Algebraic structure of the space of multiaffine maps Let $V$ be a vector space over a field $\mathbb F$ and $k$ some natural number.
It isn't hard to show that the space of multiaffine maps $V^{[k]}\to\mathbb F$ decomposes as a direct sum of vector spaces $\bigoplus_{I\subset [k]} M_I $ where $M_I$ is the space of multilinear maps $V^I\to\mathbb F$ (thought of as maps on $V^{[k]}$ which only care about the coordinates in $I$).
But there is also more structure here - if $I,J$ are disjoint we have an obvious bilinear map $M_I\times M_J\to M_{I\sqcup J}.$ These maps are also associative (when $I,J,K$ are pairwise disjoint) and commutative.
So is there a name for this type of algebraic structure? If so, where could I read about it?
 A: Let $\newcommand{\FB}{\mathrm{FB}}\newcommand{\FI}{\mathrm{FI}}\newcommand{\Vect}{\mathrm{Vect}}\FB$ be the category of finite sets and bijections, and $\FI$ the category of finite sets and injections, considered as symmetric monoidal categories under disjoint union.
The assignment $I \mapsto M_I$ is a functor $\FB\to \Vect$. The fact that we have maps $M_I \otimes M_J \to M_{I \sqcup J}$ says that this is a lax symmetric monoidal functor. Lax symmetric monoidal functors out of the category $\FB$ are often called twisted commutative algebras, or alternatively left modules over the commutative operad.
Let $N_I$ denote the space of multi-affine maps $V^I \to \mathbb F$. The assignment $I \mapsto N_I$ has an additional functoriality which $I \mapsto M_I$ lacks, namely that if $I \hookrightarrow J$ is an injection then there is an evident inclusion map $N_I \hookrightarrow N_J$. This makes $I \mapsto N_I$ a lax symmetric monoidal functor $\FI \to \Vect$. Functors out of the category $\FI$ are often called $\FI$-modules, and they have been intensely studied in the past decade or so in connection with the subject of representation stability. Lax symmetric monoidal functors out of the category $\FI$ are called commutative $\mathcal I$-algebras e.g. in work of Sagave-Schlichtkrull and Richter-Sagave (where $\mathcal I$ is their notation for the category $\FI$). For them, the interesting examples are e.g. lax symmetric monoidal functors from $\FI$ to topological spaces or chain complexes, in which case there is a Quillen model structure on the category of such functors making it Quillen equivalent to the category of $\mathbb E_\infty$-algebras.
The fact that $N_I \cong \bigoplus_{J \hookrightarrow I} M_I$ says that $I \mapsto N_I$ is the left Kan extension of $I \mapsto M_I$ along the inclusion $\FB \hookrightarrow \FI$. It is a general purely category-theoretic fact that the left Kan extension of a lax symmetric monoidal functor along a strong symmetric monoidal functor is again lax symmetric monoidal.
