Let $E,F$ and $G$ be vector spaces over the field $\Gamma$ and let $\phi:E\times F \to G$ be homogeneous map of degree $k \in \mathbb N$, i.e., $$ \phi(ax,ay) = a^k\phi(x,y), \qquad \forall x \in E, y \in F a \in \Gamma $$ Is there universal property for such maps ? i.e., is there a pair $(\odot,H)$ where $\odot$ is a homogeneous map of degree $k$ on $E\times F$ into $H$ (a vector space) such that for every homogeneous map (degree $k$) $\phi$ there is a linear $f$ such that $f\circ \odot = \phi$ ?
My attempt:
Let $C(E\times F)$ be the free vector space over the $E\times F$ and let the $N$ be the subspace generated by all elements of the form $$ (ax,ay) - a^k(x,y) $$ Now consider the canonical projection $\pi:C(E\times F) \to C(E\times F)/N$, then define the linear map $h:C(E\times F) \to G$ such that $h((x,y)) = \phi(x,y)$. It can be shown that $N \subset \ker h$. Then by the universal property of quotient maps there is a unique linear map $f:C(E\times F)/N \to G$ such that $f \circ \pi = h$. If the restriction of $\pi$ to $E\times F$ is denoted $\odot$, then this a homogeneous map of degree $k$, and it follows that $f\circ\odot=\phi$, and if $C(E\times F)/N$ is denoted $H$ then we have the pair $(\odot,H)$.
Please comment!, I would like to know whether there is any mistake, or if something like this universal property of homogeneous maps does exist at all. Thanks in advance!
Added
I have the following two comments on this construction:
I. This construction can be carried out for homogeneous maps $\phi:V_1\times\cdots\times V_n \to W$ provided the subspace $N$ of $C(V_1\times\cdots\times V_n)$ is modified accordingly, i.e., to be generated by all elements of the form $(av_1,\cdots,av_n)-a^k(v_1,\cdots,v_n)$
II. The basis of $H$ for the case of one Vector space $n=1$ with finite dimension $d > 1$ and over $\mathbb R$ or $\mathbb C$, is uncountably infinite. Since such maps are determined by their action on all lines through the origin. To see this, take a basis in $V$, then each direction is determined by $d-1$ numbers, and there is uncountably infinite directions to determine the action of $\phi$ on them.
In contrast to ($p$-)linear maps on $V$ into $V$ (for example), which requires only $d^{(p)}\cdot d$ numbers to determine a ($p$-) linear map, (finite dimensional tensor product space.)
Just to help you close out this question, I'm going to put the consensus here: this looks like a good construction of a universal property for homogeneous maps.