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Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute. There is another possible d …

I doubt that someone could learn higher category theory (and more in general higher dimensional algebra) without first studying a little of category theory, mostly because the definition given in such …

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any refere …

Direct to the point. Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that s …

Of course you can define a (just-arrow) category $\mathcal C$ like a partial algebra which consist of: a set $\mathcal C$ (namely the set of arrows of your category), a set $D_\mathcal{C} \subseteq \ …

I cannot say what exactly Voevodsky meant but here is a wild guess. Disclaimer in what follows I use heavily type theoretic notation, so you have trouble understanding feel free to ask in the comment …

Could anyone explain what higher dimensional algebra is? I tried to look on the web but I couldn't find a satisfactory definition, the ones that I found are too vague. What I'm looking for is a good …

Is there a way to recapture the additional understanding imparted by multicategories using higher categories? Well I would say so. Multicategories are basically categories whose morphisms have multi …

I think the idea should pretty much like this: once you drop the requirement for the deductive system to be freely generated from the axioms by the inference rules (i.e. you accept the existence of no …

It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way …

Another interesting reason why categories cannot be identified always with categories having functions for morphisms is given in this paper, by Peter Freyd in which is proven that there are some categ …

I am not an expert on the subject but looking at the definition I would dare to say that these are basically the axioms that characterize an homology theory, if it weren't for the fact the $\delta$ ma …

Following the previous indication of Professor Brown I want to add another possible way to see natural transformation which is a generalization of the previous definition. Given categories $\mathc …

I am not aware of Kock's works. Nevertheless Kelly provides the definition of its tensor product in the next page: it defines its tensor product $\mathcal A \otimes_{\mathcal F} \mathcal B$ as the $\ …

In every $n$-category (weak or strict) can be defined the concept of equivalence via a recursive definition: * an equivalence in a set ($0$-category) is just an identity; * for each $n \in \mathbb N$ …