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Let me expand on my comments. Functorial deformations for $\mathbf{K} (\mathcal{A})$ (the chain homotopy category) are much easier to obtain in practice than for $\textbf{Ch} (\mathcal{A})$ because K- …

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 …

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 …

This is going to be a long post, so let me state my question first and then explain what I am interested in by way of examples. Question. Is there any literature studying notions of generation under c …

Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones. We can make $\textbf{Top}$ into a simplicially enriched category as follows: Given topological spaces $X$ and $Y$ …

In [Schwänzl and Vogt, Strong cofibrations and fibrations in enriched categories], the authors refer to an earlier preprint, [Schwänzl and Vogt, Cofibration and fibration structures in enriched catego …

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$: Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then …

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 …

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 …

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 …

The quotation in the title is attributed to Frank Adams and appears in several places: In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not necessar …

Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, …

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 …

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 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 …