weighted colimit


A weighted colimit is a concept of colimit suitable for enriched category theory, dual (in the enriched sense) to the concept of weighted limit.


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

Δ(c)=(J1cC)\Delta(c) = (J \to 1 \stackrel{c}{\to} C)

so that a cocone for FF is an object of a comma category

FΔF \downarrow \Delta

where Δ:C 1C J\Delta: C^1 \to C^J is the diagonal functor obtained by pulling back along the unique functor J1J \to 1. A universal cocone is simply an initial object of FΔF \downarrow \Delta.

In enriched category theory, where one considers categories CC enriched in a “nice” monoidal category VV (generally one where VV is complete, cocomplete, closed symmetric monoidal) there is in general no VV-enriched diagonal functor Δ:CC J\Delta: C \to C^J to speak of. For example, when VV is the category Ab, we have CC IC \simeq C^I where II is the unit VV-category having one object 11 for which hom(1,1)=hom(1, 1) = \mathbb{Z}, but then for a general Ab-enriched category JJ, there is no enriched functor JIJ \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 CC has all coproducts and coequalizers. As is well known, it follows that CC has all colimits; the proof is we can write down a formula for the colimit of F:JCF: J \to C: as a coequalizer of a pair

j,kOb(J)hom(j,k)×F(j) kOb(J)F(k)colim JF\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)F(j) indexed over the set hom(j,k)hom(j, k). One of the two parallel arrows is induced by a collection of actions of the category JJ on FF, viz.

hom(j,k)×F(j)F(k)=F(f):F(j)F(k) f:jkhom(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)×F(j)F(j)hom(j, k) \times F(j) \to F(j)

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

We can think of the map hom(j,k)1hom(j, k) \to 1 also as a component of an action: where JJ acts on the terminal functor 1:JSet1: J \to Set. Or rather, dual to the way in which JJ acts covariantly on FF (so FF is a left JJ-module), we will think of JJ acting contravariantly on the terminal functor 1:JSet1: J \to Set (so that 11 becomes a right JJ-module).

Then the colimit of FF above is precisely a tensor product of the left module FF with the right module 11. More explicitly, the tensor product is the coequalizer of two arrows

j,k1(k)×hom(j,k)×F(j) j1(j)×F(j)\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 JJ on the functor 1, having components

ρ j,k×F(j):(1(k)×hom(j,k))×F(j)1(j)×F(j)\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 JJ on FF, having components

1(k)×λ j,k:1(k)×(hom(j,k)×F(j))1(k)×F(k)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 FF is a rather specialized tensor product of the form

1 JF1 \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 11 basket.

A general right JJ-module W:J opSetW: J^{op} \to Set may be called a weight (with W(j)W(j) the weight at jj). Thus instead of giving all objects jj an equal weight W(j)=1W(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.


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

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

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

W(j)C(F(j),WF),W(j) \to C(F(j), W \cdot F),

natural in jj, 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 VV. In that case, if JJ is a small VV-category (that is a VV-enriched category whose object class is small), and if F:JCF: J \to C and W:J opVW: J^{op} \to V are VV-functors, then a colimit of FF with respect to the weight WW is an object WFW \cdot F of CC together with an VV-natural isomorphism

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

(between VV-functors in the argument cc). In fact, we can dispense with the conditions that VV be complete, cocomplete, and closed, at the cost of not being able to refer to functor categories V J opV^{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 opSet^{J^{op}} on JJ (or V J opV^{J^{op}} in the enriched case) is the free (VV-enriched) cocompletion of JJ. In other words, if CC is a (VV-)category which is cocomplete in the “right” sense of the word, then every (VV-)functor F:JCF: J \to C extends, uniquely up to unique (VV-)isomorphism, to a (VV-)cocontinuous functor

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

which is given by the weighted colimit construction WWFW \mapsto W \cdot F.


To be filled in. The tensor product of functors is a general example.


Last revised on January 30, 2021 at 11:02:14. See the history of this page for a list of all contributions to it.