Abstract Nonsense

The Opposite Category

Point of Post: In this post we define, mainly for referential purposes, the opposite category and discuss some of the obvious theorems.

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Motivation

In this post we shall discuss an interesting way to create a new category out of an old one. The basic idea is to create a category with the same objects as our inital category but “turn all the arrows around” and “compose in the opposite way”. In this sense it has a very similar feel to the opposite ring. This category is interesting because it allows us to talk about contravariant functors as covariant functors, and also creates an interesting way of generating new, interesting categories. Also, some of the most deep theorems in mathematics have to do with a kind of “equivalence of categories” (which will be discussed soon enough) for which take the form “$\mathcal{C}$ is equivalent to $\mathcal{D}^{\text{op}}$” where $\mathcal{C,D}$ are common categories. This allows us to say meaningful things about the category $\mathcal{C}$ by interpreting things in $\mathcal{D}^{\text{op}}$. The actual case that comes to mind, if any reader is aware of/heard of these terms, is that $\mathbf{Aff}$, the category of affine schemes, is equivalent to $\mathbf{CRing}^{\text{op}}$–a fundamental fact in algebraic geometry.

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The Opposite Category

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Let $\mathcal{C}$ be a given category. We then define the opposite category of $\mathcal{C}$, denoted $\mathcal{C}^{\text{op}}$, to be the category with $\text{obj}(\mathcal{C}^{\text{op}})\overset{\text{def.}}{=}\text{obj}\left(\mathcal{C}\right)$, formal morphisms $X\xleftarrow{f^{\text{op}}}Y$ for every morphism $X\xrightarrow{f}Y$ in $\mathcal{C}$, and composition $g^{\text{op}}\circ f^{\text{op}}\overset{\text{def.}}{=}(f\circ g)^{\text{op}}$. It’s a straightforward task to check that $\mathcal{C}$ really is a category. Moreover, it’s clear that the map $x\mapsto x$ and $f\mapsto f^{\text{op}}$ is a functor $^{\text{op}}:\mathcal{C}\to\mathcal{C}^{\text{op}}$. It’s easy to see that $^{\text{op}}\circ^{\text{op}}:\mathcal{C}\to\left(\mathcal{C}^{\text{op}}\right)^{\text{op}}$ is a actually an isomorphism of categories, and so there are the usual identifications made between $\mathcal{C}$ and $(\mathcal{C}^\text{op})^{\text{op}}$ (i.e. often times we shall treat them as literally the same category).

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The fact that the functor $^{\text{op}}$ is an “involution” (once we make the obvious identifications) allows us to conclude, for example, that a morphism $f$ is an isomorphism if and only if $f^{\text{op}}$ is an isomorphism. We have obvious extensions of these theorems for all the types of morphisms, but I’d like to mention one in particular:

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Theorem: Let $\mathcal{C}$ be a category and $x\xrightarrow{f}y$ a morphism in $\mathcal{C}$. Then, $f$ is monic if and only if $y\xrightarrow{f^{\text{op}}}x$ is epic in $\mathcal{C}^{\text{op}}$.

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As now obvious, it’s easy to see that contravariant functors $F:\mathcal{C}\to\mathcal{D}$ are heavily related to covariant functors $F:\mathcal{C}\to\mathcal{D}^{\text{op}}$. Namely, a mapping $F:\mathcal{C}\to\mathcal{D}$ is a contaravariant functor if and only if $^\text{op}\circ F:\mathcal{C}\to\mathcal{D}^{\text{op}}$ is a covariant functor. Or, in fact, it’s easy to see that the putting the ‘op’ on the other category also gives the same theorem, namely that $F:\mathcal{C}\to\mathcal{D}$ is a covariant functor if and only if $F\circ^\text{op}:\mathcal{C}^\text{op}\to\mathcal{D}$ is a covariant functor.

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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.

December 30, 2011 -