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 $\mathcal{F}$-monad, $\mathcal{F}$-limit, and so on.

Definition

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

$$\array{D_\tau & & \overset{J_D}{\to} & & D_\lambda\\ & {}_{\Phi_\tau}\searrow & \neArrow & \swarrow_{\Phi_\lambda} \\ & & Cat }$$

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

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

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

Definition

A weight $\Phi\colon D\to \mathcal{F}$ is $l$-rigged if

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

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

Characterization

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

Theorem

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

1. $\Phi$ is $w$-rigged.
2. For any $\mathcal{F}$-monad $T$ on an $\mathcal{F}$-category $K$, the $\mathcal{F}$-functor $U_w\colon T Alg_w \to K$ creates $\Phi$-weighted limits.
3. For any 2-monad $T$ on a 2-category $K$, the functor $U_w\colon T Alg_w \to K$ (where $K$ denotes the chordate $\mathcal{F}$-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 $f,g\colon A\to B$ 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.

Related concepts

• Some limits of diagrams involving both lax and colax morphisms can also be given a $T$-algebra structure; see for instance colax/lax comma object.

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.