0
votes
1answer
138 views

Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R. Is it possible to extend this function to a smooth function on R²? ...
10
votes
1answer
642 views

Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory? Motivation: In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
1
vote
1answer
101 views

Are the pullback functors of adjoint functors also adjoint?

Given adjoint functors $F: A \to B$, $G: B \to A$, if you then take their pullback functors $F^* : Set^B \to Set^A$ and $G^* : Set^A \to Set^B$ given by pre-composition, are these two also adjoint ...
6
votes
1answer
329 views

What information is lost in $X \to \mathrm{Sh}(X)$?

Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure? Thanks, Adrian.
7
votes
0answers
156 views

What's the Hochschild homology of the category of constructible sheaves?

Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?
18
votes
0answers
555 views

The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal. Suppose that $Q$ is a presheaf on ...
4
votes
1answer
192 views

Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...
0
votes
1answer
137 views

Sheaffication using a $\lambda$-transfinite colimit

I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway. I was reading this article ...
1
vote
1answer
176 views

What is the connection between direct/inverse image of set maps and direct/inverse image functors of sheaves?

Given a function of sets $f:X\to Y$, one defines the direct image and inverse image maps: $$ f_*:\mathcal{P}(X)\to\mathcal{P}(Y) $$ $$ f^{-1}:\mathcal{P}(Y)\to\mathcal{P}(X) $$ In the usual way. ...
3
votes
0answers
170 views

Can one construct Freyd-Mitchell's embeddings that respect sheafifications?

For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...
2
votes
3answers
550 views

Why are (pre)sheaves defined as contravariant functors? Why not just reverse the arrows in the first place?

Why not just have arrows in the category of opens represent coverings instead of inclusions? It seems to me like both conventions (whether presheaves are co/contra and which of the two dual orderings ...
-1
votes
1answer
165 views

Pure Quotient and pure sub-object

Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ ...
8
votes
1answer
331 views

What is the difference between internal presheaves and presheaves on a total space?

Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$. ...
0
votes
1answer
159 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 ...
3
votes
1answer
277 views

Reference request: sheaves on closed sets

I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. Specifically, I say a ...
3
votes
0answers
82 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 ...
0
votes
0answers
203 views

Needless axiom for Grothendieck topologies?

Hi, The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a covering family. Why ...
5
votes
2answers
378 views

Are subfunctors of left exact functors also left exact?

Consider the direct image functor $f_*: Sh(X) \rightarrow Sh(Y)$, let $X$ and $Y$ be topological spaces, let $f: X \rightarrow Y$ be a continuous map, let $G \in Sh(X)$ be a sheaf. I was reading this ...
3
votes
3answers
163 views

Constants sheaves on an open subset

Let $X$ be a topological space and $U\subset X$ an open subset. Let's work in the category of sheaves of abelian groups on $X$. Consider the constant sheaf on $U$, $\mathbb{Z}_U$, given by ...
7
votes
0answers
231 views

Colimits of quasi-coherent sheaves on a ringed space

Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is ...
2
votes
0answers
125 views

Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$. I know well the plus-construction of sheafification, which is presented in Artin's paper ...
1
vote
1answer
184 views

quotient of ind scheme

Is I consider an ind scheme such as $G(k((t)))$ for a reductive connected group over $k=\bar{k}$ I have the conjugacy action of $G(k[[t]])$. In what category can I make the quotient ...
3
votes
1answer
191 views

Coverage, itself considered as a presheaf

A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for ...
6
votes
2answers
458 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
313 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 ...
2
votes
0answers
133 views

Does mapping cylinder category have enough injectives?

Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor. We define a category $C$ as follows: objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to ...
1
vote
1answer
315 views

Descent of Morphisms of Sheaves

While reading Brylinski I am trying to understand the descent of morphisms of sheaves. In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local ...
3
votes
1answer
132 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
3answers
856 views

What are the benefits of viewing a sheaf from the “espace étalé” persepctive?

I learned the definition of a sheaf from Hartshorne---that is, as a (co-)functor from the categor of open sets of a topological space (with morphisms given by inclusions) to, say, the category of ...
7
votes
3answers
1k views

Sheafification - Why does twice suffice?

Hi, I'm currently reading through "Sheaves in Geometry and logic" by Mclane-Moerdijk and this one issue has been bugging me for a long time, which I hope you could help me resolve. It is known that ...
17
votes
5answers
745 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 ...
7
votes
2answers
823 views

Is Sheafification Functor Exact?

I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology. My question is: ...
1
vote
1answer
232 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 ...
2
votes
0answers
313 views

Quantum sheaves

Are the following definitions known? Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions: (a) {0} and H lie in Sigma (b) If ...
4
votes
1answer
216 views

When do adjunctions preserve equivalence?

Let $\mathcal{C}'$ and $\mathcal{D}'$ be categories so that $\mathcal{C} \subseteq \mathcal{C}'$ and $\mathcal{D} \subseteq \mathcal{D}'$ are full subcategories. Suppose the forgetful functors ...
1
vote
1answer
358 views

Restriction of a sheaf to an infinitely small neighbourhood of a closed submanifold: how to work with this ind-sheaf?

Let $X$ be a manifold, $i: Z\to X$ is a closed embedding. For a sheaf $S$ (of abelian groups) on a manifold $X$ and each $\varepsilon>0$ we denote by $Z_\varepsilon$ the set of points of $X$ ...
1
vote
0answers
357 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
0answers
288 views

Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?

In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ ...
1
vote
1answer
398 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 ...
2
votes
1answer
322 views

Does anyone understand the notation in this equation for the sheafification of a presheaf on a site?

Hi there, I'm trying to sheafify a constant presheaf on a site, I went to http://ncatlab.org/nlab/show/sheafification, but can't understand the notation in the equation for W (in the proof for ...
14
votes
4answers
1k views

The tangent bundle to an infinite-dimensional manifold

Suppose that $A,B$ are smooth ($\mathrm C^\infty$) manifolds, and denote by $\hom(A,B)$ the set of $\mathrm C^\infty$-maps $A \to B$. It is a perfectly well-defined set, but often one wants more. ...
2
votes
0answers
190 views

Exercise concerning locally constant presheaves [closed]

Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. ...
10
votes
2answers
1k views

Locally constant sheaves for the étale topology, lack of intuition about “étale-localness”

I have started studying some étale cohomology and I am trying to build up some intuition about the concept of local for the étale topology. I can understand some nice examples (like Kummer exact ...
2
votes
2answers
289 views

The single-plus construction is not the left adjoint of the inclusion of separated presheaves?

Convention: Before I start, please note that it is not going to be sufficient to assume that the topology is subcanonical or that the site $C$ has finite limits, since the application I have in mind ...
4
votes
0answers
220 views

Topologies (and sheaves) on Cat and CAT

I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism ...
6
votes
1answer
244 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 ...
3
votes
2answers
313 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 ...
2
votes
2answers
290 views

Equivalence of two definitions of sheaves on a site

I'd like to prove that two definitions of sheaves on a site are equivalent, but I'm having trouble proving one direction. Let $C$ be a category with pullbacks. Let $(C,T)$ be a site defined through a ...
8
votes
4answers
2k views

Derived categories of coherent sheaves: suggested references?

I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least ...
1
vote
1answer
255 views

Realizing a restriction as direct/inverse image of sheaves

Consider the inclusion $j$ of ${\mathbb{R}}$ as the real axis of ${\mathbb{C}}$. On ${\mathbb{C}}$ I have a real polynomial algebra ${\mathbb{R}}[x,\bar{x}]$, where $\bar{x}$ denotes conjugation. ...