Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
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 -limits. A number of strict 2-categories admit all flexible limits, but not all strict -limits, and this is a convenient way to show that they admit all -limits.
Let be a small strict 2-category. Write for the strict 2-category of strict 2-functors, strict 2-natural transformations?, and modifications, and 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 , which is the pseudo morphism classifier? for an appropriate strict 2-monad. Therefore, for any functor , we have such that pseudonatural transformations are in natural bijection with strict 2-natural transformations .
The counit of this adjunction is a canonical strict 2-natural transformation . We say that is flexible if this transformation has a section in .
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 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
- G. J. Bird, G. M. Kelly, A. J. Power, R. H. Street, Flexible limits for 2-categories, doi