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In short: always. Indeed, given a functor $F : \mathcal{C} \to \mathcal{D}$ left adjoint to $G : \mathcal{D} \to \mathcal{C}$, the triangle identities say that the composites of the canonical morphis …

Yes: every morphism of Grothendieck toposes arises as a morphism of sites. (However, the site may depend on the morphism.) This is Corollary C2.3.10 in Johnstone's Sketches of an elephant. The argumen …

First things first: we need a more tractable definition of "continuous". Let $\mathcal{C}$ and $\mathcal{D}$ be categories, let $J$ be a Grothendieck topology on $\mathcal{C}$, and let $K$ be a Gr …

Let me expand on my comments. Assuming a morphism is flat and locally of finite presentation, it is surjective if and only if it is a universally effective epimorphism. A morphism $f : X \to Y$ of s …

Apparently, there is an abstract nonsense argument that shows $\mathbf{Sch}$ is concretisable. Here is a hands-on proof. We define $U_0 : \mathbf{Sch} \to \mathbf{Set}$ to be the functor that sends …

This is not true even for affine schemes. Let $k = \mathbb{Z}$, let $X = \operatorname{Spec} \mathbb{Z}$, let $Y = \operatorname{Spec} \mathbb{F}_p$, and let $Z \cong \operatorname{Spec} \mathbb{Z} [ …

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 …

This is ultimately the same construction as the one Simon Henry describes, but you might like the different perspective. Definition. Let $A$ be a commutative rig and let $L$ be a distributive lattice. …

If your codomain is a (2, 1)-category then lax colimits are the same as pseudocolimits, which are a strict kind of homotopy colimit. For the very special case of diagrams over a one-object groupoid, t …

This is true for abstract nonsense reasons. If $G$ is a group object in a category $\mathcal{C}$, then $G \times S$ is a group object in the slice category $\mathcal{C}_{/ S}$, and there is a natural …

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 …

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

It is well-known that the category of local rings and ring homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with …

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $\mathcal{F}$ actually is fibred in categories (and not higher categories), then you can trunc …

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 …