Search Results

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 …

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a …

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 …

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 …

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 …

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 …

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 …

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 $Y$ …

Let $\mathcal{L}$ be the language of intuitionistic propositional logic generated by some atomic propositions $t_1, t_2, \ldots$. The Lindenbaum–Tarski algebra of $\mathcal{L}$ can be regarded as a bi …

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 …

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 …

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 …

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 $\mathcal{B}$, …

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 …

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 …