nLab
rigged limit

Rigged limits

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 DD be a small strict \mathcal{F}-category. Then we have the functor \mathcal{F}-category [D,][D,\mathcal{F}] (where \mathcal{F} denotes the \mathcal{F}-category \mathcal{F}). An object of [D,][D,\mathcal{F}] is an \mathcal{F}-functor Φ:D\Phi\colon D\to \mathcal{F}, which can be identified with a pair of 2-functors Φ τ:D τCat\Phi_\tau\colon D_\tau \to Cat and Φ λ:D λCat\Phi_\lambda\colon D_\lambda\to Cat together with a 2-natural transformation

D τ J D D λ Φ τ Φ λ Cat\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,][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,)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,]Oplax(D,) [D,\mathcal{F}] \to Oplax(D,\mathcal{F})

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

Definition

A weight Φ:D\Phi\colon D\to \mathcal{F} is ll-rigged if

  1. It is a 𝒬 c D\mathcal{Q}_c^D-coalgebra, and
  2. The canonical functor Lan J DΦ τΦ λLan_{J_D} \Phi_\tau \to \Phi_\lambda is surjective on objects.

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

Characterization

Let ww denote one of ll, cc, or pp.

Theorem

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

  1. Φ\Phi is ww-rigged.
  2. For any \mathcal{F}-monad TT on an \mathcal{F}-category KK, the \mathcal{F}-functor U w:TAlg wKU_w\colon T Alg_w \to K creates Φ\Phi-weighted limits.
  3. For any 2-monad TT on a 2-category KK, the functor U w:TAlg wKU_w\colon T Alg_w \to K (where KK denotes the chordate \mathcal{F}-category on KK) creates Φ\Phi-weighted limits.

See (LS) for the proof.

Examples

The following limits are ll-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:ABf,g\colon A\to B such that ff (the domain of the 2-cell to be inserted) is tight. Here the projection to AA is tight and tightness-detecting.

  • The equifier of a parallel pair of 2-cells between a parallel pair of 1-morphisms f,g:ABf,g\colon A\to B such that ff (the domain of the 2-cells) is tight. Again, the projection to AA 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 cc-rigged. There are pp-rigged versions as well, but pp-rigged weights are almost equivalent to PIE-limits; see (LS) for details.

References

Revised on February 22, 2012 03:25:07 by Mike Shulman (128.54.32.45)