nLab monad with arities




2-Category theory



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.

Monads with arities are subsumed by relative monads.


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

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}.

By definition of density, this means 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 restricted 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,

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

is in the essential image of ν A\nu_A.

The proof of the nerve theorem, following BMW, is fairly straightforward. Consider the adjunction j !:[𝒜 op,Set][Θ T op,Set]:j *j_! : [\mathcal{A}^{op},Set] \rightleftarrows [\Theta_T^{op},Set] : j^* given by restriction and left Kan extension. The assumption that TT has arities 𝒜\mathcal{A} can be reformulated to say that the nerve ν 𝒜:𝒞[𝒜 op,Set]\nu_{\mathcal{A}} : \mathcal{C} \to [\mathcal{A}^{op},Set] is a strong monad morphism from TT to j *j !j^* j_!, i.e. there is a coherent isomorphism ν 𝒜Tj *j !ν 𝒜\nu_{\mathcal{A}} T \cong j^* j_! \nu_{\mathcal{A}}. Since ν 𝒜\nu_{\mathcal{A}} is fully faithful, this means that if we identify 𝒞\mathcal{C} with the image of ν 𝒜\nu_{\mathcal{A}}, then the monad TT gets identified with j *j !j^* j_!. But the adjunction j !j *j_! \dashv j^* is also monadic (since jj is bijective on objects), so the category of TT-algebras gets identified with the full subcategory of j *j !j^* j_!-algebras, i.e. presheaves on Θ T\Theta_T, whose underlying presheaf on 𝒜\mathcal{A} is in the image of ν 𝒜\nu_{\mathcal{A}}. This is exactly the two statements of the nerve theorem.


See BMW for more.


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

On the connection between relative monads and monads with arities:

Last revised on December 21, 2023 at 20:51:56. See the history of this page for a list of all contributions to it.