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 $\mathcal{A}$.
It follows (?) 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 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 $\nu_A$.
For now, see the paper of Berger, Melliès, and Weber below…
See the discussion at
The associated paper is
These ideas are clarified and expanded on in
Last revised on January 16, 2012 at 15:26:56. See the history of this page for a list of all contributions to it.