Let's start with the following random example: If $F$ is a presheaf, then for every chain of open subsets $U \subseteq V \subseteq W$, the morphisms $F(W) \to F(V) \to F(U)$ and $F(W) \to F(U)$ coindice. But this may be an evil (nlab link) definition, especially when the values of $F$ are categories (which occurs in my current research). So we should just impose that these morphisms are equivalent. But then we arrive at compatibility conditions between these equivalences, which are, again, equalities, and may be evil. So what is the consequence of this: Should every mathematical theory take place in a $\infty$-category? Or is 'real' mathematics basically evil? I know this question is quite imprecise, but currently I just don't see where and why this process of increasing the "depth" of category theory should end. Anyway, I have to admit that my knowledge of higher category theory is very, very rudimental.

Here is a evil question, which is part of my confusion: Let $Ring$ denote the category of rings, ring homomorphisms and $AbCat$ denote the category of abelian categories, functors. Then $Ring \to AbCat, A \to Mod_A$ should be a functor, right? If $A \to B$ is a ring homomorphism, just tensor with $B$ over $A$. But then functoriality is only satisfied up to equivalence ($2$-isomorphism) in $AbCat$. To get a honest functor, we might mod out these equivalences to get a $1$-category $\tilde{AbCat}$. But I don't think that this is the most natural way to handle this. Or we may define quasi-functors (cf. the presheaf example). Anyway, may we *think* of it as a usual functor, without turning into troubles? Or is it important, in practice, to have this higher category theoretic point of view? Or is it possible to turn this functor into a honest functor, by choosing the tensor products $M \otimes_A B$ carefully? I think the latter is interesting, although I know that nobody really cares about such evil equalities like $(M \otimes_A B) \otimes_B C = M \otimes_A C$.

`Mod_A`

into the category of presentations of A-modules (i.e. an object is a set X together with a submodule of relations in the free module on X), and this is functorial. $\endgroup$ – Tyler Lawson Sep 14 '10 at 17:51monoidal categoryis the same as a ("weak") 2-category with one object, and monoidal functors are functors along with some coherent natural isomorphisms. The coherency conditions are that the isomorphism respect the various associator structures, and sometimes fail. A notable example is that of a the representation theory of a quasi-Hopf algebra: the category of representations has a monoidal structure coming from the quasi-Hopf structure, but the forgetful map does not respect the associators. $\endgroup$ – Theo Johnson-Freyd Sep 15 '10 at 5:18