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Here is a reformulation/generalisation of G. Stefanich's counterexample, showing that sheaf-locality can fail very dramatically once we leave the realm of locally finitely presentable categories. More …

Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams. For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X) …

Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf of sets" (i.e. …

It sounds to me you seek a solution to the following problem: Given a suitable subcategory of $\textbf{Set}$, find a Grothendieck topology on $\textbf{Set}$ so that sheaves on $\textbf{Set}$ are equi …

To answer your question directly, WISC does not imply the existence of subobject classifiers. Notice that when there are only trivial covers, WISC is trivially satisfied, so it suffices to find a cate …

There is no hope for this in any subcanonical topology coarser than the fppf topology, or more generally, any subcanonical topology in which morphisms $\operatorname{Spec} C \to \operatorname{Spec} K$ …

My position is that the definition of $\mathcal{C}$-valued sheaves for completely general categories $\mathcal{C}$ is not yet a settled matter. For locally finitely presentable categories $\mathcal{C} …

There is a precise, almost literal, sense in which $f^* : \textbf{Sh} (Y) \to \textbf{Sh} (X)$ generalises the inverse image as defined in elementary set theory. Observe that open subspaces $V \subset …

I think the language of classifying toposes is helpful in understanding this view of forcing. Let $P$ be a poset. The set theorists have the intuition that forcing over $P$ adjoins a generic filter of …

Following a suggestion of Thomas Holder, we can close the gap as follows: For each object $Y$ in $\mathbf{T}$, there is a pseudonatural local geometric morphism $\mathbf{Sh}(\mathbf{T}_{/ Y}) \to \m …

The functor $i$ does not have the cover lifting property in general. If it did, then every epimorphism $X \to 1$ in $\mathbf{Sh}(\mathcal{C}, \tau)$ would be an isomorphism, or equivalently, every $\t …