$\newcommand{\lim}{\mathsf{lim}}$

# Weighted limits

(This is from [Kelly's Basic Concepts of Enriched Category Theory](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf), Chapter 3)

The weighted (or indexed) limit of a diagram $D:\mathcal J\to\mathcal B$ and a weight $\phi:\mathcal J\to\mathcal V$
$$\lim_\phi D,$$

is, if it exists, the (up to iso) unique solution of
$$\mathcal B(B,\lim_\phi D)\cong[\mathcal J,\mathcal V](\phi,\mathcal B(B,D)).$$

## The limiting cylinder of a weighted limit

$B=\lim_\phi D$ in the isomorphism above yields a cone (or ***cylinder*** as Kelly says) [^cylinder]
$$\mu:\phi\to\mathcal B(\lim_\phi D,D)$$

which is also called the *counit of the representation* (Kelly, Chapters 1.10) or the *counit cylinder* (Kelly, Chapter 3.1). I prefer ***limiting cylinder*** because then I don't have to remember whether it should be "unit" or "counit". The isomorphism above then implies that any other cylinder
$$\alpha: \phi\to \mathcal B(B,D))$$

factors through the limiting cylinder $\mu$ as $\alpha = \mathcal B(f,1)\mu$ for a unique $f:B\to\lim_\phi D$.

## References

[Kelly: Basic Concepts of Enriched Category Theory](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) (1982)

## Links

[Weighted limits and reflective subcategories](https://hackmd.io/@alexhkurz/rk_6JXgH8)

[](https://hackmd.io/@alexhkurz/HkJe2KqfL)

[^cylinder]: Strictly speaking $\phi\to\mathcal B(\lim_\phi D,D)$ is an abbreviation for the arrow
$$\mathbb I\longrightarrow \mathcal B(\lim_\phi D,\lim_\phi ID) \stackrel{\cong\ }{\longrightarrow} [\mathcal J,\mathcal V](\phi,\mathcal B(\lim_\phi D,D))$$

Following Kelly again, we will say ***map*** for elements of a homset, that is, for arrows of the form $\mathbb I\to\mathcal A(A,B)$. Note that a map $A\to B$ is indeed an element of the set $\mathcal A_0(A,B)$ obtained from applying the forgetful functor $(-)_0$ to the object $\mathcal A(A,B)$ in $\mathcal V$. See (3.4) of [Kelly's monograph](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) for more on cylinders.