# nLab monad with arities

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### 2-Category theory

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 $𝒞$ 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 $\left(𝒜↓X\right)\to 𝒞$, 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)$\begin{array}{rl}{\nu }_{𝒜}:𝒞& \to \left[{𝒜}^{\mathrm{op}},\mathrm{Set}\right]\\ X& ↦𝒞\left({i}_{A},X\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} \nu_{\mathcal{A}} : \mathcal{C} &\to [\mathcal{A}^{op}, \mathrm{Set}] \\ X &\mapsto \mathcal{C}(i_A, X) \end{aligned} \,.

A monad $\left(T,\mu ,\eta \right)$ 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 ${\nu }_{{\Theta }_{T}}:{𝒞}^{T}\to \left[{\Theta }_{T}^{\mathrm{op}},\mathrm{Set}\right]$ 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

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)