By the cobordism hypothesis, free higher monoidal categories basically describe the same thing as manifolds immersed in R^n

So, what corresponds on the category theory side to complex manifolds immersed in C^n?

@catnaroek johncarlosbaez julesh I would argue that the notion of oo-category (whatever definition you use) is not purely algebraic. For example, the (oo,1)-category of (oo,0)-categories is the oo-category of spaces (so for example, it contains a lot of information about smooth structures on spheres).