Abstract Nonsense

Crushing one theorem at a time

Adjoint Functors (Pt. III)


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

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The Limit and Colimit Functors

As a last example we are going to assume we have a category \mathcal{C} and a small category \mathcal{I} for which every diagram F:\mathcal{I}\to\mathcal{C} has a limit and a colimit. We want to define functors \lim:\mathcal{C}^\mathcal{I}\to\mathcal{C} and \text{colim}:\mathcal{C}^\mathcal{I}\to\mathcal{C} where \mathcal{C}^\mathcal{I} is the functor category. We already know how this should act on objects–it just takes a diagram F:\mathcal{I}\to\mathcal{C} and assigns to it either \lim F or \text{colim }F. The question then is exactly how to figure out what the direct limit of a morphism should be.

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To this end suppose that we have an arrow \eta:F\implies G in \mathcal{C}^\mathcal{I}. We then have arrows F(i)\xrightarrow{\eta_i}G(i) for each i an object of \mathcal{I}. Note then though that we have arrows \lim F\xrightarrow{\alpha_i}F(i)\xrightarrow{\eta_i}G(i) which by the definition of limits gives us a unique arrow \lim F\xrightarrow{\lim \eta}\lim G with \beta_i\circ\lim\eta=\eta_i\circ\alpha_i (where of course, the \alpha_i and \beta_i are the cone maps associated to \lim F and \lim G respectively). Of course, it’s easy to see that if we have the composition \eta\circ\varepsilon of two natural transformations F\overset{\varepsilon}{\implies}G\overset{\eta}{\implies}H then \lim(\varepsilon\circ\eta) satisfies

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\begin{aligned}\gamma_i\circ\lim(\eta\circ\varepsilon) &=(\eta_i\circ\varepsilon_i)\circ\alpha_i\\ &=\eta_i\circ(\varepsilon_i\circ\alpha_i)\\ &=\eta_i\circ(\beta_i\circ\lim\varepsilon)\\ &= (\eta_i\circ\beta_i)\circ\lim\varepsilon\\ &= (\gamma_i\circ\lim\eta)\circ\lim\varepsilon\\ &=\gamma_i\circ(\lim\eta\circ\lim\varepsilon)\end{aligned}

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and since \lim(\eta\circ\varepsilon) and \lim\eta\circ\lim\varepsilon agree on \gamma_i (obviously the cone maps \lim H\to H(i)) we may conclude that they are equal. Noticing finally that if 1_F:F\implies F then \alpha_i\circ\lim 1_F=1_{F(i)}\circ \alpha_i=\alpha_i=\alpha_i\circ 1_{\lim F} and thus \lim 1_F=1_{\lim F}. Thus, we may conclude that, with these definitions, \lim:\mathcal{C}^\mathcal{I}\to\mathcal{C} is a functor.

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Very similarly we may construct a functor \text{colim}:\mathcal{C}^\mathcal{I}\to\mathcal{C} which on objects is obviously \text{colim }F and on arrows sends F\overset{\eta}{\implies}G to the unique arrow \text{colim }F\xrightarrow{\text{colim }\eta}\text{colim }G with \text{colim }\eta\circ\nu_i=\omega_i\circ\eta_i where G(i)\xrightarrow{\omega_i}\text{colim }G and F(i)\xrightarrow{\nu_i}\lim F are the cocone maps.

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Ok, so with all of this defined we’d like to discuss a certain series of adjunctions involving these two functors and another one we’ve seen before. Namely, define the diagonal functor \Delta:\mathcal{C}\to\mathcal{C}^\mathcal{I} by taking each object X in \mathcal{C} to the functor \Delta_X:\mathcal{I}\to\mathcal{C} with \Delta_X(i)=X for all i and \Delta_X(s)=1_X for all arrows i\xrightarrow{s}j, and taking each arrow X\xrightarrow{f}Y to the natural transformation \Delta_X\overset{\Delta_f}{\implies}\Delta_Y with X\xrightarrow{\eta_i=f}Y for all i (it’s easy to verify this really is a natural transformation).

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In particular, we claim that \text{colim}\dashv\Delta\dashv\lim.

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To show that \text{colim }\dashv \Delta we being by defining the isomorphisms \Psi_{F,X}:\text{Hom}_{\mathcal{C}^\mathcal{I}}(F,\Delta_X)\to\text{Hom}_\mathcal{C}(\text{colim }F,X). How do we define such an isomorphism? Well, what exactly is a natural transformation F\overset{\varepsilon}{\implies}\Delta_X? Well, it’s a set of maps F(i)\xrightarrow{\varepsilon_i}\Delta_X(i)=X–and that’s it, since the commutativity is trivial since \Delta_X(s)=1_X for all morphisms s. Ok, great! Clearly then we can make \Psi_{F,X} into a bijection by merely defining \Psi_{F,X}(\varepsilon) to be the unique arrow \text{colim }F\to X with \Psi_{F,X}(\varepsilon)\circ\nu_i=\varepsilon_i. Let’s then verify that these isomorphisms are natural in each entry. To this end let \eta:G\implies F be some natural transformation. We must then verify that \Psi_{G,X}\circ\eta^\ast=(\text{colim }\eta)^\ast\circ\Psi_{F,X}. To do this let F\overset{\varepsilon}{\implies}\Delta_X be some natural transformation we see then that

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(\Psi_{G,X}\circ\eta^\ast)(\varepsilon)\circ\omega_i=\Psi_{G,X}(\varepsilon\circ\eta)\circ\omega_i=\varepsilon_i\circ\eta_i

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and

\begin{aligned}((\text{colim }\eta)^\ast\circ\Psi_{F,X})(\varepsilon)\circ\omega_i &=\Psi_{F,X}(\varepsilon)\circ\text{colim }\eta\circ\omega_i\\ &=\Psi_{F,X}(\varepsilon)\circ \nu_i\circ\eta_i\\ &=\varepsilon_i\circ\eta_i\end{aligned}

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and since our desired maps agree on \omega_i they must be equal.

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Now, for naturality in the other variable let X\xrightarrow{f}Y be an arrow in \mathcal{C}, we must verify then that f^\ast\circ\Psi_{F,X}=\Psi_{F,Y}\circ\Delta(f)^\ast. To this end, let F\overset{\varepsilon}{\implies}\Delta_X  be a natural transformation. We check then that

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(f^\ast\circ\Psi_{F,X})(\varepsilon)\circ\nu_i=f\circ\Psi_{F,X}(\varepsilon)\circ\nu_i=f\circ\varepsilon_i

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and

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(\Psi_{F,Y}\circ\Delta(f)^\ast)(\varepsilon)\circ\nu_i=\Psi_{F,Y}(\Delta(f)\circ\varepsilon)\circ\nu_i=f\circ\varepsilon_i

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Thus, we see that \text{colim }\dashv\Delta as claimed.

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Of course, dualizing the proof one can show that \Delta\dashv\lim and thus we may finally conclude that:

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Theorem: Let \mathcal{C} be a category which admits all limits (resp. colimits) for diagrams F:\mathcal{I}\to\mathcal{C}. Then, the functor \lim (resp. \text{colim}) is right adjoint (resp. left adjoint). 

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Uniqueness of Adjoints

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As a last point we note that right or left adjoints to a given functor are necessarily unique. Indeed, if F\dashv G and F\dashv H then we see that

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\text{Hom}(X,G(Y))\cong\text{Hom}(F(X),Y)\cong\text{Hom}(X,H(Y))

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for all objects X,Y. We see then by Yoneda’s lemma that G(Y)\cong H(Y) naturally for all objects Y. So, G\cong H. The other cases are similar.

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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] Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.

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

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April 15, 2012 - Posted by | Algebra, Category Theory | , , , , , , ,

2 Comments »

  1. […] and are like iterated limits that can be roughly symbolized as and respectively. Since the limit functor is a right adjoint we know that it commutes with limits and so we may conclude that, in fact, . It is easy then to […]

    Pingback by Continuous and Cocontinuous Functors (Pt. II) « Abstract Nonsense | April 15, 2012 | Reply


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