Associativity in category theory

Question

In Category Theory, the composition of morphisms must be associative. What would happen if we give up this associative law? For example Lie algebras are vector spaces with non-associative binary relation. How can we carry this idea to category theory ?

Answer 1

Lie algebras aren't just vector spaces with a non-associative binary operation. That binary operation has to satisfy a very important condition, the Jacobi relation, and be antisymmetric. It's a general fact that, the more you drop "axioms" for an algebraic structure (associativity, unit...), the less you can say about that structure, because you're describing more and more general objects.

In general when you have a "normal" algebraic notion, for example "monoid", "group", "Lie algebra"... The process of translating that into a "categorical" notion is called Categorification. This isn't really a precisely defined concept but more of a heuristic. The general idea is to replace sets by categories, functions by functors, etc. This MO question gives some information and references. This article of Baez–Dolan looks accessible.

There are basic examples of categorifications. For example, a monoid is a set $M$ with a unit $1 \in M$ and a binary operation $\cdot : M \times M \to M$. The "categorification" of this is just a category! A category is a collection of objects. The unit becomes the identity of each object, and the multiplicative law becomes the composition. The idea is that a category with a single object is a monoid, and a category is a "higher level monoid".

Similarly, a group is a monoid where all elements have inverses. The categorification of this is a groupoid, a category where all the morphisms are invertible, and indeed a groupoid with a single object is a group. There are plenty of examples like that. For example, $k$-linear categories are a categorification of $k$-vector spaces in that sense.

If you're really intent on dropping the associativity requirement, there is a notion called $A_\infty$-categories. This is a somewhat advanced notion, and I suggest that you get very comfortable with category theory and homotopy theory before trying to tackle them. The basic idea is that rather to have strict associativity $f \circ (g \circ h) = (f \circ g) \circ h$, you only want $f \circ (g \circ h)$ and $(f \circ g) \circ h$ to be in some sense "equivalent", that you can "continuously deform" one into the other. There are many things to say about $A_\infty$-categories, and if you're interested I suggest you check out the references given in the $n$Lab article, once you're comfortable with category theory.