Abstract Nonsense

Crushing one theorem at a time

Examples of Categories (Revisited)

Point of Post: In this post we more closely examine some examples of categories.

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We are going to be starting to talk a fair amount about categories, and so I thought that it would be helpful to lay out some examples, not only to give us intuition about categories, but to set down some of the notation and recurring characters. We have already defined categories (although, somewhat unsatisfactorily, but it will have to do) and motivated them.

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The Basic Examples

Remark: In all that follows the multiplication (composition) in the categories is just function composition.

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We begin by noting perhaps one of the most important of all categories, the category of all sets with \text{obj}(\mathbf{Set}) equal to the class of all sets, with the morphisms \text{Hom}_{\mathbf{Set}}(X,Y)=Y^X=\left\{f:X\to Y\right\}. We also have the category \mathbf{Set}_\ast of all pointed sets which are just ordered pairs (X,x) where X is a set and x\in X (i.e. sets with a distinguished points). The morphisms of \mathbf{Set}_\ast are just functions which respect the distinguished points, i.e. a morphism f:(X,x)\to (Y,y) is a function f:X\to Y with f(x)=y.

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These are the three main categories of groups. One has that \text{obj}(\mathbf{Grp}) is the class of all groups, \text{obj}(\mathbf{Ab}) is the class of all abelian groups, and in all cases the morphisms are just the group homomorphisms.

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This is the category of all topological spaces, with \text{obj}(\mathbf{Top}) equal to the class of all topological spaces, and \text{Hom}_{\mathbf{Top}}(X,Y) equal to the set of all continuous maps X\to Y. Similar to \mathbf{Set} we have the category \mathbf{Top}_\ast of all pointed topological spaces, which are just topological spaces with a distinguished point, and the morphisms are continuous maps which respect the distinguished points.

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These are the categories of metric spaces with \text{obj}(\mathbf{Met})=\text{obj}(\mathbf{Met}_u)=\text{obj}(\mathbf{Met}_c) the class of all metric spaces and \text{obj}(\mathbf{CompMet}) the class of all complete metric spaces. Now, one has, unlike the previous examples, a difference in which morphisms we take for the different categories. In particular \text{Hom}_{\mathbf{Met}}, \text{Hom}_{\mathbf{CompMet}} consist of the contractions between the spaces, \text{Hom}_{\mathbf{Met}_u} consists of the uniformly continuous maps between the spaces, and \text{Hom}_{\mathbf{Met}_c} consists of the continuous maps between the spaces. Also lurking in here is the category \mathbf{Ban} of all Banach spaces with contractions as morphisms.

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These are the ring categories. Their objects, listed in order of appearance, are the class of all unital rings, class of all rings (not necessarily unital–or at least with no specified 1), commutative unital rings, and commutative rings. Once again, there is a difference in the morphisms. \text{Hom}_{\mathbf{Rng}},\text{Hom}_{\mathbf{CRng}} consist of all ring homomorphisms and \text{Hom}_{\mathbf{Ring}},\text{Hom}_{\mathbf{CRing}} consists of all the unital ring homomorphisms.

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Where in the above we have fixed a particular ring. We then define, obviously, \text{obj}(R\text{-}\mathbf{Mod}) to be the class of all left R-modules, where we require the modules to be unital if R is. Not shockingly \mathbf{Mod}\text{-}R is just the category of all right R-modules. In both cases the morphisms are just R-morphisms.

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\mathbf{Vec}_k, \mathbf{FinDimVec}_k

Here k is a fixed field.  The objects in these categories are the vector spaces over k and the finite dimensional vector spaces over k respectively. In both cases the morphisms are just the linear transformations between the spaces.

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This is the category of all monoids with monoid homomorphisms.

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Degenerate Examples

The above are what are called concrete categories which are, roughly, categories where the objects are sets and the morphisms are maps between the sets. That said, there are a lot of examples of categories which are nothing like concrete categories. For example, every monoid M is canonically a category by defining \mathcal{C} to be the category with \text{obj}(\mathcal{C})=\{M\}, \text{Hom}_\mathcal{C}(M,M)=M, and the composition is just the multiplication in the monoid–clearly \text{id}_M=e (where e is the identity element of M). In fact, it’s not hard to see that, in a sense which can be made more precise, every category is just a quasimonoid, or a “monoid” where the multiplication between any two elements isn’t necessarily defined. Similarly, every group G is a category \mathcal{C} with \text{obj}(\mathcal{C})=\{G\}, \text{Hom}_\mathcal{C}(G,G)=G, and the composition is just G-multiplication. We also have discrete categories which are just categories where \text{Hom}(X,X)=\{\text{id}_X) and \text{Hom}(X,Y)=\varnothing if X\ne Y. It’s not hard to see that discrete categories are reallly just their underlying sets (classes). For example, we could take a category \mathcal{C} with \text{obj}(\mathcal{C})=\{\bullet\} (just some thing) and the morphisms are given by \underset{\circlearrowright}{\bullet}, i.e there is just one morphism the identity, so we can see that \mathcal{C} is just the discrete category on a set with one object. We can also create more categories in this dot manner by saying suppose we had a category consisting of two (distinct) objects and the following morphisms \underset{\circlearrowright}{\bullet}\to\underset{\circlearrowleft}{\bullet} so that there is the identity maps on the two objects and a non-identity map between them. We can generalize this obviously to create a category with some n objects,  labeled \{x_1,\cdots,x_n\} with the identity morphism on each x_i and precisely one morphism x_i\to x_j for i>j and no others. We denote this category by \mathbf{n} (so that we have already discussed \mathbf{1,2}).

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Another interesting example is the category \mathbf{Mat}_R where R is a commutative unital ring R and \text{Hom}(m,n) is the set of all m\times n matrices over R.

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As a last thought we note that we can take the product of categories as follows. If \mathcal{C},\mathcal{D} are two categories we can form the product category \mathcal{C}\times\mathcal{D} where the objects are ordered pairs of objects of \mathcal{C},\mathcal{D} and morphisms are ordered pairs of morphisms (f,g) where f is a \mathcal{C}-morphisms and g is a \mathcal{D}-morphism and we compose the morphisms componentwise.

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[1] Mac, Lane Saunders, Lane Saunders Mac, and Lane Saunders Mac. Categories for the Working Mathematician. New York [etc.: Springer, 1988. Print.

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

[3] Adámek, Jiří, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: Wiley, 1990. Print.


November 2, 2011 - Posted by | Algebra, Category Theory | , , ,


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