nLab flexible limit

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 $2$-limits. A number of strict 2-categories admit all flexible limits, but not all strict $2$-limits, and this is a convenient way to show that they admit all $2$-limits.

Definition

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

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

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

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

Examples

All PIE-limits are flexible. This includes products, inserters, equifiers by definition, and also descent objects, iso-inserters, comma objects, Eilenberg–Moore objects, and so on. In fact, PIE-limits have a characterization similar to the definition above of flexible limits: they are the coalgebras for $Q$ 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
• products, iso-inserters, powers, splitting of idempotents
• products, powers, splitting of idempotents, pullbacks of normal isofibrations?
• products, powers, splitting of idempotents, pullbacks of amnestic isofibrations

Related pages

• Flexible limits may be characterised as the persistent double limits.
• semiflexible limit
• PIE-limit
• coherence theorem for bicategories with finite limits

References

• Greg J. Bird, Max Kelly, John Power, Ross Street, Flexible limits for 2-categories, Journal of Pure and Applied Algebra 61 1 (1989) 1-27 [doi:10.1016/0022-4049(89)90065-0]

• John Bourke. Accessible aspects of 2-category theory. Journal of Pure and Applied Algebra 225.3 (2021): 106519.