nLab
rigged limit

Idea

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 $ℱ$-monad, $ℱ$-limit, and so on.

Definition

Let $D$ be a small strict $ℱ$-category. Then we have the functor $ℱ$-category $[D,ℱ]$ (where $ℱ$ denotes the $ℱ$-category $ℱ$). An object of $[D,ℱ]$ is an $ℱ$-functor $\Phi :D\to ℱ$, which can be identified with a pair of 2-functors ${\Phi }_{\tau }:D_{\tau }\to \mathrm{Cat}$ and ${\Phi }_{\lambda }:D_{\lambda }\to \mathrm{Cat}$ together with a 2-natural transformation

$$\begin{array}{ccccc}{D}_{\tau }& & \stackrel{J_D}{\to }& & D_{\lambda }\\ & {}_{{\Phi }_{\tau }}\searrow & ⇗& {\swarrow }_{{\Phi }_{\lambda }}\\ & & \mathrm{Cat}\end{array}$$\array{D_\tau & & \overset{J_D}{\to} & & D_\lambda\\ & {}_{\Phi_\tau}\searrow & \neArrow & \swarrow_{\Phi_\lambda} \\ & & Cat }

whose components are full embeddings (objects of $ℱ$).

The tight morphisms in $[D,ℱ]$ are $ℱ$-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 $ℱ$-category $\mathrm{Oplax}(D,ℱ)$ 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

$$[D,ℱ]\to \mathrm{Oplax}(D,ℱ)$$[D,\mathcal{F}] \to Oplax(D,\mathcal{F})

has a left adjoint, which induces an $ℱ$-comonad ${𝒬}_c^D$ on $[D,ℱ]$.

Definition

A weight $\Phi :D\to ℱ$ is $l$-rigged if

1. It is a ${𝒬}_c^D$-coalgebra, and
2. The canonical functor ${\mathrm{Lan}}_{J_D}{\Phi }_{\tau }\to {\Phi }_{\lambda }$ is surjective on objects.

We obtain definitions of $c$-rigged and $p$-rigged weights if we replace $\mathrm{Oplax}(D,ℱ)$ by $\mathrm{Lax}(D,ℱ)$ and $\mathrm{Pseudo}(D,ℱ)$, respectively.

Characterization

Let $w$ denote one of $l$, $c$, or $p$.

Theorem

For an $ℱ$-weight $\Phi$, the following are equivalent.

1. $\Phi$ is $w$-rigged.
2. For any $ℱ$-monad $T$ on an $ℱ$-category $K$, the $ℱ$-functor $U_w:T{\mathrm{Alg}}_w\to K$ creates $\Phi$-weighted limits.
3. For any 2-monad $T$ on a 2-category $K$, the functor $U_w:T{\mathrm{Alg}}_w\to K$ (where $K$ denotes the chordate $ℱ$-category on $K$) creates $\Phi$-weighted limits.

See (LS) for the proof.

Examples

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 $f,g:A\to B$ 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 $f,g:A\to B$ 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.

References

• 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.