nLab weighted colimit

Idea

A weighted colimit (also called indexed colimit or mean tensor product in older texts) is a concept of colimit suitable for enriched category theory, dual (in the enriched sense) to the concept of weighted limit.

Motivation

Recall that a colimit of a diagram in a category $C$, that is, of a functor $F: J \to C$, is given by a universal cocone for $F$. A cocone for $F$ is a natural transformation from $F$ to a constant diagram

$$\Delta(c) = (J \to 1 \stackrel{c}{\to} C)$$

so that a cocone for $F$ is an object of a comma category

$$F \downarrow \Delta$$

where $\Delta: C^1 \to C^J$ is the diagonal functor obtained by pulling back along the unique functor $J \to 1$. A universal cocone is simply an initial object of $F \downarrow \Delta$.

In enriched category theory, where one considers categories $C$ enriched in a “nice” monoidal category $V$ (generally one where $V$ is complete, cocomplete, closed symmetric monoidal) there is in general no $V$-enriched diagonal functor $\Delta: C \to C^J$ to speak of. For example, when $V$ is the category Ab, we have $C \simeq C^I$ where $I$ is the unit $V$-category having one object $1$ for which $hom(1, 1) = \mathbb{Z}$, but then for a general Ab-enriched category $J$, there is no enriched functor $J \to I$ to pull back along (or, there may be many, but none stand out as canonical). This shows that the usual notion of colimit doesn’t adapt particularly well to the general enriched setting.

The more flexible notion of weighted colimit (also called an indexed colimit in some of the older accounts) was introduced by Borceux (and Kelly?) as giving the right notion of colimit for enriched category theory.

Weighted colimits in ordinary category theory

First we reformulate ordinary colimits in the language of tensor products, in a way that suggests more general weighted colimits.

Assume for the moment that the receiving category $C$ has all coproducts and coequalizers. As is well known, it follows that $C$ has all colimits; the proof is we can write down a formula for the colimit of $F: J \to C$: as a coequalizer of a pair

$$\sum_{j, k \in Ob(J)} hom(j, k) \times F(j) \overset{\to}{\to} \sum_{k \in Ob(J)} F(k) \to colim_J F$$

where the cartesian product on the left refers to a coproduct of copies of $F(j)$ indexed over the set $hom(j, k)$. One of the two parallel arrows is induced by a collection of actions of the category $J$ on $F$, viz.

$$hom(j, k) \times F(j) \to F(k) = \langle F(f): F(j) \to F(k) \rangle_{f: j \to k}$$

and the other is induced by a collection of projections

$$hom(j, k) \times F(j) \to F(j)$$

each of which is the application of the functor $- \times F(j): Set \to C$ to the unique map $!: hom(j, k) \to 1.$

We can think of the map $hom(j, k) \to 1$ also as a component of an action: where $J$ acts on the terminal functor $1: J \to Set$. Or rather, dual to the way in which $J$ acts covariantly on $F$ (so $F$ is a left $J$-module), we will think of $J$ acting contravariantly on the terminal functor $1: J \to Set$ (so that $1$ becomes a right $J$-module).

Then the colimit of $F$ above is precisely a tensor product of the left module $F$ with the right module $1$. More explicitly, the tensor product is the coequalizer of two arrows

$$\sum_{j, k} 1(k) \times hom(j, k) \times F(j) \overset{\to}{\to} \sum_j 1(j) \times F(j)$$

where one arrow is induced from a right action of $J$ on the functor 1, having components

$$\rho_{j, k} \times F(j): (1(k) \times hom(j, k)) \times F(j) \to 1(j) \times F(j)$$

and the other is induced from a left action of $J$ on $F$, having components

$$1(k) \times \lambda_{j, k}: 1(k) \times (hom(j, k) \times F(j)) \to 1(k) \times F(k)$$

From this standpoint, the colimit of $F$ is a rather specialized tensor product of the form

$$1 \otimes_J F$$

and the unsuitability of this notion for general enriched categories could be thought of as a case of putting all one’s eggs in the $1$ basket.

A general right $J$-module $W: J^{op} \to Set$ may be called a weight (with $W(j)$ the weight at $j$). Thus instead of giving all objects $j$ an equal weight $W(j) = 1$, we vary the weight and get a more general notion of colimit (just as weighted averages generalize ordinary averages). More importantly, this notion of weighted colimit makes perfect sense in the context of enriched categories.

Definition

Let $J$ be a small category. Given a functor $W: J^{op} \to Set$ (the weight) and a functor $F: J \to C$ (the diagram), the weighted colimit or tensor product is an object $W * F$ of $C$ together with an isomorphism

$$C(W * F, c) \cong Set^{J^{op}}(W, C(F-, c))$$

that is natural in objects $c$ of $C$. (By the Yoneda lemma, such an isomorphism is induced by a uniquely determined transformation

$$W(j) \to C(F(j), W * F),$$

natural in $j$, which is a weighted analogue of the universal cocone.)

The notion of weighted colimit carries over in straightforward fashion to categories enriched in a complete, cocomplete, closed symmetric monoidal category $V$. In that case, if $J$ is a small $V$-category (that is a $V$-enriched category whose object class is small), and if $F: J \to C$ and $W: J^{op} \to V$ are $V$-functors, then a colimit of $F$ with respect to the weight $W$ is an object $W * F$ of $C$ together with an $V$-natural isomorphism

$$C(W * F, c) \cong V^{J^{op}}(W, C(F-, c))$$

(between $V$-functors in the argument $c$). In fact, we can dispense with the conditions that $V$ be complete, cocomplete, and closed, at the cost of not being able to refer to functor categories $V^{J^{op}}$, without which the notion is conceptually harder to express.

A leitmotif playing in the background is that the category of weights $Set^{J^{op}}$ on $J$ (or $V^{J^{op}}$ in the enriched case) is the free ($V$-enriched) cocompletion of $J$. In other words, if $C$ is a ($V$-)category which is cocomplete in the “right” sense of the word, then every ($V$-)functor $F: J \to C$ extends, uniquely up to unique ($V$-)isomorphism, to a ($V$-)cocontinuous functor

$$\widetilde{F}: V^{J^{op}} \to C$$

which is given by the weighted colimit construction $W \mapsto W * F$.

Examples

• A conical colimit is a weighted colimit where $W(j)=I$ (the monoidal unit of $V$, e.g. $W(j)=1$ in the case $V=Set$). Then $F$ is the diagram.

• A copower is a weighted colimit where $J=1$, the one object category, and $W$ picks out the object of $V$ and $F$ picks out the object of $C$.

• The tensor product of functors is a general example.

Cocompleteness

An enriched category admits coends if it admits conical colimits and copowers. It admits all weighted colimits if it admits coends and copowers. Explicitly, for $W: J^{op} \to Set$ and $F: J \to C$:

$$W * F \cong \int^{j \in J} W(j) \cdot F(j)$$

Thus a category with all conical colimits and copowers is cocomplete.

Related entries

• homotopy weighted colimit

• weighted limit

References

• Very nice descriptions on the n-Cafe by:
  • John Baez
  • Todd Trimble
• Notes by Emily Riehl based on a talk by Mike Shulman Weighted Limits and Colimits