10
$\begingroup$

Every weighted limit can be constructed from conical limits and cotensors. However, yesterday, a friend of mine, asked a question that may be rephrased as follows.

What is the reason that in the world of $\mathbf{Set}$-enriched categories every weighted limit can be constructed from conical limits (and trivial cotensors with $1$), and in the world of $\mathbf{Cat}$-enriched categories every weighted limit can be constructed from conical limits and cotensors with $2$?

Is it directly related to the fact that every set can be built upon $1$ and every category can be built upon $2$?

Is it possible to generalise these results to arbitrary (sufficiently well-behaved) monoidal category? For example, let us say that a symmetric monoidal closed category $\mathbb{V}$ is cocomplete and there exists a set $F$ of objects from $\mathbb{V}$ such that every object in $\mathbb{V}$ is a colimit of some objects from $F$. Is it true that every $\mathbb{V}$-weighted limit can be expressed via conical limits and cotensors with objects from $F$?

$\endgroup$

1 Answer 1

12
$\begingroup$

Yes, is directly related to that fact, as you surmise. The cotensor $X^K$, for $K\in \mathbb{V}$, preserves (co)limits in the variable $K$, that is we have

$$ X^{\mathrm{colim}_i K_i} \cong \lim_i X^{K_i}. $$

Even better, if $\lim_i X^{K_i}$ exists, then it automatically has the universal property to be $X^{\mathrm{colim}_i K_i}$. Therefore, if $\mathbb{C}$ is a $\mathbb{V}$-category with all small conical limits, then the class of objects in $\mathbb{V}$ for which $\mathbb{C}$ admits cotensors is closed under (conical) colimits. For instance, $\mathbf{Set}$ is the colimit-closure of $\{1\}$ and $\mathbf{Cat}$ is the colimit-closure of $\{2\}$. This is why those particular cotensors suffice.

A good way to check that $\mathbb{V}$ is the colimit-closure of a subcategory, by the way, is to show that that subcategory is a strong generator. As long as $\mathbb{V}$ is complete and cocomplete and extremally well-copowered, that's sufficient, as sketched in these notes. Note that being the colimit-closure of a subcategory is a weaker statement than every object being a colimit of objects in the subcategory; it allows iterated formation of colimits.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.