Consider a finite toset (cool word) $A=\{a_1<\ldots<a_N\}$. Let an ordered monomial be a finite formal product $b_1\ldots b_M$ with $b_i\in A$ and $b_i\le b_{i+1}$.
Consider the following category. Every object is a pair $(L,v)$ of a complex vector space $L$ and a nonzero $v\in L$. Additionally we choose some $X(v)\in L$ for every ordered monomial $X$ in such a way that $1(v)=v$. These objects can also be viewed as linear maps $$\varphi:S^*(\mathbb Ca_1)\otimes\ldots\otimes S^*(\mathbb Ca_N)\to L$$ with $\varphi:1\otimes\ldots\otimes 1\mapsto v.$ (Remark: $X(v)$ is just a notation! $X$ does not act on the whole $L$.)
A morphism from $(L,v)$ to $(L',v')$ is a linear map $\psi:L\to L'$ such that for every ordered monomial $X$ we have $\psi:X(v)\mapsto X(v')$.
This somewhat rudimentary construction has one nice property: there are tensor products. The tensor product of $(L,v)$ and $(L',v')$ is $(L\otimes L', v\otimes v')$ where for every ordered monomial $X=b_1\ldots b_M$ one has $$X(v\otimes v')=\sum X_1(v)\otimes X_2(v').$$ The above sum ranges over all partitions of $\{b_1,\ldots,b_M\}$ into two sets with $X_1$ and $X_2$ being the ordered products of the elements in each set.
To make this less abstract let me briefly mention my motivation. Consider a semisimple Lie algebra $\mathfrak g$ with negative root vectors spanning the subalgebra $\mathfrak n_-$. If we let $A$ be the set of these negative root vectors with some ordering, then every highest weight module $L$ with highest weight vector $v$ can be viewed as an object $(L,v)$ in our category. Here $X(v)$ is simply the action of the monomial $X\in\mathcal U(\mathfrak n_-)$ on $v$. I'm working with certain degenerations of these objects where the described structure is more-or-less what remains of the action.
Be as it may, this whole business seems a bit suspicious to me. I would appreciate it if someone would tell me how this can be reduced something more conventional, where this has been studied or why this doesn't really make sense.
