21
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 ...
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21
votes
Accepted
Are there (enough) injectives in condensed abelian groups?
Indeed, there are no nonzero injective condensed abelian groups.
Let $I$ be an injective condensed abelian group. We can find some surjection
$$ \bigoplus_{j\in J} \mathbb Z[S_j]\to I$$
for some index ...
- 21.3k
16
votes
Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?
To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)...
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16
votes
Accepted
Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?
Yes, your counterexample seems to be correct — there’s an error in Def C2.2.1 as printed. This issue is mentioned in the n-lab’s article on dense sub-site (current revision permalink), which notes a ...
- 21.3k
15
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 ...
- 9,111
13
votes
Accepted
Tensor product of sites
The category $H$ can be described as the category of $E$-valued sheaves on $D$, or $F$-valued sheaves on $C$.
You get a site by taking the category $C \times D$ and taking the topology generated by ...
- 42.4k
11
votes
Accepted
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 ...
- 13.5k
9
votes
Accepted
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 $...
- 42.4k
8
votes
Accepted
Points of the big Zariski site
Let's simplify and consider the presheaf topos.
I asked the same question over at the nForum a while back. There Marc Hoyois reminded me of the following quite general fact: The category of topos-...
- 3,302
7
votes
"Covering-flat" part in definition of morphism of sites
Short answer: A morphism of sites as you define it (a functor which preserves covers and the fibre products showing up in the gluing condition) gives rise to an adjunction
$$f_s:\mathrm{Sh}(\mathbf{C})...
- 163
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....
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6
votes
Accepted
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 ...
- 63.1k
4
votes
Resources for topos theory
Here are some references on Steven Vickers works to look at topoi from other "geometric" and logical sides.
S. Vickers, Locales and Toposes as Spaces Chapter 8 in M. Aiello, I. Pratt-...
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4
votes
Accepted
Has this "backwards" perspective on toposes been studied?
Actually, the closure operator of a topology is a finite colimit preserving monad on a powerset.
4
votes
Accepted
Necessary and sufficient conditions for all sheaves on a site to be continuous functors?
Given a category $C$, and a familly of co-cone in $C$ (you can take all colimit cocone in $C$ if you want - the family doesn't even have to be small) there is a smallest topology on $C$ so that ...
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3
votes
Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"
I have found a very elementary proof, where the presheaves are allowed to take values in any category $\mathcal{A}$ with enough limits (however Dylan’s argument can be used to show this in the same ...
2
votes
Accepted
Does the (Vistoli-)sheafification induce isomorphism?
You cannot prove this since it is not true. The map $F \to F^+$ is an isomorphism if and only if $F$ is already a sheaf. (Also notice that Vistoli, of course, does not claim that $F \to F^+$ is an ...
2
votes
Accepted
What to call a morphism of sites inducing an equivalence on categories of sheaves?
Johnstone's Sketches of an Elephant (2 volumes) is a standard reference which uses "Morita equivalence" in this way. In fact, Jonstone systematically uses "Morita equivalence" in a similar way across ...
- 63.9k
2
votes
Accepted
Stacks for the extensive topology?
This a genral fact: the coproducts in an extensive category are disjoint so the sheaf/stack condition with respect to a cover by the coproduct injection exactly say that the coproduct is sent to a ...
- 42.4k
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