mathcal{C} a category. Then, if \mathcal{C} has an initial, terminal, or zero object then so does \mathcal{C}^\mathcal{S}.

Proof: Let i be an initial object of \mathcal{C} and consider the constant functor c_i:\mathcal{S}\to\mathcal{C}. We claim that c_i is an initial object in \mathcal{C}^\mathcal{S}. Indeed, let F be a functor \mathcal{S}\to\mathcal{C} for each object x in \mathcal{S} let \eta_x be the unique arrow i\to F(x). We claim that \eta is a natural transformation. Indeed, suppose that x\xrightarrow{f}y is an arrow in \mathcal{S} we then must prove that F(f)\circ\eta_x=\eta_y\circ 1_i, or that \eta_y=F(f)\circ\eta_x, but since F(f)\circ\eta_x is an arrow i\to G(y) it must be equal to \eta_y by the definition of initial object, and so naturality follows. Thus, we have at least one morphism \eta:c_i\implies F, moreover it’s clear it’s the only one since any other morphism would have to give a different arrow i\to F(x) for some x, contradicting the initial property of i. Thus, c_i is initial in \mathcal{C}^\mathcal{S} as desired.

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The case for terminal and zero objects follows similarly by considering the same cosntant functors. \blacksquare

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As mentioned in an earlier post, we see that the process of taking functor categories preserves any \mathbf{Ab}-category structure the target category might have. In particular, suppose that we are given an \mathbf{Ab}-category \mathcal{C} and a small category \mathcal{S}. To make \mathcal{C}^\mathcal{S} into an abelian category, we need to be able to add natural transformations. Namely, given two functors F,G:\mathcal{S}\to\mathcal{C} we know that \text{Hom}_{\mathcal{C}^\mathcal{S}}(F,G) is \text{Nat}(F,G) and so turning this into an abelian group amounts to the addition of natural transformations F\implies G. This is simple though, because we can just add “object wise”. Namely, suppose we have two natural transformations \eta,\nu:F\implies G, we can then add them by declaring (\eta+\nu)_x=\eta_x+\nu_x where + is the predefined addition on \text{Hom}_\mathcal{C}(F(x),G(x)). We need to prove that this really defines a natural transformation F\implies G. Since \eta_x+\nu_x really is an arrow F(x)\to G(x) it suffices to prove naturality. To this end suppose that x\xrightarrow{f}y is an arrow in \mathcal{S} we need to verify that (\eta_y+\nu_y)\circ F(f)=G(f)\circ(\eta_x+\nu_x). But,

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(\eta_y+\nu_y)\circ F(f)=\eta_y\circ F(f)+\nu_y\circ G(f)=G(f)\circ\eta_x+G(f)\circ\nu_x=G(f)\circ(\eta_x+\nu_x)

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so that \eta+\nu defined by (\eta+\nu)_x=\eta_x+\nu_x really is a natural transformation F\implies G. To check that this notion of the addition of natural transformations defines an \mathbf{Ab}-category structure on \mathcal{C}^\mathcal{S} it suffices to check that \eta\circ(\nu+\varepsilon)=\eta\circ\nu+\eta\circ\varepsilon but this follows immediately since this is true “object wise” (i.e. \eta_x\circ(\nu_x+\varepsilon_x)=\eta_x\circ\nu_x+\eta_x\circ\varepsilon_x). From the above and our previous observation about the preservation of zero objects we have the following:

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Theorem: Let \mathcal{S} be a small caetgory and \mathcal{C} an \mathbf{Ab}-category. Then, \mathcal{C}^\mathcal{S} with addition of natural transformations (\eta+\nu)_x=\eta_x+\nu_x is an \mathbf{Ab}-category. Moreover, if \mathcal{C} is preadditive then so is \mathcal{C}^\mathcal{S}.

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If we further suppose that \mathcal{S} is an \mathbf{Ab}-category it’s easy to see that:

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Theorem: Let \mathcal{S} be a small \mathbf{Ab}-category, and \mathcal{C} an \mathbf{Ab}-category. Then, the category \text{Add}(\mathcal{S},\mathcal{C}) of all additive functors \mathcal{S}\to\mathcal{C} is an \mathbf{Ab}-category. Moreover, if \mathcal{C} is preadditive then so is \text{Add}(\mathcal{S},\mathcal{C}).

Proof: We know that \text{Add}(\mathcal{S},\mathcal{C}) is a full subcategory of \mathcal{C}^\mathcal{S} from where it follows that \text{Add}(\mathcal{S},\mathcal{C}) is an \mathbf{Ab}-category (since full subcategories of \mathbf{Ab}-categories are \mathbf{Ab}-subcategories). To prove that \text{Add}(\mathcal{S},\mathcal{C}) is preadditive, assuming \mathcal{C}, is equivalent to checking that the constant functor c_i:\mathcal{S}\to\mathcal{C} for an initial object i of \mathcal{C} (as described above) is additive. But, this follows immediately since if f,g are arrows x\to y then F(f+g)=1_i=1_i+1_i=F(f)+F(g) since 1_i, as previously discussed, is the additive identity of \text{End}_\mathcal{C}(i). \blacksquare

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Miscellany

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We discuss two last things which come up often when discussing functor categories. The first is the evaluation functor associated to a functor category. Namely, suppose we are given some functor category \mathcal{C}^\mathcal{S}, there is then an obvious way to construct a functor E:\mathcal{C}\times\mathcal{C}^\mathcal{S}\to\mathcal{C} defined by E(x,F)=F(x) and given morphisms x\xrightarrow{f}y in \mathcal{C} and \eta:F\implies G in \mathcal{C}^\mathcal{S} we define E(f,F)=\eta_y\circ F(f)=G(f)\circ\eta_x. It’s easy to verify that this is actually a functor.

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The second bit of miscellany I’d like to mention is the following theorem:

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Theorem: Let \mathcal{S},\mathcal{T} be two small categories and \mathcal{C} a category. Then, \mathcal{C}^{\mathcal{S}\times\mathcal{T}}\cong\left(\mathcal{C}^\mathcal{S}\right)^\mathcal{T}\cong\left(\mathcal{C}^\mathcal{T}\right)^\mathcal{S}.

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The proof of this is annoying, but fairly straightforward. It can be found on page 97 of [6].

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References:

[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.

[5] Rotman, Joseph J. Introduction to Homological Algebra. Springer-Verlag. Print.

[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.

 

 

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January 7, 2012 - Posted by | Algebra, Category Theory | , , ,

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