Posted on 20221015 · last modified: 20230313 · 16 min read · maths
Contents
If you’ve been doing category theory for any amount of time, you’ll
probably have stumbled upon enriched category theory as a way of
expressing categorical ideas internal to some context other than
Set. Reading into it, you might have come across these foreign
sounding concepts like weighted (co)limits and wondered what that was
all about—and then got lost for a few days, trying to decipher what
Kelly is talking about and why symbols resembling tensor
products are suddenly being thrown around. At least that’s what
happened to me.
After scouring the internet for good resources, I found two really
enlightening blog posts: one by Todd Trimble and the
other by John Baez—and they’re too good not to share.
Plus, people always say that you don’t understand a concept unless you
can explain it to someone else, so here’s my shot at it!
I will assume familiarity with basic notions of category theory (limits,
colimits, adjunctions, monoidal categories, …), as well as elementary
abstract algebra (in particular, rings and modules). If you’re not
comfortable with these and have a lot of time to kill, I recommend
Category Theory in Context by Emily Riehl for the former and
A Course in Algebra by Ernest Vinberg for the latter.
Really, it’s good if you have heard about enriched category theory
before, as this is where weighted colimits tend to naturally crop up a
lot; also because I can’t possibly do the topic justice in a single blog
post. I will still try, of course, but be warned. Even if you’re not
familiar with enriched categories, however, this post might still be of
interest. Weighted colimits do also appear in ordinary category theory,
so feel free to substitute for whenever you
feel like it. On top of that, most of the main
part of the text doesn’t use enrichment at all.
Before we start I must note that—moreso than elsewhere—these are very
much not my own thoughts. I’m just retelling the story in order to
understand it better myself. Sources and resources for everything are
linked at the end. The key insights come from the already
mentioned blog posts by Trimble and Baez,
as well as the accompanying (resulting) nLab
article.
Enriched category theory¶
Before diving into the gory details, enriched category theory is perhaps best explained a bit more intuitively at first. In short, instead ordinary categories—whose homsets are always sets—one studies socalled categories, whose homobjects are objects in some “environmental” category . This category is what replaces , so it will usually be assumed to have some very nice properties. For the purposes of this blog post, I will assume that is a (small) complete and cocomplete closed symmetric monoidal category.This is also sometimes called a cosmos.
If you don’t know what some of these
words mean, you can read that as “It’s an environment with enough
structure so that a large chunk of ordinary category theory makes sense
internal to it.”
In addition, I would also like to fix a category
for the rest of this blog post. For the moment, you can
think of this like an ordinary category such that for any two objects
and in , we have that is an object in .
Naturally, all the usual axioms of a category—like associativity and
unitality of morphisms—ought to hold in this new setting; however,
expressing these laws is a little bit more involved now. The fact that
is an object in means that it’s a
“black box”—we can’t peek into it anymore! Writing is no longer legal, so we somehow have to make do with
not talking about individual morphisms. As such, a little bit more care
has to be taken for the precise definition of an enriched category to
make sense.
Before we get to that, however, a few examples should do wonders for
seeing just how widespread the concept really is in mathematics.
Thankfully—lest the world explodes—categories enriched in
are exactly ordinary categories. An equally familiar example should be
: the category of finitedimensional vector spaces over
a field . It is easy to verify that the linear maps between two
vector spaces are again a vector space, and hence is,
much like the category of sets, enriched over itself. So whenever you
do linear algebra, you’re in the setting of enriched category theory
already! Categories enriched over , usually called
linear categories, are plentiful “in the wild”; for example,
representation theorists might know Tannakian categories, or the
Temperley–Lieb category.
Other examples of enriched categories include 2categoriesIn the strict sense.
as those
enriched over , and preadditive categories, which are
enriched over . Last, but certainly not least, rings can
also be seen as categories; namely, they have just a single object
and forms an abelian group—stay
tuned for more on this.
With all of these examples in mind, let us explore the technical
definition of a category enriched over . Formally, our
fixed consists of:
 A collection of objects .
 For , a homobject .
 For , a composition map in :
 For an identities map .
Functors and natural transformations¶
We furthermore need analogues for functors and natural transformations—they now also come with a  prefix. The functor laws get a bit more complicated, as we need to draw commutative diagrams and can’t simply express this property in an equation like anymore—remember that we can’t talk about individual arrows. However, most of the intuition one already has about functors and natural transformations should carry over just fine. I will leave the technical definitions of enriched functors and natural transformations as exercises to the reader; they are relatively straightforward to write down and not all that important for what follows. Thinking further, the upshot one will arrive at is that, in order to do enriched category theory, we not only need analogues for functors and natural transformations, but also for all the other basic notions of ordinary category theory. Since limits and colimits are among the most important constructions, people naturally started to think about how one could express them in the enriched language—this is precisely what lead to the development of weighted colimits! One interesting thing I want to highlight about enriched functors of the form is the induced arrow on morphisms that they always come with; namely, such a functor induces an assignment . Because is symmetric monoidal, we can use its tensor–hom adjunction and rewrite the above to look more like an action: Likewise, a functor is equipped with an action from the other side: This already frames functors as little more than generalised modules, and we will explore this connection in more detail later on.Copowers¶
One more important technical detail has to be covered before we get to the fun stuff: copowers. The basic idea is that in any ordinary—nonenriched—closed monoidal category , we have the tensor–hom adjunction (also called currying) . More explicitly, this means that there is a natural isomorphism If we’re in an enriched setting, we want to somehow replace the tensor product of the monoidal category with some action, say , while retaining an analogue of the above isomorphism. As such, the copower of by is an object , such that for all , there is a natural isomorphism Above I have slightly abused notation; now denotes the internal hom of , instead of the external one.Do remember that is an object in now!
If is clear from the context, one also often writes
.
The best thing about copowers is their existence when it comes to
and ordinary categories. If has
coproducts, there is a canonical copower .If the category is locally small. I will ignore
those kinds of technicalities for the purposes of this post.
For all and , it is given by
The fact that this is a copower follows from
for all . Because of their closeness to the tensor
product, people sometimes call copowers “tensors” and write them with
the same symbol as they write the tensor product.
Weighted colimits¶
Onto the main dish. The key idea is to reframe an ordinary colimit in terms of “looking like a monoidal product”. The weighted colimit then becomes something akin to the tensor product over a kalgebra . We like rings and modules, so let’s explore this further. To recap, when looking at bimodules and over some algebra (ring) we can define the tensor product of and over , in symbols , as the coequaliser where the two parallel arrows are induced by the left and right actions and , respectively. For ease of notation, I will often write coequalisers like the above one as Categorifying this notion, the ring can be seen as a oneobject category enriched over with object . The multiplication is recovered as function composition in , and addition is given by the abelian structure. A right module is then an enriched functor and similarly a left Rmodule is an enriched functor . Inserting the definition discussed above, we have that consists of a single object and a single arrow . Likewise, we obtain and in . Thus, we have induced maps Let us forget about enrichment for a while and just study ordinary categories now. The second observation we need is the wellknown fact that any colimit can be represented as a coequaliser. Suppose to be a cocomplete category . Given a functor we can express its colimit as Note that we can use what we learned about (valued) copowers above and write as , or even , as is a set in this case. Behold: What’s left is to define the two parallel arrows.I still mostly follow Trimble and the nLab here. A more explicit
description—in the case of limits—is given, for example, in
Riehl’s Category Theory in Context, Theorem 3.2.13.
 One arrow is induced by the “projection” . Note that is really a copower, so the existence of such an arrow is not immediately clear. Starting with the unique map to the terminal object, we apply it to the functor , in order to obtain
 The other arrow is induced by a collection of actions of on , indexed by arrows in ; i.e.,
Definition¶
Still in the nonenriched setting, let me now give you the formal definition of a weighted colimit. Suppose to be a small category. Let be a presheaf—the weight—and suppose we have a functor . The weighted colimit of comprises an object , equipped with a natural (in ) isomorphism Note that, by the Yoneda lemma, the above isomorphism is uniquely determined by a natural transformation , induced by the identity on . As promised, this is exactly the representation we arrived at above. A pair of an object and a natural transformation on their own; i.e., without the universal property, is what one would normally call a weighted cocone.Enriched weighted colimits¶
The enriched definition is now exactly the same! If is a small category and we have functors and , then we can define the weighted colimit of as an object , equipped with a natural (in ) isomorphism This is the power of the formalism we developed—the definition extends in a straightforward way to the enriched setting. This may now be used to great effect: among other things weighted colimits can be used to define the right notion of enriched coend.Examples¶
It’s probably about time for some examples. For the first two, let us focus on cocones only; not thinking about the universal property at first is perhaps a little easier to understand—at least it was for me. I learned these from Richard Garner during bcqt 2022.
Let our diagram category have two objects and one nontrivial
morphism; i.e., . Further, assume that the weight picks outBy sending to , to , and to the arrow , which exists because is the terminal set.the unique arrow in . Suppose that the functor sends to and to . Again by the Yoneda lemma we have that a cocone is given by a natural transformation . In this restricted setting, an arrow just picks out two morphisms. Thus, the whole thing amounts to the commutativity of the following diagram:

A slightly more complicated example is the following. Assume again
that as
above, only this time our choice of enrichment is not ,
but . This means that the weight is now a functor
from to . Suppose it picks
out the arrow
where the source an target are understood to be categories. In this
setting, a weighted cocone becomes something 2categorical. We still
pick out arrows and , but since the category we are looking at
contains a nontrivial isomorphism, the commutative diagram also
becomes more complicated. Namely, we required the commutativity of

A subcategory of is said to be dense in
if we can, in some sense, approximate the objects of
well enough with objects in These things are actually all other the place: for example, the Yoneda embedding itself is dense if the category is essentially small. Further, if you like higher category theory, you will appreciate that the simplex category being dense in immediately follows from the fact that the nerve functor from to is fully faithful.(think of the density of inside ). Dense categories are nice because they often tell us a lot about their super categories and are sometimes easier to reason about. For example, the category of finitedimensional (left)comodules of any (possibly infinitedimensional) Hopf algebra is dense inside the category of all comodules, which makes them much easier to work with than modules. Formally, is dense in if the restricted Yoneda embedding along the inclusion functor is still fully faithful. Another way of saying this is that every object is the weighted colimit of . Indeed, the isomorphism we have for a weighted colimit specialised to our situation looks like for all , which is exactly what it means for the above arrow to be fully faithful.
 A variant of 1.: in the case of the weight being , we obtain a notnecessarilycommutative diagram.
 A variant of 2.: in the case that the weight is (i.e., we only have an arrow between 0 and 1 and not necessarily an isomorphism), we get an ordinary (noninvertible) 2cell as the weighted cocone.
Conclusion¶
And that’s it! I’ve found this intuition very helpful in trying to wrap my head around these concepts—hopefully other people will too. As a parting gift, I leave you with some more things to think about. First, one of the most important examples of weighted colimits—and coends, of course—is the tensor product of functors. If you ever wanted to be a ninja, now is the time! It’s a fun operation to think about and play around with, and I would invite you to do just that. Lastly, the category of weights is actually very special: it is the free cocompletion of . Every functor extends uniquely (up to unique isomorphism) to a cocontinuous functor to by the assignment ; note the tensor product of functors!.(Re)sources¶

Monoidal Category Theory:
 Saunders Mac Lane: “Natural associativity and commutativity”. In: Rice Univ. Stud. 49.4 (1963), pp. 28–46. issn: 00354996.
 Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik: “Tensor categories”. In: Vol. 205. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015, pp. xvi+343.
 nLab: monoidal category

Enriched Category Theory:
 Max Kelly: “Basic concepts of enriched category theory”. In: London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp. (ISBN:9780521287029). Republished as: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1136 (link)
 nLab: enriched category
 Copowers:

Weighted Colimits:
 Todd Trimble on the nCategory Café
 John Baez on the nCategory Café
 nLab: weighted colimit
 Richard Garner: Bicategories; lecture series at bcqt 2022, Leeds.
 Emily Riehl: “Weighted Limits and Colimits”; lecture notes.