5
$\begingroup$

Motivation

Let $\mathcal{C}$ be the category of commutative unital rings, and let $R \in \mathcal{C}$. The category of commutative unital R-algebras can be defined as follows:

  • Objects: morphisms of $\mathcal{C}$ of the form $R \to A$
  • Morphisms: commutative triangles in $\mathcal{C}$ of the form $\begin{array}{ccc} R & \to & A\\ & \searrow & \downarrow \\ & & B \end{array}$

Question

Is this construction useful in general? Are there other examples of the above construction (or its dual) yielding a useful/interesting/well-known category, if we let $\mathcal{C}$ be some category other than $\operatorname{CRing}$?

  • 0
    Aside: one normally reserves the notation $\hom_{\mathcal{C}}(R, -)$ for the representable functor $X \mapsto \hom_{\mathcal{C}}(R, X)$. – 2017-02-14
  • 0
    Of course you are right. What I had in mind was $\bigcup_{A \in \mathcal{C}} \operatorname{Hom}_{\mathcal{C}}(R,A)$. Fixed. – 2017-02-14

1 Answers 1

4

These constructions which are generally known as slice categories are very useful, and are special cases of comma categories. Tons of mathematical objects satisfy universal properties which boil down to being an initial or terminal object in some comma category.

To elaborate, let $\mathcal{C}$ be a category and let $A$ be an object of $\mathcal{C}$. We then form the coslice category $(A\downarrow\mathcal{C})$ whose objects are morphisms $f:A\to X$ in $\mathcal{C}$, and whose morphisms: $$(f:A\to X)\xrightarrow{\ \phi \ }(g:A\to Y)$$ are morphisms $\phi:X\to Y$ such that $\phi\circ f=g$. If $i:A\to Q$ is initial in this category, then that means that for every object $f:A\to X$, there is a unique map $\phi:Q\to X$ such that $f=\phi\circ i$. So, in these categories, the initial objects tend to capture the notion of unique factorization of maps, which pops up everywhere.

I've explained the universal property of quotient spaces from this point of view in this question.

You can find a couple more examples of comma categories here.

  • 0
    Thank you for your answer! Do you know of any other examples of (co)slice categories? All I could find in the links you provided was the pointed sets/spaces category. Also, any reference you can provide to more examples of use of (co)slice categories (e.g. in some proof/theory which would be significantly less concise if it made no use of these categories) is most welcome! – 2017-02-14
  • 0
    @posilon Happy to help! You can find a couple more examples in [this question](http://math.stackexchange.com/questions/994256/whats-a-good-motivating-example-for-the-concept-of-a-slice-category/994302#994302), where Martin Brandenburg also explains their use in algebraic geometry. – 2017-02-14