Mike: Are weighted limits and weighted colimits really different enough that they need entirely separate pages?
Todd: I saw a lot of requests for weighted colimits from other pages; thus prompted, I started writing this.
I haven’t been following discussion about duplication of material. I’m guessing it irritates some people, which I don’t understand really. My own guess is that in the present case, it might help having separate pages, rather than force non-experts to deduce what they need to know about weighted colimits by staring at the article on weighted limits. But I’m generally open to discussion.
Toby: For pedagogical purposes, the examples might well be different. In any case, since this is written somewhat differently from weighted limit, we could also consider which is better and edit the other to match (or not, if we want to give its approach a chance too).
Recall that a colimit of a diagram in a category , that is, of a functor , is given by a universal cocone for . A cocone for is a natural transformation from to a constant diagram
so that a cocone for is an object of a comma category
where is the diagonal functor obtained by pulling back along the unique functor . A universal cocone is simply an initial object of .
In enriched category theory, where one considers categories enriched in a “nice” monoidal category (generally one where is complete, cocomplete, closed symmetric monoidal) there is in general no -enriched diagonal functor to speak of. For example, when is the category Ab, we have where is the unit -category having one object for which , but then for a general Ab-enriched category , there is no enriched functor 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.
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 has all coproducts and coequalizers. As is well known, it follows that has all colimits; the proof is we can write down a formula for the colimit of : as a coequalizer of a pair
where the cartesian product on the left refers to a coproduct of copies of indexed over the set . One of the two parallel arrows is induced by a collection of actions of the category on , viz.
and the other is induced by a collection of projections
each of which is the application of the functor to the unique map
We can think of the map also as a component of an action: where acts on the terminal functor . Or rather, dual to the way in which acts covariantly on (so is a left -module), we will think of acting contravariantly on the terminal functor (so that becomes a right -module).
Then the colimit of above is precisely a tensor product of the left module with the right module . More explicitly, the tensor product is the coequalizer of two arrows
where one arrow is induced from a right action of on the functor 1, having components
and the other is induced from a left action of on , having components
From this standpoint, the colimit of is a rather specialized tensor product of the form
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 basket.
A general right -module may be called a weight (with the weight at ). Thus instead of giving all objects an equal weight , 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 be a small category. Given a functor (the weight) and a functor (the diagram), the weighted colimit or tensor product is an object of together with an isomorphism
natural in , 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 . In that case, if is a small -category (that is a -enriched category whose object class is small), and if and are -functors, then a colimit of with respect to the weight is an object of together with an -natural isomorphism
(between -functors in the argument ). In fact, we can dispense with the conditions that be complete, cocomplete, and closed, at the cost of not being able to refer to functor categories , without which the notion is conceptually harder to express.
A leitmotif playing in the background is that the category of weights on (or in the enriched case) is the free (-enriched) cocompletion of . In other words, if is a (-)category which is cocomplete in the “right” sense of the word, then every (-)functor extends, uniquely up to unique (-)isomorphism, to a (-)cocontinuous functor
which is given by the weighted colimit construction .
To be filled in. The tensor product of functors is a general example.