nLab semiflexible limit

Idea

A semiflexible limit is a strict 2-limit whose weight is semiflexible (defined below). The semiflexible weights are precisely those weights whose limits are bilimits.

All flexible limits, hence also PIE-limits and strict pseudo-limits, are semiflexible.

The class of semiflexible weights is a saturated class of 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 semiflexible if this transformation admits a pseudo-section? (equivalent is an equivalence) in $[D,Cat]$.

Examples

• Every flexible limit is semiflexible.

Related pages

• flexible limit
• PIE-limit

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 and Richard Garner, On semiflexible, flexible and pie algebras, Journal of Pure and Applied Algebra 217.2 (2013): 293-321.