nLab saturated class of limits

Idea

A class $\mathcal{X}$ of limits is saturated if it is closed under the “construction” of other limits out of limits in $\mathcal{X}$.

Definition

Let $V$ be a Bénabou cosmos, and let $\mathcal{X}$ be a class of $V$-functors $\Phi\colon D\to V$ where $D$ is a small $V$-category. Note that the domain $D$ may be different for different elements of $\mathcal{X}$.

Any such $\Phi\colon D\to V$ can serve as a weight for defining the weighted limit of a $V$-functor $D\to C$, for any $V$-category $C$. We say that a $V$-category is $\mathcal{X}$-complete if it admits all such limits, for all $\Phi\in\mathcal{X}$, and that a $V$-functor between $\mathcal{X}$-complete $V$-categories is $\mathcal{X}$-continuous if it preserves all such limits.

The saturation of $\mathcal{X}$ is the class of all weights $\Phi\colon D\to V$ (with $D$ small) such that

1. Any $\mathcal{X}$-complete $V$-category admits $\Phi$-weighted limits, and
2. Any $\mathcal{X}$-continuous $V$-functor preserves $\Phi$-weighted limits.

It is an open question whether the second condition is implied by the first in general.

Finally, $\mathcal{X}$ is saturated if it is its own saturation.

The conical case

When $V=Set$, we frequently discuss only conical limits, i.e. limits whose weight $\Phi\colon D\to Set$ is the constant functor $\Delta_D 1$ at the terminal set. These give the classical notion of limit in a category.

In this case, we may consider instead classes $\mathcal{J}$ of small categories; we write $\Delta_{\mathcal{J}}$ for the class of weights $\{ \Delta_C 1 | C \in \mathcal{J}\}$. We say that a category $D$ lies in the saturation of $\mathcal{J}$ if the weight $\Delta_D 1$ lies in the saturation of $\Delta_{\mathcal{J}}$, and that $\mathcal{J}$ is saturated if it is its own saturation.

Note that in practically all cases, the saturation of $\Delta_{\mathcal{J}}$ will contain weights that are not of the form $\Delta_D 1$. Moreover, even when $V=Set$ there are nontrivial saturated classes of weights that do not contain any nontrivial conical weights, such as the saturation of the weight for “cartesian squares” $A\times A$.

However, for conical weights the answer to the above open question is known to be affirmative. On the one hand, if $\mathcal{X}$ is a class of $Set$-weights such that every $\mathcal{X}$-complete category is also $\Delta_D 1$-complete, then every $\mathcal{X}$-continuous functor is also $\Delta_D 1$-continuous. See AK for a proof of this.

On the other hand, if $\mathcal{J}$ is a class of $Set$-categories and $\Phi$ is a $Set$-weight such that every $\Delta_{\mathcal{J}}$-complete category is also $\Phi$-complete, then every $\Delta_{\mathcal{J}}$-continuous functor is also $\Phi$-continuous. In fact, this is still true if instead of $\Delta_{\mathcal{J}}$ we consider a class of weights all of which take only nonempty sets as values. See KP for a proof of this.

Characterization

The main theorem of AK (which introduced the notion under the name “closure”) is the following.

Theorem

$\Phi\colon D\to V$ lies in the saturation of $\mathcal{X}$ if and only if it lies in the closure of the representables under $\mathcal{X}$-weighted colimits in $[D,V]$.

Examples

The following examples are all for $V=Set$, restricted to the conical case.

• The class of small products is not saturated; its saturation includes all $\Delta_D 1$ where $D$ has local initial objects. Similarly, the class of finite products is not saturated; the saturation of this coincides with the saturation of the finite class containing only terminal objects and binary products.

• The class of L-finite limits is saturated with respect to conical weights (but not in the sense of the definition above); it is the saturation of the class of finite limits. It is also the saturation of the finite class containing only terminal objects and pullbacks, and the saturation of the class containing only finite products and equalizers.

• The class of connected limits is saturated with respect to conical weights. It is the saturation of the class consisting of wide pullbacks and equalizers. Similarly, the class of L-finite connected limits is the saturation of the finite class of pullbacks and equalizers. See also pullback and wide pullback for their saturations.

There are also interesting examples for other $V$.

• When $V=Cat$, the classes of PIE-limits, flexible limits, and semiflexible limits are saturated. The first class is, essentially by definition, the saturation of the class containing products, inserters, and equifiers. The second class can be proven to be the saturation of the class containing products, inserters, equifiers, and splitting of idempotents. A similar characterisation of the third is not known.

• When $V=F$ is the category of fully faithful functors, so that a $V$-category is an F-category, the class of $w$-rigged weights is saturated (for any of $w=p$, $l$, or $c$ denoting pseudo, lax, or colax).

• For any $V$, the class of absolute colimits is saturated. When $V=Set$, this is the saturation of the splitting of idempotents.

It is also worth mentioning some non-examples.

• For $V=Set$, the class of finite limits is not saturated; its saturation is the class of L-finite limits.

• For $V=Cat$, the class of strict pseudo-limits is not saturated; it does not even contain the representables. (The same is true for strict lax limits.) It is unclear precisely what its saturation looks like.

References

• M. H. Albert and G. Max Kelly: The closure of a class of colimits, J. Pure. App. Alg. 51 (1988) 1-17 [doi:10.1016/0022-4049(88)90073-4]

• Max Kelly and Robert Paré: A note on the Albert-Kelly paper ‘The closure of a class of colimits’, JPAA 51 (1988) 19-25 [doi;10.1016/0022-4049(88)90074-6]

• G. Max Kelly and V. Schmitt, Notes on enriched categories with colimits of some class, Theory and Applications of Categories 14 (2005) 399–423 [tac:14-17arXiv:math/0501383]

• Charles Rezk, Free colimit completion in $\infty$-categories &lbrackarXiv:2210.08582&rbrack (2022).