Properties not shared by the category of associative algebras and the category of non associative algebras What (interesting) properties are not shared between these two categories:


*

*The category of associative algebras

*The category of (not necessarily associative) algebras


Motivation: Some people define an algebra to always be associative. I want to understand what this changes at the level of categories.
 A: The categories of associative and non-associative algebras have a lot in common: both are defined by operations and equations, so in particular they are cocomplete monadic categories. This implies that they are exact and have a regular projective generator. They are also both protomodular, because their theory contains a group operation (the addition). If you do not require the existence of a unit, they are pointed, and in particular, semi-abelian.
Now in a semi-abelian category, one can call normal the subobjects which are kernels; and there are some operations that one can define on subobjects (normal or not), such as joins and meets. Among these operations, one has commutators, which can be defined as an operation on subobjects; and it's often useful to know whether the normal subobjects are closed under commutators or not (this is explained in more details in this paper).
And this is where the categories of associative and not necessarily associative algebras behave differently. Indeed, in both categories normal subobjects are just ideals, and the commutator of two subalgebras $A,B$ is their product. And while the product of two ideals is again an ideal in the associative case, it is no longer true for a non-associative algebra.
In fact, the closure of normal subobject under commutators is a consequence of a property called algebraic coherence; it turns out that the category of associative algebras is algebraically coherent (and so are the categories of Lie and Leibniz algebras), but the category of not necessarily associative algebras is not (and neither is the category of Jordan algebras).
