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I believe is more a matter of tastes, personally I find easier and simpler n-fold categories than categories. For me n-fold categories are more natural and so are easier, for different reasons: for …

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 …

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 \ …

Maybe this doesn't address completly to the question, but I think it's a start. Functor categories and higher categories are quite different objects, the only relation that I can think of is that usu …

If I understand correctly your question you are looking for some definition of limit-sketches such that if two sketches $\mathcal T$ and $\mathcal T'$ are Morita-equivalent (that is the categories of …

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 suppose that there are possibly many different answer to this question. Here is the one I got. Being a reflector is equivalent to being an inclusion that has a left adjoint. In general being a co …

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 …

By Yoneda for every natural transformation $\tau \colon \mathbf C(c,-) \to F$, where $F \colon \mathbf C \to \mathbf{Set}$ and $\mathbf C(c,-)$ is the covariant $\hom$-functor, is a family of function …

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 $n \in \mathbb N$ …

Yesterday I was wandering for the $n$-lab and I've found the definition of $n$-poset. Following this post it seems that a $n$-poset should be a $(n,n+1)$-category. Now an $(n,r)$-category should be a …

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 …

A little preliminary: I'm an undergraduate student and I started to study category theory as self-taught at the beginning of second year of university, mostly because of my interest in logic and found …

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 …

Well the two monads are quite different: in May definition you deal with an actual monad in $\mathbf {Cat}$ (i.e. a strict-$2$-category) while in the second case you work with monads in the bicategory …