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?



Algebraic structure of the space of multiaffine maps
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.


