Examples of Weighted (co)limits in Algebra

Question

When reading about weighted (co)limits in enriched category theory, most examples are given in the context of $\mathsf{Cat}$-enriched categories (ie. 2-limits in 2-category theory) or $\mathsf{sSet}$-enriched categories (ie. homotopy (co)limits), see for example here.

I find the latter rather hard to work with and while 2-limits are comparatively easy to grasp, they still belong to the category theory bubble (categories for the sake of categories). But there are other easy bases of enrichment (notably $\mathsf{Ab}$, $\mathsf{Mod}_R$ or $\mathsf{Poset}$), for which to my surprise I could not find any discussions of weighted (co)limits. Hence

Besides the tensor product, what are interesting examples of weighted (co)-limits in $\mathsf{Ab}$- or $\mathsf{Mod}_R$-enriched categories?

I've started to compute some random weighted limits, but would like to know whether there are some obvious candidates I didn't see yet. Maybe we are in a similar situation as in $\mathsf{Set}$-enriched categories in that limits and weighted limits have the same strength, in which case a warning would be very welcome. Anyway thank you very much for your considerations.

Answer 1

$\require{AMScd}$One example is 4.2.1 here (although not exactly in a category of R-modules, but in a category of chain complexes): the cone of a chain map $f : X_* \to Y_*$ is a weighted colimit.

Of course, this is not exactly an answer to your question, because this works in a sense you have not abandoned homotopy theory, which is where weighted co/limits become powerful, by virtue of the Bousfield-Kan formula.

In the case of posets, what I would do is to review Kelly's "Elementary observations on 2-categorical limits" and specialise the usual shapes of weights and diagrams to the case of a $\text{Pos}$-enriched diagram, regarding posets as categories (but I expect that the absence of more than 1 2-cell between any two given 1-cells trivialises a lot some constructions). On this line, my advice would be to familiarise with "2-category theory done in a posetal universe", for example reading about monoidal topology or quantales/quantaloids.

Maybe we are in a similar situation as in $\mathsf{Set}$-enriched categories in that limits and weighted limits have the same strength, in which case a warning would be very welcome. Anyway thank you very much for your considerations.

This question is interesting on its own right: as a rule of thumb, every base of enrichment that allows for a "construction of elements" allows to reduce weighted co/limits to conical ones, over said category of elements. By "construction of elements" I mean a fully faithful functor $$ {\cal V}\text{-Cat}({\cal A},{\cal V}) \hookrightarrow \text{DFib}/{\cal A} $$ obtained as follows: to a $\cal V$-presheaf $F$ on $\cal A$ you associate the pullback $$ \begin{CD} E(F) @>>> {\cal V}_\bullet \\ @VVV @VVUV\\ {\cal A} @>>F> {\cal V} \end{CD} $$ along a "universal fibration" $U : {\cal V}_\bullet \to {\cal V}$ (for ${\cal V} = \text{Set}$, $U$ is the forgetful functor from pointed sets).

Answer 2

As you probably know, weighted limits exist as soon as you have cotensors (i.e. weighted limits of singleton diagrams) and conical limits. Since the forgetful functor $\textbf{Ab} \to \textbf{Set}$ is conservative and preserves limits, unenriched limits in any $\textbf{Ab}$-enriched category are automatically (enriched) conical limits. But because $\textbf{Ab}$ has the special property that every object is an iterated (unenriched) colimit of the unit object, conical limits suffice to construct cotensors! Thus, any $\textbf{Ab}$-enriched category whose underlying unenriched category has limits of small diagrams automatically has weighted limits of small diagrams too! Put it another way, maybe the reason why there is not much discussion of weighted limits in $\textbf{Ab}$-enriched categories is because they can be reduced to ordinary limits. All this also works for $R\textbf{-Mod}$ instead of $\textbf{Ab}$, where $R$ is an arbitrary commutative ring.

The situation is different for $\textbf{Cat}$, $\textbf{Poset}$, $\textbf{Top}$, $\textbf{sSet}$ etc. because the relevant forgetful functor is not conservative and also because the unit object is not an iterated-colimit-generator. It seems to me that in practice the use of general weighted limits is avoided by rewriting constructions using cotensors and conical limits instead.