nLab semiflexible limit



2-Category theory

Limits and colimits



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.


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 semiflexible if this transformation admits a pseudo-section? (equivalent is an equivalence) in [D,Cat][D,Cat].



Created on June 12, 2024 at 09:35:02. See the history of this page for a list of all contributions to it.