Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A rigged limit is a 2-limit which is created in 2-categories of algebras and lax, colax, or pseudo morphisms for a 2-monad.
In order to characterize these most precisely, however, it turns out to be convenient to generalize from 2-categories to F-categories, using the corresponding notions of -monad, -limit, and so on.
Let be a small strict -category. Then we have the functor -category (where denotes the -category ). An object of is an -functor , which can be identified with a pair of 2-functors and together with a 2-natural transformation
whose components are full embeddings (objects of ).
The tight morphisms in are -natural transformations in the usual sense of enriched category theory, whereas its loose morphisms are 2-natural transformations between loose parts.
We also have an -category with the same objects, whose loose morphisms are oplax natural transformations between loose parts which are strictly 2-natural on tight morphisms, and whose tight morphisms are those whose components are all tight. The inclusion
has a left adjoint, which induces an -comonad on .
A weight is -rigged if
- It is a -coalgebra, and
- The canonical functor is surjective on objects.
We obtain definitions of -rigged and -rigged weights if we replace by and , respectively.
Let denote one of , , or .
For an -weight , the following are equivalent.
- is -rigged.
- For any -monad on an -category , the -functor creates -weighted limits.
- For any 2-monad on a 2-category , the functor (where denotes the chordate -category on ) creates -weighted limits.
See (LS) for the proof.
The following limits are -rigged.
The 2-limit of any diagram of tight morphisms which is also a limit as a diagram of loose morphisms. This includes any product and any power.
The oplax limit of any diagram of loose morphisms.
The inserter of a parallel pair such that (the domain of the 2-cell to be inserted) is tight. Here the projection to is tight and tightness-detecting.
The equifier of a parallel pair of 2-cells between a parallel pair of 1-morphisms such that (the domain of the 2-cells) is tight. Again, the projection to is tight and tightness-detecting.
The Eilenberg-Moore object of a loose monad. Here the canonical forgetful morphism is tight and tightness-detecting.
Each has a fairly obvious dual version which is -rigged. There are -rigged versions as well, but -rigged weights are almost equivalent to PIE-limits; see (LS) for details.