Familiar categorical limits viewed elegantly as weighted limits?

Question

Are there any limits in ordinary category theory that are more elegantly seen as weighted limits?

In $\mathsf{Set}$-enriched category theory, one can say that the limit of $\mathbf{J} \xrightarrow{D} \mathscr{A}$ weighted by $\mathbf{J} \xrightarrow{W} \mathsf{Set}$ can be equivalently expressed as an ordinary limit of $D$ precomposed with the projection functor from the elements of $W$. And an ordinary limit is of course a weighted limit with constant weight.

I'm looking for examples along the lines of the following. The kernel pair of $f:a \to a' \in \operatorname{Ar}\mathscr{A}$ is the limit of $\mathbf{2} \xrightarrow{f} \mathscr{A}$ weighted by $\mathbf{2} \to \mathsf{Set}$ landing on the arrow sending the doubleton to the singleton. (Riehl's Categorical Homotopy Theory, page 100)

Of course colimits are welcome as well; they're limits just like you and me.

Answer 1

Are there any limits in ordinary category theory that are more elegantly seen as weighted limits?

There are many examples of constructions in category theory which are best seen as weighted co/limits, but the notion of weighted co/limits is inherently higher-dimensional, so plain 1-category theory and plain 1-dimensional limits can't capture the intrinsic "2-dimensional aspect" hidden in these constructions.

The slogan is $$ \text{limits} : \text{category theory} = \text{weighted limits} : \text{enriched category theory} $$ so that the "right" notion of co/limit when you "fatten" category theory into enriched category theory fattens the weight accordingly.

Among Riehl's examples, I think those coming from algebraic topology are the most inspiring to see weighted co/limits in action:

1. The mapping cone of $f\colon X\to Y$ is the weighted colimit of $f$, regarded as a functor $\{0\to 1\}\to {\bf Top}$, with the weight $W\colon \{0\to 1\}\to \bf sSet$ sending $i\in\{0,1\}$ to $\Delta[i]$.
2. The suspension of a space $X$ is the homotopy colimit of the diagram $\text{pt}\leftarrow X \to \text{pt}$, and homotopy co/limits can be viewed as weighted co/limits in nice model categories (a nice category of spaces is a nice model category), since
3. every homotopy colimit $\text{ho}\varinjlim X_i$ of a diagram $X\colon J\to \bf Spaces$ can be written as a weigthed colimit $$ \{N(J_{/-}), X\} = \int^{j\in J} N(J_{/j})\otimes X_j $$ (you can also regard this as a "replacement" procedure: there is a canonical weak equivalence $N(J_{/j}) \to \text{pt}$ in a suitable model category of functors $J\to \bf sSet$).
4. The lax colimit of a functor $F\colon {\cal A}\to \bf Cat$ coincides with the weighted colimit of $F$ by the functor ${\cal A} \to \bf Cat$ sending $a$ to the slice ${\cal A}_{a/}$ (arrows $a\to x$ as objects); but this is precisely the Grothendieck construction associated to $F$ (since there is an embeddng ${\bf Set}\to {\bf Cat}$, when $F$ has discrete values this amounts to the "category of elements" construction for the presheaf $F\colon \mathcal A\to \bf Set$)!
5. Several examples of 2-categorical limits like inserters (equalize two 1-cells up to a 2-cell), equifiers (equalize two parallel 2-cells), comma objects (the lax notion of pullback) admit a description in terms of weigthed co/limits: you can read the wonderfully clear paper by kelly, "Elementary observations on 2-categorical limits".
6. Co/ends are co/limits of functors $T\colon {\cal A}^\text{op}\times {\cal A}\to \cal B$, weighted by the hom functor $\hom : {\cal A}^\text{op}\times {\cal A}\to \bf Set$. This is described in Riehl's book and in my notes (Example 4.11, but I copied from Emily!) on coend calculus.

Whew! Hope this long list helped a little bit. :-)

Answer 2

A nice example is the colimit expressing an arbitrary presheaf as a colimit of representables. It's convenient to express this as a weighted colimit for some purposes, and necessary to describe the appropriate generalization to enriched categories.