0
votes
1answer
151 views

Cocontinuous functor out of the terminal category

Let $\mathcal{C}$ be a small finitely complete category equipped with a Grothendieck (pre)topology $\tau$. For $\ast$ the terminal category (one object, one morphism), denote by $i: \ast \to ...
5
votes
2answers
283 views

Is the site of (smooth) manifolds hypercomplete?

By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
3
votes
0answers
77 views

How does the machinery of left-exact comonads generalize from sheaves to stacks?

Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...
6
votes
2answers
449 views

Can one characterize those sheaves which have Hausdorff etale spaces?

Given a sheaf of sets $F$ on a space $X,$ under the equivalence of categories between etale spaces over $X$ and sheaves over $X,$ $F$ is associated to a local homeomorphism $$E\left(F\right) \to X$$ ...
7
votes
0answers
303 views

Are there simple conditions on a category C which guaranty that Ind(C) is a Grothendieck topos?

The category of finite sets is not a Grothendieck topos, but its Ind category Ind(Finite-Sets) = Sets is a Grothendieck topos. Similarly, given a pro-finite group G, the Grothendieck topos of discrete ...
3
votes
1answer
126 views

Which limits are preserved by prolongation of presheaves?

Let $F:C \to D$ be a full and faithful functor between small categories. Then we get a triple of adjoint functors $F_! \dashv F^* \dashv F_*$, with $$F_!:Set^{C^{op}} \to Set^{D^{op}}.$$ Notice that ...
16
votes
5answers
702 views

Sheafification via hypercovers

The sheafification of a presheaf on a site is often constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is ...
1
vote
1answer
231 views

Do Categorical Quotients Preserve Covering Maps?

Before asking a question, please let me write down settings. SETTINGS: Let $C$ be a category with fiber products and $B$ be a closed subcategory of $C$ (i.e. $B$ contains any isomorphism of $C$, and ...
1
vote
0answers
351 views

Fine and acyclic sheaves on locales

Hey all. Here's the thing, so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and ...
1
vote
1answer
392 views

Question about the definition of a sheaf cohomology group for a sheaf using tensor products of sheaves

In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a ...
3
votes
1answer
440 views

Question on the interpretation of a presheaf category as a co-completion

The category of presheaves $Pre(C)$ on a small category $C$ is the category of functors $C^{op}\to Sets$. Since the category of sets is co-complete and every presheaf is a colimit of representable ...
6
votes
1answer
240 views

Automorphisms of constant sheaves

Let E be a Grothendieck topos, such as the category of sheaves of sets on a topological space. Then there is a unique geometric morphism $(\Delta \dashv \Gamma)\colon E\to \mathrm{Set}$, where ...
14
votes
6answers
3k views

How should one think about sheafification and the difference between a sheaf and a presheaf

The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
3
votes
2answers
300 views

Colimits of covers

Suppose I have category $C$ equipped with a Grothendiek pretopology of covers, and let $y:C \to Sh(C)$ be the Yoneda embedding into sheaves and $y/c:C/c \to Sh(C)/y(c)\cong Sh(C/c)$. How can I show ...
0
votes
0answers
109 views

Are sieves in locally small categories still sets?

In "Sheaves in Geometry and Logic", M&M define a sieve of an object $C$ as a downward-closed set of arrows $S$ with codomain $C$. They go on to say that for a locally small category, a sieve of ...
3
votes
0answers
200 views

Descent of singular cohomology

When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the ...
16
votes
1answer
1k views

Cohomology of sheaves in different Grothendieck topologies

Suppose I have a sheaf $\mathcal{F}$ on the (small) ├ętale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. ...
7
votes
1answer
332 views

Category of copresheaves over commutative monoids

Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets. ...
5
votes
1answer
496 views

Sites which are stacks over themselves

A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a ...
6
votes
1answer
513 views

Encoding fuzzy logic with the topos of set-valued sheaves

One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset ...
10
votes
2answers
666 views

Does sheafification preserve sheaves for a different topology?

Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 \cup T_2$ (by which I mean the smallest Grothendieck topology on $C$ containing $T_1$ and $T_2$). ...