Understanding Weighted Colimits as Tensor Products of Modules

Posted on 2022-10-15  ·  last modified: 2023-03-13  ·  16 min read  ·  maths

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 \mathsf{Set} for \mathcal{V} 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—more-so 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 hom-sets are always sets—one studies so-called \mathcal{V} -categories, whose hom-objects are objects in some “environmental” category \mathcal{V} . This category is what replaces \mathsf{Set} , so it will usually be assumed to have some very nice properties. For the purposes of this blog post, I will assume that (\mathcal{V}, \otimes, 1) 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 \mathcal{V} -category \mathcal{C} 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 a and b in \mathcal{C} , we have that \mathcal{C}(a, b) \mathrel{\vcenter{:}}= \mathrm{Hom}_{\mathcal{C}}(a, b) is an object in \mathcal{V} . 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 \mathcal{C}(a,b) is an object in \mathcal{V} means that it’s a “black box”—we can’t peek into it anymore! Writing f \in \mathcal{C}(a,b) 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 wide-spread the concept really is in mathematics. Thankfully—lest the world explodes—categories enriched in \mathsf{Set} are exactly ordinary categories. An equally familiar example should be \mathsf{vect}_k : the category of finite-dimensional vector spaces over a field k . It is easy to verify that the linear maps between two vector spaces are again a vector space, and hence \mathsf{vect}_k 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 \mathsf{vect}_k , usually called k -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 2-categories
In the strict sense.
as those enriched over \mathsf{Cat} , and preadditive categories, which are enriched over \mathsf{Ab} . Last, but certainly not least, rings can also be seen as categories; namely, they have just a single object \star and \mathrm{Hom}(\star,\star) 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 \mathcal{V} . Formally, our fixed \mathcal{C} consists of:
  • A collection of objects \mathrm{ob}\, \mathcal{C} .
  • For x, y \in \mathcal{C} , a hom-object \mathcal{C}(x, y) \in \mathcal{V} .
  • For x, y, z \in \mathcal{C} , a composition map in \mathcal{V} : \circ_{x, y, z} \colon \mathcal{C}(y, z) \otimes \mathcal{C}(x, y) \longrightarrow \mathcal{C}(x, z).
  • For x \in \mathcal{C} an identities map e_x \colon 1 \longrightarrow \mathcal{C}(x,x) .

Further, this data has to satisfy appropriate associativity and unitality conditions:

axioms for an enriched category

In the above diagrams, \alpha , \lambda , and \rho respectively denote the associativity, left, and right unitality constraints of \mathcal{V} .

If these diagrams remind you of a monoidal category, they absolutely should! Much like you can think of ordinary categories as multi-object monoids, a decent mental model for \mathcal{V} -categories is to think of them as multi-object monoidal categories.

Functors and natural transformations

We furthermore need analogues for functors and natural transformations—they now also come with a \mathcal{V} - 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 F(f \circ g) = Ff \circ Fg 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 \mathcal{C} \longrightarrow \mathcal{V} is the induced arrow on morphisms that they always come with; namely, such a functor F induces an assignment \mathcal{C}(a, b) \longrightarrow \mathcal{V}(F a, F b) . Because \mathcal{V} is symmetric monoidal, we can use its tensor–hom adjunction and rewrite the above to look more like an action:

\mathcal{C}(a, b) \otimes F a \longrightarrow F b.

Likewise, a \mathcal{V} -functor F \colon \mathcal{C}^{\mathrm{op}} \longrightarrow \mathcal{V} is equipped with an action from the other side:

F b \otimes \mathcal{C}(a, b) \longrightarrow F a.

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—non-enriched—closed monoidal category (\mathcal{A}, \otimes_{\mathcal{A}}, 1_{\mathcal{A}}) , we have the tensor–hom adjunction (also called currying) {-} \otimes b \,\dashv\, [b, {-}] . More explicitly, this means that there is a natural isomorphism

\mathcal{A}(a \otimes_{\mathcal{A}} b, c) \cong \mathcal{A}(a, [b, c]), \qquad \text{for } a, b, c \in \mathcal{A}.

If we’re in an enriched setting, we want to somehow replace the tensor product of the monoidal category with some action, say \cdot \colon \mathcal{C} \times \mathcal{V} \longrightarrow \mathcal{C} , while retaining an analogue of the above isomorphism. As such, the copower of c \in \mathcal{C} by v \in \mathcal{V} is an object c \cdot v \in \mathcal{C} , such that for all b \in \mathcal{C} , there is a natural isomorphism

\mathcal{C}(c \cdot v, b) \cong \mathcal{V}(v, \mathcal{C}(c, b)).

Above I have slightly abused notation; \mathcal{V}({-}, {-}) now denotes the internal hom of \mathcal{V} , instead of the external one.
Do remember that \mathcal{C}(a,b) is an object in \mathcal{V} now!
If \mathcal{V} is clear from the context, one also often writes [{-},{-}] .

The best thing about copowers is their existence when it comes to \mathsf{Set} and ordinary categories. If \mathcal{A} has coproducts, there is a canonical copower \cdot \colon \mathsf{Set} \times \mathcal{A} \longrightarrow \mathcal{A} .
If the category \mathcal{A} is locally small. I will ignore those kinds of technicalities for the purposes of this post.
For all X \in \mathsf{Set} and a \in \mathcal{A} , it is given by

X \cdot a \mathrel{\vcenter{:}}= \coprod_{x \in X} 1_{\mathcal{A}} \otimes_{\mathcal{A}} a \cong \coprod_{x \in X} a.

The fact that this is a copower follows from

\mathcal{A}(X \cdot a, b) = \mathcal{A}\left(\coprod_{x \in X} a, b\right) \cong \prod_{x \in X} \mathcal{A}(a, b) \cong \mathsf{Set}(X, \mathcal{A}(a, b)),

for all b \in \mathcal{A} . 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 k-algebra R . We like rings and modules, so let’s explore this further.

To recap, when looking at bimodules A and B over some k -algebra (ring) R we can define the tensor product of A and B over R , in symbols A \otimes_R B , as the coequaliser

A \otimes_R B \mathrel{\vcenter{:}}= \mathrm{coeq} \left( A \otimes R \otimes B \rightrightarrows A \otimes B \right),

where the two parallel arrows are induced by the left and right actions \rhd \colon A \otimes R \longrightarrow A and \lhd \colon R \otimes B \longrightarrow B , respectively.

For ease of notation, I will often write coequalisers like the above one as

A \otimes R \otimes B \rightrightarrows A \otimes B \longrightarrow A \otimes_R B. \tag{1}

Categorifying this notion, the ring R can be seen as a one-object category enriched over \mathsf{Ab} with object 1 . The multiplication is recovered as function composition in R(1, 1) , and addition is given by the abelian structure. A right R -module A is then an enriched functor A \colon R^{\mathrm{op}} \longrightarrow \mathsf{Ab} and similarly a left R-module is an enriched functor B \colon R \longrightarrow \mathsf{Ab} . Inserting the definition discussed above, we have that A consists of a single object A1 and a single arrow A1 \otimes R(1, 1) \longrightarrow A1 . Likewise, we obtain B1 and R(1,1) \otimes B1 \longrightarrow B1 in \mathcal{V} . Thus, we have induced maps

A1 \otimes R(1,1) \otimes B1 \rightrightarrows A1 \otimes B1.

Let us forget about enrichment for a while and just study ordinary categories now. The second observation we need is the well-known fact that any colimit can be represented as a coequaliser. Suppose \mathcal{D} to be a cocomplete category . Given a functor F \colon \mathcal{J} \longrightarrow \mathcal{D} we can express its colimit as

\coprod_{a, b \in \mathcal{J}} \coprod_{f \in \mathcal{J}(a, b)} F a \rightrightarrows \coprod_{b \in \mathcal{J}} F b \longrightarrow \mathrm{colim}_\mathcal{J} F.

Note that we can use what we learned about ( \mathsf{Set} -valued) copowers above and write \coprod_{f \in \mathcal{J}(a, b)} F a as \mathcal{J}(a, b) \cdot F a , or even \mathcal{J}(a, b) \times F a , as \mathcal{J}(a,b) is a set in this case. Behold:

\coprod_{a, b \in \mathcal{J}} \mathcal{J}(a,b) \times F a \rightrightarrows \coprod_{b \in \mathcal{J}} F b \longrightarrow \mathrm{colim}_\mathcal{J} F. \tag{2}

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.

  1. One arrow is induced by the “projection” \pi_2 \colon \mathcal{J}(a, b) \times F a \longrightarrow F a . Note that \mathcal{J}(a, b) \times F a is really a copower, so the existence of such an arrow is not immediately clear. Starting with the unique map ! \colon \mathcal{J}(a, b) \longrightarrow \{\star\} to the terminal object, we apply it to the functor {-} \times F j \colon \mathsf{Set} \longrightarrow \mathcal{C} , in order to obtain

    \pi_2 \!\mathrel{\vcenter{:}}=\; ! \times F a \colon \mathcal{J}(a,b) \times F a \longrightarrow \{\star\} \times F a \cong F a.

  2. The other arrow is induced by a collection of actions of \mathcal{J} on F , indexed by arrows f \colon a \longrightarrow b in \mathcal{J} ; i.e.,

    \begin{align*} (\mathcal{J}(a,b) \times F a \longrightarrow F b) &= \left( \coprod_{f \in \mathcal{J}(a,b)} F a \longrightarrow F b \right) \\ &= \langle Ff \colon Fa \longrightarrow F b \rangle_{f \in \mathcal{J}(a,b)}. \end{align*}

So that’s the story with expressing colimits as coequalisers. We now need to completely reframe this in terms of actions. For the second arrow we are already done: F can be seen as a left \mathcal{J} -module.

Using the symmetry of the Cartesian product \times of sets, the arrow \mathcal{J}(a, b) \longrightarrow \{\star\} can be reinterpreted as the components of a right action of \mathcal{J} on the terminal functor \mathbb{T} \colon \mathcal{J} \longrightarrow \mathsf{Set} that sends every object to the one-element set \{\star\} :

(\mathbb{T}b \times \mathcal{J}(a,b) \longrightarrow \mathbb{T}a) = (\{\star \} \times \mathcal{J}(a,b) \longrightarrow \{\star\}) \cong (\mathcal{J}(a,b) \longrightarrow \{\star\}).

Putting these two observations together, we really have two induced arrows, each with type signature

\mathbb{T} b \times \mathcal{J}(a, b) \times F a \longrightarrow \mathbb{T} a \times F a.

Inserting these into Equation  (2) , this yields

\coprod_{a, b \in \mathcal{J}} \mathcal{J}(a,b) \times F a \cong \coprod_{a, b \in \mathcal{J}} \mathbb{T} b \times \mathcal{J}(a, b) \times F a \rightrightarrows \coprod_{a \in \mathcal{J}} \mathbb{T} a \times F a \cong \coprod_{a \in \mathcal{J}} F a.

This is exactly the way the tensor product of bimodules is defined in Equation  (1) , hence it is tempting to write the resulting coequaliser as \mathbb{T} \otimes_{\mathcal{J}} F . As such, a colimit of a functor F over \mathcal{J} can be seen as a tensor product of functors with the terminal functor. Now, the terminal functor is not very interesting—what if we replace it with something more complicated? Well, that’s exactly the point where weighted colimits come into play! Using a weight W instead of \mathbb{T} , we would end up with something like

\coprod_{a, b \in \mathcal{J}} W b \times \mathcal{J}(a, b) \times F a \rightrightarrows \coprod_{a \in \mathcal{J}} W a \times F a \longrightarrow W \otimes_{\mathcal{J}} F.

Because this looks like a tensor product—and it’s universal, due to it being a colimit—it should also support some form of currying: given an arrow W \otimes_{\mathcal{J}} F \longrightarrow c , for an object c \in \mathcal{C} , we should be able to obtain a map W \Longrightarrow \mathcal{C}(F, c) . Now’s your time to guess what exactly a weighted colimit will be!

Definition

Still in the non-enriched setting, let me now give you the formal definition of a weighted colimit. Suppose \mathcal{J} to be a small category. Let W \colon \mathcal{J}^{\mathrm{op}} \longrightarrow \mathsf{Set} be a presheaf—the weight—and suppose we have a functor F \colon \mathcal{J} \longrightarrow \mathcal{A} . The W -weighted colimit of F comprises an object W \otimes_{\mathcal{J}} F \in \mathcal{A} , equipped with a natural (in a \in \mathcal{A} ) isomorphism

\mathcal{A}(W \otimes_{\mathcal{J}} F, a) \cong [\mathcal{J}^{\mathrm{op}}, \mathsf{Set}] (W, \mathcal{A}(F, a)).

Note that, by the Yoneda lemma, the above isomorphism is uniquely determined by a natural transformation W \Longrightarrow \mathcal{A}(F, W \otimes_{\mathcal{J}} F) , induced by the identity on W \otimes_{\mathcal{J}} F . As promised, this is exactly the representation we arrived at above.

A pair of an object c \in \mathcal{A} and a natural transformation W \Longrightarrow \mathcal{A}(F, c) on their own; i.e., without the universal property, is what one would normally call a W -weighted cocone.

Enriched weighted colimits

The enriched definition is now exactly the same! If \mathcal{J} is a small \mathcal{V} -category and we have \mathcal{V} -functors F \colon \mathcal{J} \longrightarrow \mathcal{C} and W \colon \mathcal{J}^{\mathrm{op}} \longrightarrow \mathcal{V} , then we can define the W -weighted colimit of F as an object W \otimes_{\mathcal{J}} F \in \mathcal{C} , equipped with a \mathcal{V} -natural (in c \in \mathcal{C} ) isomorphism

\mathcal{C}(W \otimes_{\mathcal{J}} F, c) \cong [\mathcal{J}^{\mathrm{op}}, \mathcal{V}] (W {-}, \mathcal{C}(F {-}, c)).

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.

  1. Let our diagram category have two objects and one non-trivial morphism; i.e., \mathcal{J} \mathrel{\vcenter{:}}= \{ \varphi \colon a \longrightarrow b \} . Further, assume that the weight W picks out
    By sending a to \{0, 1\} , b to \{1\} , and \varphi to the arrow \{0, 1\} \longrightarrow \{1\} , which exists because \{1\} is the terminal set.
    the unique arrow \{ 0, 1 \} \longrightarrow \{ 1 \} in \mathsf{Set} . Suppose that the functor F \colon \mathcal{J} \longrightarrow \mathcal{C} sends a, b \in \mathcal{J} to x, y \in \mathcal{C} and \varphi to \theta \colon x \longrightarrow y .

    Again by the Yoneda lemma we have that a cocone is given by a natural transformation W \Longrightarrow \mathcal{C}(F, c) . In this restricted setting, an arrow Wa \longrightarrow \mathcal{A}(Fb, c) just picks out two morphisms. Thus, the whole thing amounts to the commutativity of the following diagram:

    necessary commutative diagram

    In more plain language, the following equation must hold:

    (x \xrightarrow{\;\;\theta\;\;} y \xrightarrow{\;\;g\;\;} c) = (x \xrightarrow{\;\;\theta\;\;} y \xrightarrow{\;\;f\;\;} c).

  2. A slightly more complicated example is the following. Assume again that \mathcal{J} = \{ \varphi \colon a \longrightarrow b \} as above, only this time our choice of enrichment is not \mathsf{Set} , but \mathsf{Cat} . This means that the weight W is now a functor from \mathcal{A}^{\mathrm{op}} to \mathsf{Cat} . Suppose it picks out the arrow

    \{ 0 \;\; 1 \} \hookrightarrow \{ 0 \cong 1 \},

    where the source an target are understood to be categories. In this setting, a weighted cocone becomes something 2-categorical. We still pick out arrows f and g , but since the category we are looking at contains a non-trivial isomorphism, the commutative diagram also becomes more complicated. Namely, we required the commutativity of

    necessary commutative diagram

    Instead of the requiring \theta \circ g to equal \theta \circ f , we now only require the existence of an invertible 2-cell that mediates between the two.

  3. A subcategory \mathcal{D} of \mathcal{E} is said to be dense in \mathcal{E} if we can, in some sense, approximate the objects of \mathcal{E} well enough with objects in \mathcal{D}
    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 \mathsf{Cat} immediately follows from the fact that the nerve functor from \mathsf{Cat} to \mathsf{sSet} is fully faithful.
    (think of the density of \mathbb{Q} inside \mathbb{R} ). 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 finite-dimensional (left-)comodules of any (possibly infinite-dimen­sio­nal) Hopf algebra is dense inside the category of all comodules, which makes them much easier to work with than modules.

    Formally, \mathcal{D} is dense in \mathcal{E} if the restricted Yoneda embedding along the inclusion functor \iota \colon \mathcal{D} \hookrightarrow \mathcal{E}

    \mathcal{E} \longrightarrow [\mathcal{E}^{\mathrm{op}}, \mathsf{Set}] \xrightarrow{\;[\iota, \mathsf{Set}]\;} [\mathcal{D}^{\mathrm{op}}, \mathsf{Set}]

    is still fully faithful. Another way of saying this is that every object e \in \mathcal{E} is the \mathcal{E}(\iota, e) -weighted colimit of \iota . Indeed, the isomorphism we have for a weighted colimit specialised to our situation looks like

    \mathcal{E}(e, a) \cong [\mathcal{D}^{\mathrm{op}}, \mathsf{Set}] (\mathcal{E}(\iota, e), \mathcal{E}(\iota, a)),

    for all a \in \mathcal{E} , which is exactly what it means for the above arrow to be fully faithful.

Exercise: Try to find a weight W such that a W -weighted cocone recovers the normal, unweighted, notion.

Exercise: As you can imagine 1. and 2. can be used to produce all kinds of relations between f and g . As such, prove the following statements:

  • A variant of 1.: in the case of the weight being \{0, 1\} \xrightarrow{\;\;\mathrm{id}\;\;} \{0, 1\} , we obtain a not-necessarily-commutative diagram.

  • A variant of 2.: in the case that the weight is \{ 0 \} \hookrightarrow \{ 0 \longrightarrow 1 \} (i.e., we only have an arrow between 0 and 1 and not necessarily an isomorphism), we get an ordinary (non-invertible) 2-cell 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 [\mathcal{J}^{\mathrm{op}}, \mathcal{V}] is actually very special: it is the free cocompletion of \mathcal{J} . Every functor G \colon \mathcal{J} \longrightarrow \mathcal{A} extends uniquely (up to unique isomorphism) to a cocontinuous functor [\mathcal{J}^{\mathrm{op}}, \mathcal{V}] to \mathcal{A} by the assignment W \mapsto W \otimes_{\mathcal{J}} G ; 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: 0035-4996.

    • 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. 1-136 (link)

    • nLab: enriched category

  • Copowers:

  • Weighted Colimits: