symmetric monoidal (∞,1)-category of spectra
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.
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
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.
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 $j$,
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.
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.
See the discussion at
The associated paper is
These ideas are clarified and expanded on in
Last revised on December 30, 2018 at 01:32:29. See the history of this page for a list of all contributions to it.