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First, there is indeed nothing mathematically very deep in this observation, and I agree that the word "breakthrough" might be exaggerated. But on the other hand lots of very deep ideas look trivial o …

I've heard about two ways mathematicians describe Feynman diagrams: They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in a monoidal clo …

The answer is yes, in fact one has a lot better than bi-interpretability, as shown by the corollary at the end. It follows by mixing the comments by Martin Brandenburg and mine (and a few additional d …

The problem is that the question is highly dependent on the definition of $n$-groupoids. The notion of strict $n$-groupoid is very clear and precise but we know very well (and Grothendieck knew that) …

In some of my works I need to prove some results within the internal logic of categories with not much structures (like pretoposes or even just categories with finite limits). The kind of things I wan …

You believe the 1-category is interesting but somehow not the bicategory, yet one could say the exact same thing one dimension below: Why is this category even interesing while you could just consider …

I just realized that my comment actually answer completely the case of a Cartesian closed category with finite coproducts, and not just extensive categories. So I'm posting it as an answer. The short …

Yes there is: the formal locale of p-adic integer is simply defined as the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a pro-finite locale). So internally in any topos a continuous function …

First, it is important to distinguish between the problem related to the foundation you are using from the problems that are inherent to category theory. For example, the distinction between $\mathbb{ …

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales). For the record, this is proved by, starting fo …

By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict to separab …

The bicategory of locally presentable categoires, and left adjoint functor between them, is monoidal closed for the Kelly tensor product. My question is what are the dualizable objects for this monoid …

First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone The Art of pointless thinking which gives a lot of insight on how locale theory works. Here are some o …

This follows from two Facts: 1) A category monadic over Set/S is always an exact category. That is it has quotient by equivalence relation that are effective and universal. It is in particular a regu …

First - yes, for symmetric set-operads this functor is "injective", though it is not fully faithful. It is faithful on general maps and fully faithful on isomorphisms. Its image can easily be characte …