Abstract Nonsense

Subcategories and Skeleta

Point of Post: In this post we discuss the notion of a subcategory, and give a few examples showing the differences between the different kind of subcategories. We also define the notion of Skeleta

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Motivation

Now that we know what categories are we’d like to define the appropriate sub notion, of when a smaller category sits inside a given category. While the notion shall be easy to digest, it’s important to note that there are some subtleties in the subcategory notion. Namely, one may have the impression (having never seen subcategories before) that subcategories must share the same “flavor” as the ambient category. More precisely, there is no need for the subcategory to have more structure than the ambient category (examples of which we shall discuss) nor is there a reason why the morphisms in the subcategory have to be the same as the morphisms in the larger category. These differences are what distinguishes a full subcategory from a non-full subcategory. We then discuss the notion of skeleta which, intuitively, can be thought of as nonredundant description of the entire category, up to isomorphism. In other words, it should contain all the possible isomorphism types of the category, but not contain redundancies, i.e. two distinct objects of the same isomorphism type.

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Subcategory

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Let $\mathcal{C}$ be a given category. Then, a category $\mathcal{S}$ is called a subcategory of $\mathcal{C}$ if $\text{obj}(\mathcal{S})\subseteq\text{obj}(\mathcal{C})$ (where, as usual, we have to be a little tip-toe-through-the-tulips like about the whole class vs. set thing, but inclusion is still fine), for every $A,B\in\text{obj}(\mathcal{S})$ one has $\text{Hom}_\mathcal{S}(A,B)\subseteq\text{Hom}_\mathcal{C}(A,B)$, the identities of objects in $\mathcal{S}$ and the identities of objects in $\mathcal{C}$ are equal, and the composition in $\mathcal{S}$ is just the restriction of the composition in $\mathcal{C}$ to $\mathcal{S}$. Let’s look at some examples,

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The category $\mathbf{Met}_c$ of metric spaces with contractions as morphisms is a subcategory of $\mathbf{Top}$. This is because every metric space is a topological space, and every contraction is automatically continuous.

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The category $\mathbf{Ring}$ of unital rings with unital ring homomorphisms is a subcategory of $\mathbf{Rng}$, the category of rings with ring homomorphisms.

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The category $\mathbf{Ab}$ of abelian groups is a subcategory of $\mathbf{Grp}$ the category of groups which in turn is a subcategory of the category $\mathbf{Mon}$ of monoids.

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The category $\mathbf{FinVect}_k$ of finite-dimensional $k$-spaces is a subcategory of the category $\mathbf{Vect}_k$ of $k$-spaces.

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We further call a subcategory $\mathcal{S}$ of a category $\mathcal{C}$ full if $\text{Hom}_\mathcal{S}(A,B)=\text{Hom}_\mathcal{C}(A,B)$ for all $A,B\in\text{obj}(\mathcal{C})$. Note that in the above, the first two examples are not full subcategories (there are continuous maps which aren’t contractions, and there are ring homomorphisms [the zero map] which aren’t unital) but the rest are.

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We call a subcategory $\mathcal{S}$ of $\mathcal{C}$ isomorphism closed if whenenver $B\in\text{obj}(\mathcal{C})$ is such that there exists $A\in\text{obj}(\mathcal{S})$ of the same isomorphism type, then $B\in\text{obj}(\mathcal{S})$.

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We call a subcategory $\mathcal{S}$ of $\mathcal{C}$ isomorphism dense if for every $B\in\text{obj}(\mathcal{C})$ there exists some $A\in\text{obj}(\mathcal{S})$ of the same isomorphism type. It’s obviously true that $\text{obj}(\mathcal{S})=\text{obj}(\mathcal{C})$ if and only if $\mathcal{S}$ is isomorphism dense and isomorphism closed. These definitions should remind one of the topological notions of closed and dense, and it’s clear that in both contexts being closed and dense implies that the subcategory/subspace is the full category/space.

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A subcategory $\mathcal{S}$ of $\mathcal{C}$ is called a skeleton if it’s full and isomorphism dense, and furthermore no two distinct objects of $\mathcal{S}$ are of the same isomorphism type. Thus, skeleta represent a “transversal” of the isomorphism types of $\mathcal{C}$.

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Assuming the axiom of choice for classes, it’s not hard to see that:

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Theorem: Every category has a skeleton.

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As alluded to, this just comes by applying the axiom of choice to full subcategories not containing distinct objects of the same isomorphism type. To get a feel for skeleta, we luckily have (with such an ample sample space, this isn’t surprising) simple example of a skeleton. Namely, if we look at the category $\mathbf{FinVect}_\mathbb{R}$ of finite-dimensional real vector spaces it’s not hard to see that the full subcategory $\mathcal{S}$ of $\mathbf{FinVect}_\mathbb{R}$ with $\text{obj}\left(\mathcal{S}\right)=\left\{\mathbb{R}^n:n\in\mathbb{N}\cup\{0\}\right\}$ is a skeleton for $\textbf{FinVect}_\mathbb{R}$.

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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.

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