Weighted limits in the $Cat$-category of categories

Question

What is a weighted limit in the $Cat$-category of categories, functors and natural transformations?

I can find the general definition of a weighted limit for enriched categories in Kelly's book or Borceux's handbook. But I get lost when I try to specialize, simplify and beautify it.

I expect an elementary definition in the form: "given $W, F : I \to Cat$, the weighted limit $\{W, F\}$ is the category $C$ and the functors... blah blah... such that there exists a unique... blah blah... such that the following diagram commutes: ... blah blah..."

Answer 1

Is coend-calculus elementary enough to you? If yes, supposing the involved powers exist, there is a canonical isomorphism $$ \{W,F\}\cong \int_c Fc^{Wc} $$ for $F\colon C\to A$, and $W\colon C\to V$, where $A,C\in V\text{-Cat}$. This, together with the fact that $$ \int_c Fc^{Wc} \cong \text{eq}\Big(\prod_{c\in C}Fc^{Wc}\rightrightarrows \prod_{c\to d} Fd^{Wc}\Big) $$ allows you to prove a number of intriguing properties of the weigthed limit $\{W,F\}$ in a completely formal way and it allows you to notice that weigthed limits are all around you:

1. When $V$ admits a "category of elements" construction, then weighted limits can be reduced to conical ones: if $\Sigma\colon Elts(W)\to C$ is the forgetful functor, then $$ \{W,F\}\cong \varprojlim_{(c,x)\in Elts(W)} F\circ \Sigma. $$
2. the (ptwise) right Kan extension $\text{Ran}_GF$ is the weighted limit $\{\hom(1,G),F\}$
3. when weighted limits always exist in $A$, the correspondence $(W,F) \mapsto \{W,F\}$ is a bifunctor: $$ \{\sim,\approx\}\colon \big(V^C \big)^\text{op}\times A^C\longrightarrow A. $$
4. There is an iso $\{\varinjlim_J W_j, F\}\cong \varprojlim_J \{W_j,F\}$ valid for any small diagram of weights $J\to [C,V]\colon j\mapsto W_j$.
5. Ends are weighted limits: given $H\colon C^\text{op}\times C\to D$ the hom functor plays the role of a weight $\hom_C(\sim,\approx)\colon C^\text{op}\times C\to V$ so that $$ \{\hom_C,H\}\cong \int_{(c,c')\in C^\text{op}\times C } H(c,c')^{\hom(c,c')}\overset{\text{Fubini}}\cong \int_c \Big( \int_{c'}H(c,c')^{\hom(c,c')}\Big)\stackrel{\text{Yoneda}}{\cong}\int_c H(c,c). $$

This language encodes both an intuition and a set of practical rules of manipulations, and holds in every $V$-category (with sufficient tensors). You are interested in the case when $V=Cat$ though. You have already been advised to read Kelly's "Elementary observations". I advise you to rephrase the universal properties of the weighted limits described in that paper in terms of this characterization. I try to write down the details of a specific example (4.2 in Kelly's elementary obs.): let $V=Cat$, and $C = \{0\to 2\leftarrow 1\}$; let $W,F$ be defined respectively by the diagrams $$ \begin{array}{ccc} && {\bf 1}\\ &(W)&\downarrow\\ {\bf 1} &\to& {\bf 2} \end{array} $$ and $$ \begin{array}{ccc} && A\\ &(F)&\downarrow g\\ B &\underset{f}\to& D \end{array} $$ where ${\bf 2}=\{a\to b\}$, and the two arrows are the canonical "face maps" choosing/avoiding $a$ or $b$. You can have fun showing that there are two maps $$ B\times A\times D^{\bf 2}\rightrightarrows B^{\bf 2}\times A^{\bf 2}\times B\times A\times D^{\bf 2} $$ one induced by $F$- and the other induced by $W$-action on arrows of $C$, so that their equalizer is precisely the category "something in $A$, something in $B$, plus an arrow $fb\to ga$". This is the comma category $(f/g)$, which yes, can also be characterized as a 2-pullback, giving you more intuition about the fact that various notions of "lax" limit can be interpreted as weighted limits.

Additional fun: find the category of elements (i.e. the "Grothendieck construction") of $W$ and see if $(f/g)$ is the limit of $F\Sigma$ over $Elts(W)$ (hint: it is).

Cheers!