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20 votes

Resources for topos theory

For a beginner, the more accessible textbooks seem to be the following two. Francis Borceux, Handbook of Categorical Algebra, Volume 3. Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and ...
Dmitri Pavlov's user avatar
17 votes
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Voevodsky's Triangulated Categories of Motives and their Relationships

I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint. Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with ...
Mikhail Bondarko's user avatar
14 votes

Resources for topos theory

Dmitri has mentioned two fantastic references, which are very complete and well written. I will mention two short references for those that want to get the general idea, before approaching a complete ...
Ivan Di Liberti's user avatar
13 votes

Voevodsky's Triangulated Categories of Motives and their Relationships

I won't embark on the difficult question of what one wants out of a category of motives, but I can make some comments on what might motivate the various choices of topologies. Nisnevich (aka ...
Shane Kelly's user avatar
13 votes
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Are all Grothendieck topologies on Set equivalent?

(Much of this has basically been said by someone in the comments already.) Here is a way of making examples of topologies on Set. Let $\mathcal C$ be a class of sets. Define a topology on Set by ...
Tom Goodwillie's user avatar
13 votes

Classical point-set topology using Grothendieck topologies

Point-set topology is used to formalize the intuition of continuity and of convergence. It finds its ideal applications for example in Analysis. The notion of Grothendieck topology is designed to ...
Qfwfq's user avatar
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11 votes
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Why care about Grothendieck topology?

Etale topology, required to define etale cohomology, is not a topology in the usual sense. It is Grothendieck topology only. In the category of topological manifolds, an etale cover of $X$ is a ...
Bugs Bunny's user avatar
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11 votes
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When is a basis of a topological space a Grothendieck pretopology?

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By a “base” in this answer I mean what appears to be the most common definition: a ...
Dmitri Pavlov's user avatar
10 votes
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Applications of $h$-topology and $h$-descent

I guess a relevant part of the philosophy is that schemes are smooth locally in the $h$-topology. So, if we are in characteristic zero, we can use resolution of singularities to produce an $h$-...
Matthias Wendt's user avatar
10 votes
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Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"

$\DeclareMathOperator\im{im}$Actually I couldn't quite figure out how to do a Ken Brown sort of argument, but here's an argument that works: Let $L$ denote the usual sheafification functor, a la ...
Dylan Wilson's user avatar
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9 votes

Voevodsky's Triangulated Categories of Motives and their Relationships

Some comments regarding how these categories are related: In reasonable situations, all your categories should satisfy the gluing property for an open subscheme and closed complement. This property ...
Tom Bachmann's user avatar
  • 1,951
9 votes

Needless axiom for Grothendieck topologies?

The only important axiom in order to define a notion of sheaf is the stability under pullback. There is a proposition in SGA4 saying that if you have a family of sieves only satisfying the pullback ...
Simon Henry's user avatar
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9 votes
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Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site?

Take $U=\coprod_{i∈I}Y(U_i)$, where $Y\colon C\to\mathop{\rm Presh}(C,{\rm Set})$ is the Yoneda embedding. We have a canonical morphism $U→Y(X)$. The Čech groupoid of $J_c$ can now be defined as the ...
Dmitri Pavlov's user avatar
9 votes
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Relationship between canonical topology on a topos and its site of definition

This is essentially correct, and there is no need for the topology to be subcanonical. But let me clarify: Whether the topology is subcaninical or not, we have the following: given any family of maps $...
Simon Henry's user avatar
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8 votes
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Subsheaves of Spec K, K a field

There is no hope for this in any subcanonical topology coarser than the fppf topology, or more generally, any subcanonical topology in which morphisms $\operatorname{Spec} C \to \operatorname{Spec} K$ ...
Zhen Lin's user avatar
  • 14.9k
7 votes

Classical point-set topology using Grothendieck topologies

I'm not quite sure what you're asking, but under one way to interpret the question, an answer is that the theory of locales is a well-developed alternative to the classical theory of topological ...
Mike Shulman's user avatar
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7 votes
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Subobject classifier for sheaves on large sites with WISC

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 ...
Zhen Lin's user avatar
  • 14.9k
6 votes

How to construct cup-product in a general site?

Yes. This is treated in detail in Section 8.4 of Jardine's book “Local homotopy theory”. See also the introduction to Chapter 8 there for a historical comment on cup products and Godement resolutions....
Dmitri Pavlov's user avatar
6 votes

Is there a category of topological spaces such that open surjections admit local sections?

Maybe this is not quite satisfactory answer, but the category of zero-dimensional Polish spaces and their continuous maps has the required property: each open continuous map between Polish zero-...
Taras Banakh's user avatar
6 votes

Grothendieck topologies on $\mathbb{C}$

The subpresheaf $\mathcal S\subseteq\mathcal C^0$ is not a subsheaf. Indeed, let $U,V\subseteq\mathbb C$ be two disjoint open sets, and consider the function $f:U\cup V\to\mathbb C$ assigning $0$ to ...
John Pardon's user avatar
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6 votes

Is there a way to "puncture" a topos?

A more general question is how do you form the image of a geometric morphism f : E -> F, as a subtopos of F, and how do you form its complement? Recall that if U is a subobject of 1 in F then it ...
Gavin Wraith's user avatar
6 votes
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Group scheme with an isotrivial maximal torus

Edit. I realized after the original post that there are even easier examples. These examples also show that the Quillen–Suslin Theorem fails already for smooth affine quadric hypersurfaces (in some ...
6 votes
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Is the slice of a subcanonical site also subcanonical?

Isn't this very basic? If $\{a_i \to b\}$ are compatible morphisms in $\mathcal{C}/c$, then these are compatible morphisms in $\mathcal{C}$, hence they glue to a unique morphism $a \to b$, and this is ...
Martin Brandenburg's user avatar
5 votes
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Exercise on "locality" in topos theory

Let $ \chi : X \rightarrow \Omega$ be the characteristic function of $U$. By definition of a subobject classifier, the characteristic function of the pullback of $U$ by $U_i \rightarrow X$ is just ...
Simon Henry's user avatar
  • 40.5k
5 votes
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Coverages that are not pretopologies

Thanks to a prompt by Emilio Minichiello I was reminded of my claim above However, the coverage of good open covers on $\mathbf{Mfld}$ does satisfy the 'covers compose' axiom of a pretopology, which ...
David Roberts's user avatar
  • 34.3k
5 votes

Can we just use effective descent morphisms (pure morphisms) as covers?

Every faithfully flat morphism is of effective descent. However, the topology consisting of all faithfully flat morphisms is not subcanonical (i.e. it is not the case that every representable functor ...
Jonathan Beardsley's user avatar
5 votes
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When can a scheme be recovered from its descent groupoid?

In any topos, if $Y \rightarrow X$ is an epimorphism then: $$Y \times_X Y \rightrightarrows Y \rightarrow X $$ is indeed a colimit diagram. If you have a site $S$ and a cover $Y \rightarrow X$ ...
Simon Henry's user avatar
  • 40.5k
5 votes
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Do coherent sheaves on rigid analytic spaces form an abelian category?

The answer is yes. An abelian category is a category $\mathcal C$ with the following properties: $\mathcal C$ is additive. $\mathcal C$ has kernels and cokernels. Images and coimages coincide. That ...
Tim Campion's user avatar
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5 votes
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Proof without sieves: Equivalent grothendieck topologies have the same sheaves

Let me break down the statement you are trying to prove into two independent facts. This answer is not really in the spirit of the question since I will make maximal use of sieves, but for such ...
Marc Hoyois's user avatar
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5 votes
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The Grothendieck topology of closed immersions on schemes

The topology you mention is called the "closed topology" $J_\mathrm{cl}$ in "Points in algebraic geometry" by Gabber–Kelly. See also the comparison diagram by Pieter Belmans. Note ...
Jens Hemelaer's user avatar

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