Weighted Limit of a Functor

Given a Copresheaf $W:\mathcal{K}\to\mathrm{Sets}{.}$ and a functor $F:\mathcal{K}\to\mathcal{C}$, a $W$-weighted limit of $F$ consists of

• An object $L:\mathcal{C}$
• For each $k:\mathcal{K}$ and $w:W(k)$, a projection $\pi_{w,k}:L\to F(k)$
• For all $k,k^{\prime}:\mathcal{K}$, $f:{{(}\to{k}},k^{\prime})$, and $w:W(k)$, the following diagram commutes

• For every other such diagram $(X,p_{w,k})$, there exists a unique universal map $u:X\to L$ with $\pi_{w,k}\circ u=p_{w,k}$ for every $w,k$.

Examples

• Every Limit of a functor $F:\mathcal{K}\to\mathcal{C}$ can be expressed as a limit weighted by the Terminal Copresheaf. The converse is also true: every weighted limit can be expressed as a limit.
• Limits over Comma Categories are often best expressed as weighted limits. The canonical example of this is the construction of Ends, where the end of a functor $F:\mathcal{C}mathrm{op}\times\mathcal{C}\to\mathcal{D}$ can neatly be expressed as a limit of $F$ weighted by the Hom Functor ${{-}\to{=}}:\mathcal{C}^{mathrm{op}}\times\mathcal{C}\to\mathrm{Sets}{.}$.
• Power Objects $\prod_{i\in I}X$ are weighted limits over the Terminal Category that are weighted by a constant functor $*\mapsto I$.