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.

Definition

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

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

(1)$$\begin{array}{rl}{\nu }_{𝒜}:𝒞& \to [{𝒜}^{\mathrm{op}},\mathrm{Set}]\\ X& \mapsto 𝒞(i_A,X)\end{array}.$$\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 $𝒞$ is said to have arities $𝒜$ if ${\nu }_{𝒜}\circ T$ sends canonical $𝒜$-cocones to colimiting cocones.

Nerve Theorem

The nerve theorem consists of two statements:

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

It follows (?) that the nerve functor ${\nu }_{{\Theta }_T}:{𝒞}^T\to [{\Theta }_T^{\mathrm{op}},\mathrm{Set}]$ 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 $j:𝒜\to {\Theta }_T$ be the free algebra functor. A presheaf $P:{\Theta }_T^{\mathrm{op}}\to \mathrm{Set}$ is in the essential image of ${\nu }_{\Theta }$ if and only if the restriction along $j$,

(2)$$P\circ j:A^{\mathrm{op}}\to Set$$P\circ j : A^{op} \to \Set

is in the essential image of ${\nu }_A$.

Examples

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

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

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