nLab
monad with arities

Context

Algebra

2-Category theory

Contents

Idea

A monad with arities is a monad that admits a generalized nerve construction. This allows us to view its algebras as presheaves-with-properties in a canonical way.

This generalized nerve construction also generalizes the construction of the syntactic category of a Lawvere theory.

Definition

Let 𝒞\mathcal{C} be a category, and i A:𝒜𝒞i_A : \mathcal{A} \subset \mathcal{C} a subcategory. As explained at dense functor, for any object XX of 𝒞\mathcal{C}, there is a canonical cocone over the forgetful functor (𝒜X)𝒞(\mathcal{A} \downarrow X) \to \mathcal{C}, which we call the canonical 𝒜\mathcal{A}-cocone at XX. The subcategory 𝒜𝒞\mathcal{A} \subset \mathcal{C} is called dense if this cocone is colimiting for every object XX of CC.

If 𝒞\mathcal{C} be a category and i A:𝒜𝒞i_A : \mathcal{A} \subset \mathcal{C} is a dense subcategory, then the 𝒜\mathcal{A}-nerve functor is given by

(1)ν 𝒜:𝒞 [𝒜 op,Set] X 𝒞(i A,X). \begin{aligned} \nu_{\mathcal{A}} : \mathcal{C} &\to [\mathcal{A}^{op}, \mathrm{Set}] \\ X &\mapsto \mathcal{C}(i_A, X) \end{aligned} \,.

A monad (T,μ,η)(T,\mu,\eta) on 𝒞\mathcal{C} is said to have arities 𝒜\mathcal{A} if ν 𝒜T\nu_{\mathcal{A}} \circ T sends canonical 𝒜\mathcal{A}-cocones to colimiting cocones.

Nerve Theorem

The nerve theorem consists of two statements:

I. If 𝒜\mathcal{A} is dense in 𝒞\mathcal{C} and if TT is a monad with arities 𝒜\mathcal{A} on 𝒞\mathcal{C}, then 𝒞 T\mathcal{C}^T has a dense subcategory Θ T\Theta_T given by the free TT-algebras on objects of 𝒜\mathcal{A}.

It follows (?) that the nerve functor ν Θ T:𝒞 T[Θ T op,Set]\nu_{\Theta_T} : \mathcal{C}^T \to [\Theta_T^{op}, \mathrm{Set}] is full and faithful. This allows us to view TT-algebras as presheaves (on Θ T\Theta_T) with a certain property. The second part of the nerve theorem tells us what this property is.

II. Let j:𝒜Θ Tj: \mathcal{A} \to \Theta_T be the free algebra functor. A presheaf P:Θ T opSetP : \Theta_T^{op} \to \mathrm{Set} is in the essential image of ν Θ\nu_{\Theta} if and only if the restriction along jj,

(2)Pj:A opSet P\circ j : A^{op} \to \Set

is in the essential image of ν A\nu_A.

Examples

For now, see the paper of Berger, Melliès, and Weber below…

References

See the discussion at

The associated paper is

  • Mark Weber, Familial 2-functors and parametric right adjoints (2007) (tac)

These ideas are clarified and expanded on in

Revised on January 16, 2012 15:26:56 by Urs Schreiber (89.204.130.179)