nLab
saturated class of limits

Context

Category theory

Enriched category theory

Limits and colimits

Saturated classes of limits

Idea

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

Definition

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

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

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

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

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

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

The conical case

When V=SetV=Set, we frequently discuss only conical limits, i.e. limits whose weight Φ:DSet\Phi\colon D\to Set is the constant functor Δ D1\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 𝒥\mathcal{J} of small categories; we write Δ 𝒥\Delta_{\mathcal{J}} for the class of weights {Δ C1C𝒥}\{ \Delta_C 1 | C \in \mathcal{J}\}. We say that a category DD lies in the saturation of 𝒥\mathcal{J} if the weight Δ D1\Delta_D 1 lies in the saturation of Δ 𝒥\Delta_{\mathcal{J}}, and that 𝒥\mathcal{J} is saturated if it is its own saturation.

Note that in practically all cases, the saturation of Δ 𝒥\Delta_{\mathcal{J}} will contain weights that are not of the form Δ D1\Delta_D 1. Moreover, even when V=SetV=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×AA\times A.

However, for conical weights the answer to the above open question is known to be affirmative. On the one hand, if 𝒳\mathcal{X} is a class of SetSet-weights such that every 𝒳\mathcal{X}-complete category is also Δ D1\Delta_D 1-complete, then every 𝒳\mathcal{X}-continuous funtor is also Δ D1\Delta_D 1-continuous. See AK for a proof of this.

On the other hand, if 𝒥\mathcal{J} is a class of SetSet-categories and Φ\Phi is a SetSet-weight such that every Δ 𝒥\Delta_{\mathcal{J}}-complete category is also Φ\Phi-complete, then every Δ 𝒥\Delta_{\mathcal{J}}-continuous functor is also Φ\Phi-continuous. In fact, this is still true if instead of Δ 𝒥\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

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

Examples

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

  • The class of small products is saturated, as is the class of finite products. The latter is the saturation of the finite class containing only terminal objects and binary products.

  • The class of L-finite limits is saturated; 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. 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 VV.

It is also worth mentioning some non-examples.

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

  • For V=CatV=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

  • Albert and Kelly, “The closure of a class of limits”, J. Pure. App. Alg. 51 (1988), 1–17
  • Max Kelly and Robert Paré, “A note on the Albert-Kelly paper ‘The closure of a class of limits’“, JPAA 51 (1988), 19–25

Revised on February 22, 2012 22:30:48 by Mike Shulman (71.136.234.110)