nLab
flexible limit

Context

2-Category theory

Limits and colimits

Contents

Idea

A flexible limit is a strict 2-limit whose weight is cofibrant. This implies that flexible limits are also 2-limits (in the non-strict sense, which for us is the default – recall that these are traditionally called bilimits).

Furthermore, all PIE-limits and therefore all strict pseudo-limits are flexible; thus any strict 2-category which admits all flexible limits also admits all 22-limits. A number of strict 2-categories admit all flexible limits, but not all strict 22-limits, and this is a convenient way to show that they admit all 22-limits.

Definition

Let DD be a small strict 2-category. Write [D,Cat][D,Cat] for the strict 2-category of strict 2-functors, strict 2-natural transformations?, and modifications, and Ps(D,Cat)Ps(D,Cat) for the strict 2-category of strict 2-functors, pseudonatural transformations, and modifications. The inclusion

[D,Cat]Ps(D,Cat) [D,Cat] \to Ps(D,Cat)

(as a wide subcategory) has a strict left adjoint QQ, which is the pseudo morphism classifier? for an appropriate strict 2-monad. Therefore, for any functor Φ:DCat\Phi \colon D\to Cat, we have QΦ:DCatQ\Phi \colon D\to Cat such that pseudonatural transformations ΦΨ\Phi \to \Psi are in natural bijection with strict 2-natural transformations QΦΨQ\Phi \to \Psi.

The counit of this adjunction is a canonical strict 2-natural transformation q:QΦΦq\colon Q\Phi \to \Phi. We say that Φ\Phi is flexible if this transformation has a section in [D,Cat][D,Cat].

Examples

All PIE-limits are flexible. This includes products, inserters, equifiers by definition, and also descent objects, iso-inserters, comma objects, and so on. In fact, PIE-limits have a characterization similar to the definition above of flexible limits: they are the coalgebras for QQ regarded as a 2-comonad.

The splitting of idempotents is flexible, but not PIE. Moreover, in a certain sense it is the “only” such. Precisely, flexible limits are the saturation of each of the following classes of limits:

  • PIE-limits together with splitting of idempotents
  • PIE-limits together with splitting of idempotent equivalences
  • strict pseudo-limits together with splitting of idempotents
  • strict pseudo-limits together with splitting of idempotent equivalences

References

  • G. J. Bird, G. M. Kelly, A. J. Power, R. H. Street, Flexible limits for 2-categories, doi

Revised on October 6, 2012 00:17:39 by Mike Shulman (192.16.204.210)