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The algebras for this monad can be described in essentially the same way: they are sets in which it makes sense to to take "convex combinations" of countably many elements. More precisely, an algebra …

Your claim is incorrect because you truncated the simplicial diagram too much. Indeed, if what you said were true, then the isomorphism class of a group would be determined by its cardinality, but thi …

I would just say that the inclusion preserves covering families (in the naïve sense). You don't need Grothendieck pretopologies to make sense of this – just plain coverages (in the sense of Johnstone; …

There are several possible definitions of initial object in a 2-category $\mathfrak{K}$; which one is appropriate depends on your applications. A 2-category has an underlying ordinary category, so we …

There is definitely discussion of internal presheaves – the whole of section B2.5 is about them! In particular, the result you seek is Corollary 2.5.8: [Let $\mathcal{S}$ be a cartesian category with …

This "tensor product" is also known as the weighted colimit in enriched category theory. The short answer is that all the isomorphisms you are interested in always exist, provided the objects you are …

First things first: $\check{H}{}^n(U, \mathscr{F})$ (resp. $H^n(U, \mathscr{F})$) are same whether you regard $\mathscr{F}$ as an $\mathscr{O}$-module or as an abelian sheaf, so we may simplify things …

Hirschhorn's book, Model categories and their localizations, is a very thorough reference with many basic results explicitly stated and proved. The result you want is implied by axiom SM7 for simplici …

There is in fact no difference between the two definitions if you take your site to be the category of affine schemes – while it is true that the forgetful functor from sheaves to presheaves does not …

I will write $[B, C]$ instead of $\underline{\mathrm{Hom}}(B, C)$. Recall the tensor–hom adjunction: $$\mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, [B, C])$$ Thus there is a canonical bijection …

Fundamentally, working in NBG is not much different from working in ZFC, except that you are allowed one level of freedom to form collections of sets that are not themselves sets. As such, you still h …