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.

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)

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]