# Weighted limits in the $Cat$-category of categories

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..."

• Weighted limits in $\mathcal{V}$ considered as a $\mathcal{V}$-enriched category are just hom-objects of the respective $\mathcal{V}$-enriched functor categories. – Zhen Lin May 23 '16 at 18:12
• Nice remark but I am looking for something more concrete. – Bob May 23 '16 at 19:31
• That is very concrete and elementary. Unless you think that hom-categories are not? – Zhen Lin May 23 '16 at 23:06
• @Bob, Also, the general definition that you refer to defines the weighted limit to be a representation of a certain $\textbf{Cat}$-valued $2$-functor. So you should make sure you understand exactly what a representation is. You should then be able to translate between the "representation" and the "elementary" expressions of the universal property. – Alexander Campbell May 24 '16 at 1:53
• The weighted limit $\{W, F\}$ is the hom-category $Hom(W, F)$ in the 2-category $[I, \mathfrak{Cat}]$. – Zhen Lin May 24 '16 at 7:29

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!

• Your answer is certainly full of very interesting remarks, but it does not answer my question. What is the universal property satisfied by weighted limits in the $Cat$-category of categories? – Bob Jun 2 '16 at 11:52
• As you maybe know, the bare definition asks that $\{W,F\}$ represents a certain functor. There's nothing different (apart the shape of $W$, in principle) from the old notion of conical limit you're used to. Or rather, I don't understand what you mean by "what is". – Fosco Jun 2 '16 at 12:20
• You can also define a conical limit as the representation of a functor, but still there is a more elementary definition that does not involve the notion of representation. In the same way, I would like to see a definition of a weighted limit that does not involve the notion of representation: "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..." – Bob Jun 2 '16 at 13:57
• It is possible to state such a result, but I think it's easy to deduce it simply re-reading the universal property you apparently master in full detail. Kelly's "Elementary observations" paper insists especially on the point that the universal property of a weighted co/limit is inherently 2-dimensional, meaning that the formula $\forall\dots \exists!\ldots : \dots$ must be read on 2-dimensional cells, not on 1-cells. – Fosco Jun 3 '16 at 17:26
• (PS: don't take my comment as a way to avoid answering you! I too have to ask thousand of questions like yours, before I grok a new idea. Feel free to ask again what you want, but take some time to go through Kelly's def of 2-dimensional universal prop!) – Fosco Jun 3 '16 at 17:40