I'm Looking at a paper using higher category theory. Would you please answer these questions ?

1) A weak n-category (or $\infty$-category) has objects and k-morphisms. What is an invertible morphism ?

2) What is the relation between topological categories (hom sets has topology and composition is continuous) and ($\infty$,1)-categories ?

3) I know what a symetric monoidal category is. Roughly speaking: what additional constraint/data do we have in a symetric monoidal n-category ($\infty$-category) ?


The answers to all three questions depend strongly on the particularities of the formalism of higher categories used. In any case, the answers are rather complicated, and I will just give the gist of the idea for each. An invertible morphism is a morphism which has an inverse morphism. In the context of weak $n$-categories that usually means there is a morphism which when composed with the original morphism is, up to an invertible morphism of one level higher, an identity morphism. For the second question, ot depends what exactly do you mean by an $(\infty, 1)$-category. If you mean that in the sense of Lurie, then there are suitable model categories on tological categories and on simplicial sets which are Quillen equivalent. Monoidal categories involve a notion of product for morphisms. So, in a higher category a monoidal structure gives a product for the various levels of morphisms. Now, these products need to satisfy some associativity (and commtativity if you consider symmetric monoidal categories), and these can be described as the commutativity up to an invertible morphsim of one level higher. It can get very tricky.

Of course, given the plethora of definitions of $n$-categories, and the inherent intricacies, a definitive all-encompassing answer to your questions is, at this stage of the development of the theory of $n$-categories, beyond reach.


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