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 $\mathcal{F}$-monad, $\mathcal{F}$-limit, and so on.
Let $D$ be a small strict $\mathcal{F}$-category. Then we have the functor $\mathcal{F}$-category $[D,\mathcal{F}]$ (where $\mathcal{F}$ denotes the $\mathcal{F}$-category $\mathcal{F}$). An object of $[D,\mathcal{F}]$ is an $\mathcal{F}$-functor $\Phi\colon D\to \mathcal{F}$, which can be identified with a pair of 2-functors $\Phi_\tau\colon D_\tau \to Cat$ and $\Phi_\lambda\colon D_\lambda\to Cat$ together with a 2-natural transformation
whose components are full embeddings (objects of $\mathcal{F}$).
The tight morphisms in $[D,\mathcal{F}]$ are $\mathcal{F}$-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 $\mathcal{F}$-category $Oplax(D,\mathcal{F})$ 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 $\mathcal{F}$-comonad $\mathcal{Q}_c^D$ on $[D,\mathcal{F}]$.
A weight $\Phi\colon D\to \mathcal{F}$ is $l$-rigged if
We obtain definitions of $c$-rigged and $p$-rigged weights if we replace $Oplax(D,\mathcal{F})$ by $Lax(D,\mathcal{F})$ and $Pseudo(D,\mathcal{F})$, respectively.
Let $w$ denote one of $l$, $c$, or $p$.
For an $\mathcal{F}$-weight $\Phi$, the following are equivalent.
See (LS) for the proof.
The following limits are $l$-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 $f,g\colon A\to B$ such that $f$ (the domain of the 2-cell to be inserted) is tight. Here the projection to $A$ is tight and tightness-detecting.
The equifier of a parallel pair of 2-cells between a parallel pair of 1-morphisms $f,g\colon A\to B$ such that $f$ (the domain of the 2-cells) is tight. Again, the projection to $A$ 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 $c$-rigged. There are $p$-rigged versions as well, but $p$-rigged weights are almost equivalent to PIE-limits; see (LS) for details.
Stephen Lack, “Limits for lax morphisms”. Appl. Categ. Structures,
13(3):189–203, 2005
Stephen Lack and Mike Shulman, “Enhanced 2-categories and limits for lax morphisms”, arXiv.
Last revised on February 27, 2018 at 11:48:12. See the history of this page for a list of all contributions to it.