The Unapologetic Mathematician

Mathematics for the interested outsider

Enriched Categories

I’d like to move on now to another way of blending various structures. We’ve seen that in certain situations the set of morphisms between two objects in a category naturally has deeper structure itself. For example, the set of homomorphisms between two abelian groups is itself an abelian group, because abelian groups are modules over the commutative ring \mathbb{Z}. More generally, the set of homomorphisms between two R-modules naturally has the structure of a \mathbb{Z}-module, and sometimes more.

We need a good way of talking about this sort of thing, where we replace hom sets by “hom objects” in some other category \mathcal{V}. When this happens we say that our category is “enriched” over \mathcal{V}. So to rephrase what I said above, the category of R-modules is enriched over \mathbf{Ab}. Similarly, locally small categories are enriched over \mathbf{Set}.

When we talk about categories — which usually for us means locally small categories — we are implicitly using a number of properties of \mathbf{Set}. In particular, to set up compositions we need to be able to take pairs of morphisms, which the cartesian product handles for us nicely: \hom_\mathcal{C}(B,C)\times\hom_\mathcal{C}(A,B). We also need to be able to pick out a special morphism in each set of endomorphisms to be the identity, which we can take to be the image of a function from a one-point set to the set of endomorphisms \hom_\mathcal{C}(C,C) sort of like we did for monoid objects.

For setting up the relations a category must satisfy we need to be able to build triples from pairs in two ways:
(\hom_\mathcal{C}(C,D)\times\hom_\mathcal{C}(B,C))\times\hom_\mathcal{C}(A,B)\cong
\hom_\mathcal{C}(C,D)\times(\hom_\mathcal{C}(B,C)\times\hom_\mathcal{C}(A,B))
We also need to pair a morphism with a (unique) identity morphism:
\hom_\mathcal{C}(A,B)\cong\hom_\mathcal{C}(A,B)\times\{*\}\rightarrow\hom_\mathcal{C}(A,B)\times\hom_\mathcal{C}(A,A)

What are the important properties of the category of sets that make it useful for these purposes? It’s just the fact that \mathbf{Set} equipped with finite products (including a singleton set as terminal object) is a monoidal category! So let’s take a monoidal category \mathcal{V} — a useful example to have always at hand is \mathbf{Ab} — and try to use it for enrichment. As we proceed, we’ll write \mathcal{V}_0 for the underlying regular category (that is, forget that \mathcal{V} is monoidal).

So, given such a monoidal category \mathcal{V} we’ll define a \mathcal{V}-category \mathcal{C} to consist of a class of objects \mathrm{Ob}(\mathcal{C}), and for each pair (A,B) of objects a “hom-object” \hom_\mathcal{C}(A,B)\in\mathcal{V}_0. For each triple of objects (A,B,C) there is a composition \circ:\hom_\mathcal{C}(B,C)\otimes\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(A,C). For each object A there is an “identity”, described by an arrow i:\mathbf{1}\rightarrow\hom_\mathcal{C}(A,A).

I’ll be spending some time on this, so let’s leave it at the definition for now. Go through and unpack it for the case of an \mathbf{Ab}-category, and see what the definition says such a thing should look like.

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August 13, 2007 - Posted by | Category theory

11 Comments »

  1. [...] Categories II So we have the basic data of a category enriched over a monoidal category . Of course, what I left out were the relations that have to hold. And [...]

    Pingback by Enriched Categories II « The Unapologetic Mathematician | August 14, 2007 | Reply

  2. [...] Here’s another example of an enriched category. This one is extremely important, and to a certain extent it’s been my goal in my coverage of [...]

    Pingback by 2-Categories « The Unapologetic Mathematician | August 16, 2007 | Reply

  3. [...] Underlying Category In the setup for an enriched category, we have a locally-small monoidal category, which we equip with an “underlying set” [...]

    Pingback by The Underlying Category « The Unapologetic Mathematician | August 20, 2007 | Reply

  4. [...] Internal Monoidal Product As we’re talking about enriched categories, we’re always coming back to the monoidal category . This has an underlying category , which [...]

    Pingback by The Internal Monoidal Product « The Unapologetic Mathematician | August 28, 2007 | Reply

  5. [...] So far in our treatment of enriched categories we’ve been working over a monoidal category , and we latter added the assumption that is [...]

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  6. [...] Let’s consider two categories enriched over a monoidal category — and — and assume that is equivalent to a small category. [...]

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  7. [...] said before that the category of vector space is enriched over itself. That is, if we have vector spaces and over the field , the set of linear [...]

    Pingback by Matrices II « The Unapologetic Mathematician | May 22, 2008 | Reply

  8. Please email any recent reference or event regarding

    Category Theory and Interoperability

    thank you

    Comment by Dr. Charalampos MELETIS | May 31, 2008 | Reply

  9. [...] category of matrices is actually enriched over the category of vector spaces over . This means that each set of morphisms is actually a [...]

    Pingback by The Category of Matrices I « The Unapologetic Mathematician | June 2, 2008 | Reply

  10. [...] represent the functor. Now, I believe that this means that the category of schemes over is enriched over itself, though I’ve never seen anyone say that, so I’m not sure if it’s [...]

    Pingback by The Hilbert Scheme « Rigorous Trivialities | July 19, 2008 | Reply

  11. [...] a monoid is just a category with a single object. Similarly, an -algebra is just like a monoid but enriched over the category of vector spaces over . That is, it’s a one-object category with an [...]

    Pingback by Category Representations « The Unapologetic Mathematician | October 27, 2008 | Reply


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