General Algebra (Universal Enveloping Algebras Generalizations)

Question

I'm trying to generalize the concept of a Universal Enveloping Algebra as much as possible, I'm trying to do it categorically but my category theory is weak since I didn't take a course that covered it yet. Is this construction acceptable in any way?

Let $C$ be a concrete category with an embedding functor $F:\mathrm{Assoc}_1\rightarrow C$ where $\mathrm{Assoc}_1$ denotes the category of Associative Algebras with unity, for an object $X \in obj(C)$ we call a pair $(U(X),i)$ an universal enveloping algebra of $X$ of type $C$ if:

1. $U(X) \in \operatorname{obj}(\mathrm{Assoc}_1)$ (it is an algebra)
2. $i$ is a morphism in $C$ between $X \rightarrow F(U(X))$ (enveloping)
3. For every morphism $\rho \in \operatorname{hom}(C)$ between $X$ and an element $F(V)$ with $V\in \operatorname{obj}(C)$ there is a unique morphism $\tilde{\rho}:U(X)\rightarrow V$ in $\operatorname{hom}(\mathrm{Assoc_1})$ such that $F(\tilde{\rho})\circ i = \rho$ (universal)

And use as an example the category of Vector Spaces $\mathrm{Vec}$ and the tensor algebra as a universal enveloping algebra of type $\mathrm{Vec}$. I didn't find many results that generalized a definition of it outside of uses in other specific areas and failure of existence in some cases.

Answer 1

Your generalisation is not only perfectly acceptable, it is basically one of the core concepts of category theory.

First, let me restate your definition a little bit. Let $F:D\to C$ be a functor ($C$ and $D$ can be any categories) and let $X$ be an object of $C$. We define the category $X\downarrow F$ to be the category whose objects are pairs $(Y,\alpha)$ where $Y$ is an object of $D$ and $\alpha:X\to F(Y)$ is an arrow in $C$, and the morphisms $(Y,\alpha)\to (Z,\beta)$ are morphisms $f:Y\to Z$ such that $F(f)\circ \alpha=\beta$. Then an "universal enveloping object of $X$ of type $C$" $(U(X),i)$ is an object of $X\downarrow F$ with the property that for all other objects $(V,\rho)$, there exists a unique morphism $\tilde{\rho}:(U(X),i)\to (V,\rho)$ in $X\downarrow F$; in other words, it is an initial object of $X\downarrow F$.

If such universal enveloping objects exist for all objects of $C$, then there is a canonical way to define morphisms $U(X)\to U(X')$ in $D$ for all $h:X\to X'$ in $C$ : you can simply use the universal property with the morphism $X\to X'\to F(U(X'))$. Then $U$ is in fact a functor, and the universal property you've described means that $U$ is a left adjoint functor of $F$.

There are numerous examples of this. You've already mentioned the tensor algebra as left adjoint to $\mathrm{Assoc}_1\to \mathrm{Vec}$. You can also take the forgetful functor $F:\mathrm{Ab}\to \mathrm{Grp}$, and then define $U(X)$ to be the abelianised group and $i:X\to F(U(X))$ the quotient map; this will have the same property. Another common example is "free-forgetful" adjunction : you can take the forgetful functor $\mathrm{Vec}\to\mathrm{Set}$, and then the free vector space on a set $X$, together with the canonical embedding of $X$ into it, will satisfy the same property again. For more examples, see this question.