nLab monad with arities

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.

Monads with arities are subsumed by relative monads.

Definition

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

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

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

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

Nerve Theorem

The nerve theorem consists of two statements:

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

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

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

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

is in the essential image of $\nu_A$.

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

Examples

• Every p.r.a. monad has arities. In particular, therefore, every polynomial monad has arities.

• The free groupoid monad on the category of directed graphs with involution has arities, although it is not p.r.a. The arities are the full subcategory of involutive graphs on the finite connected acyclic graphs.

See BMW for more.

Related pages

• A monad with arities is equivalently an internal monad in the 2-category of categories with arities.

• A monad with arities $i_A$ is equivalent to an $i_A$-relative monad.

References

See the discussion at

• Tom Leinster, How I learned to love the nerve construction (2008) (blog)

The associated paper is

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

These ideas are clarified and expanded on in

• Paul-André Melliès. Segal condition meets computational effects. (2010, 25th Annual IEEE Symposium on Logic in Computer Science).

• Clemens Berger, Paul-André Melliès, Mark Weber, Monads with Arities and their Associated Theories (2011) (arXiv:1101.3064)

• Nathanael Arkor, Dylan McDermott, The nerve theorem for relative monads (2025), Theory and Applications of Categories, [arXiv:2404.01281]

On the connection between relative monads and monads with arities:

• Thorsten Altenkirch, James Chapman, Tarmo Uustalu, Monads need not be endofunctors, Logical Methods in Computer Science 11 1:3 (2015) 1–40 [arXiv:1412.7148, pdf, doi:10.2168/LMCS-11(1:3)2015]