Weighted limits, ends, and Day convolution

Part 1

8th December, 2017
See all parts of this series: Part 1 Part 2 Part 3

A motto of category theory is that ‘Kan extensions are everywhere’. As a simplification of this, ‘(co)limits are in a lot of places’. By rephrasing the definition of a limit we end up with something that looks invitingly generalisable. This is how we can stumble across the idea of a weighted limit. In this post I’m going to assume that you are already convinced of the usefulness and omnipresence of limits and not talk too much (if at all) about why they are interesting in their own right.

Edit. If you actually want to learn this stuff then have a look at either of Emily Riehl’s books (Categorical Homotopy Theory and Category Theory in Context). I honestly don’t think there is a better teaching of this anywhere else in the literature.

This is the first in a series of notes that are basically summaries of various pages on the nLab, along with a few other sources. There is nothing original here, except any mistakes.

Motivation

We start by rephrasing what it means for an object to be the limit of a functor in terms of representable presheaves.

For sets

Let F\colon I^\mathrm{op}\to\mathsf{Set} be a presheaf on a small category I. Then the limit of F over I is the hom-set

\lim_I F := [I,\mathsf{Set}] (\underline{\{*\}},F)

where [I,\mathsf{Set}] is the functor category, and \underline{\{*\}} is the constant functor to a singleton set. Since the covariant hom functor commutes with set-valued limits, we can use the Yoneda lemma, as always, to work with \hat{Y}(\lim_I F) = \mathrm{Hom}_{\mathsf{Set}}(-,\lim_I F) and see that

\mathrm{Hom}_{\mathsf{Set}}(S,\lim_I F) \cong \lim_I\mathrm{Hom}_{\mathsf{Set}}(S,F(-))

for any set S.

For small categories

If we now find ourselves in the more general case of having a functor  F\colon I^\mathrm{op}\to\mathcal{C} for some arbitrary (small) category \mathcal{C} then we can use the above trick: we define the presheaf \widehat{\lim}_I F by using the set-valued limit above, i.e.

(\widehat{\lim}_I F)(c) \cong \hat{\mathcal{C}}(\hat{Y}(c), \widehat{\lim}_I F) := \lim_I\mathsf{Set}(\hat{Y}(c),F(-))

where \hat{\mathcal{C}} is the presheaf category [\mathcal{C}^\mathrm{op},\mathsf{Set}], and we write \mathcal{D}(x,y) to mean \mathrm{Hom}_\mathcal{D}(x,y). But by the definition of the limit given at the start, this can be rewritten (‘’setting F = \mathsf{Set}(\hat{Y}(c),F(-))’’) as

(\widehat{\lim}_I F)(c) \cong \hat{I}\big(\underline{\{*\}},\mathsf{Set}(\hat{Y}(c),F(-))\big).

It can then be shown that the limit of F, as defined in any other classical way, is exactly an object representing the presheaf \widehat{\lim}_I F, i.e.

\mathcal{C}(c, \lim_I F)\cong\hat{I}\big(\underline{\{*\}},\mathsf{Set}(\hat{Y}(c),F(-))\big)

Generalisation

There are two things in the above that look interesting to try to generalise:

1. Consider something more complicated than terminal cones: replace \underline{\{*\}} with some arbitrary functor W\in\hat{I} (that will be called the weight);
2. Work in \mathcal{V}-enriched categories rather than just \mathsf{Set}-enriched (i.e small) ones.

This leads to the following definition. Let \mathcal{C} be a \mathcal{V}-enriched category, and W\colon I^\mathrm{op}\to\mathcal{V} and F\colon I^\mathrm{op}\to\mathcal{C} functors. The W-weighted limit of F is (if it exists) an object \lim^W F\in\mathcal{C} that represents (for c\in\mathcal{C}) the functor

[I,\mathcal{V}]\big(W,\mathcal{C}(c,F(-))\big)\colon\mathcal{C}^\mathrm{op}\to\mathcal{V}.

In particular, if \mathcal{C}=\mathcal{V} then you can show that \lim^W F\cong [I,\mathcal{V}](W,F).

In the next post we’ll have a look at why this generalisation is of any interest, and how we can use it to define (co)ends.