I’m interested in generalized smooth spaces. I know there are several spaces such as Deffeological space, Frölicher space, Chen space, etc… and there are some papers compare them. However, I haven’t found one which compares smooth topos to them. (I don’t have literacy to do it on my own yet.) What are characteristics of smooth … Read more
In SGA7 Exposé XIII, Deligne introduces an algebraic theory of nearby cycle sheaves and vanishing cycle sheaves on schemes over the S, where S is the spectrum of a Henselian discrete valuation ring. Let i:s↪S denote the closed point of S. As S is Henselian, pullback along i induces an isomorphism from the finite étale … Read more
Let X be a scheme. Let (\mathrm{Sch}/X)_{ét} and X_{ét} be the big and small étale sites, resp. Is the restriction functor from \operatorname{Sh}((\mathrm{Sch}/X)_{ét}) to \operatorname{Sh}(X_{ét}) an equivalence of categories? I’m guessing it isn’t, but I haven’t been able to come up with a counterexample. Answer AttributionSource : Link , Question Author : Avi Steiner , … Read more
Is there a model of Intuitionistic Higher-Order Logic in which the following axiom is true? Covering Axiom: Any true statement of the form ∀x∈A,∃y∈B,ϕ(x,y) gives rise to an open covering of A, and a continuous mapping from each set in the covering to B, such that ϕ(x,f(x)) is true. I understand that there are difficulties … Read more
Categories of sheaves in [C,Set]Cat (functor category) are equivalently left exact reflective subcategories of presheaf toposes. Categories of sheaves on a topos [C,Set]Cat are also in correspondence with Lawvere Tierney topologies on [C,Set]Cat. Is there a topos Ω_ such that geometric morphisms from T to Ω_ are in correspondence with Lawvere-Tierney topologies on T? This … Read more
I wanted to have a better understanding of the geometric interpretation of $\infty$-toposes, and in particular learn something about étale morphisms, but I got stuck trying to unravel two points in the proof of HTT 6.3.5.13. At the end of the proof, it is claimed that the $\infty$-category of Cartesian sections of $\mathcal{Z}” \times_S (N(\Delta)^{op} … Read more
Let X be a topological space. Then X is irreducible if and only if in the internal language of Sh(X), ⊥ is not true and we have one side of De Morgan’s law, that is, for all formulae ϕ,ψ, we have ¬(ϕ∧ψ)⇒¬ϕ∨¬ψ. (Proposition 7.9 of “Using the internal language of toposes in algebraic geometry” by … Read more
Suppose I have category C equipped with a Grothendiek pretopology of covers, and let y:C→Sh(C) be the Yoneda embedding into sheaves and y/c:C/c→Sh(C)/y(c)≅Sh(C/c). How can I show that if F:J→C/c is any functor such its diagram consists only of elements of covering families, then: (y/c)∘lim→F=lim→(y/c)∘F? For example, this is true if C=Top (topological spaces) and … Read more
Is there a name for the class of geometric morphisms whose counits are epic (equivalently, whose direct images are faithful)? This notably includes the class of inclusions/embeddings of toposes. By Example A4.6.2 in Johnstone’s Sketches of an Elephant: A Topos Theory Compendium, all such morphisms are localic, but since Johnstone doesn’t include a name there … Read more
Certain Galois toposes can be written as lim where (G_i)_{i \in I} is an inverse system of discrete groups. (The limit is a strict limit in the 2-category of Grothendieck toposes and geometric morphisms.) What are the objects and morphisms in this topos? If all groups G_i in the diagram are finite and the transition … Read more
