Weighted limits and completeness

Question

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 $F$?

Answer 1

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.