Presheaves are defined by using neighborhoods of a point. Is there a way to restrict this construction to path connected neighborhoods of points? What is the name of the object which assigns other objects to path connected nbds. of points? Thank you. (Note: I am not interested in restricting presheaves to path connected open sets)
(pre)sheaves on path connected neigh
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0Absolutely. I do not see any problems with this construction since the definition of a presheaf does not use the fact that we are assigning objects to *open sets* in a topological space. In other words, the fact that a presheaf is defined on a *topological space* is not logically relevant to the definition of a presheaf. We could choose any set $A$ and assign abelian groups to subsets of $A$ (in a manner that respects certain conditions) and we would have a presheaf on $A$. – 2011-06-29
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0@Amitesh Datta: Does the set of closed sets of a topological space fulfill these conditions? – 2011-06-29
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0@Rasmus Please see http://en.wikipedia.org/wiki/Sheaf_(mathematics)#Presheaves Do you notice that the word "open" in this definition is not relevant in that the axioms satisfied by the open sets in a topological space is not relevant (except that the empty set is open)? – 2011-06-29
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0@Amitesh: Thank you for your comment. I see your point. – 2011-06-29
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0@Amitesh Datta: As soon as you also want to define a sheaf you need the domain of your functor to be a (Grothendieck) topology. – 2011-06-29
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0@Rasmus I know that. However, the question asks about *presheaves* and not *sheaves* and hence I made my comment. – 2011-06-30
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0@Amitesh: Agreed. Are there any instance where one is interested in defining presheaves on something but not sheaves? Maybe I should ask this as a new question. – 2011-06-30
1 Answers
A presheaf can be defined on any category: it is just another name for a contravariant functor from that category to any other category. In particular, if you have a topological space $X$ you can associate to it a category $\mathcal Open (X)$ whose objects are the open subsets $U\subset X$ and the morphisms the inclusions $U\subset V$. A presheaf in the usual sense is then indeed a functor from $\mathcal Open (X)$ to another category. Now you have a full subcategory $\mathcal Pathconopen (X)\subset \mathcal Open (X)$ obtained by taking only path connected open subsets of $X$ and their inclusions, and you can call a conravariant functor from it a presheaf . All fine and well. However it is impossible to say when a presheaf is a sheaf in that context: the problem is that if a path connected open set $U\subset X$ is covered by path connected open subsets $U_i$, the intersections $U_i\cap U_j$ will in general not be path connected and the sheaf conditions thus don't make sense.
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0Georges: Bredons book on sheaf theory uses sheaves to define (co)homology. If one uses the general contravariant definition of presheaves then is it possible to define some kind of limit of presheaves so that it becomes possible to do (co)homology theory as Bredon does? May I learn the most general name of this type of construction? Thank you. – 2011-06-29
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0Let $U=\{z\in\mathbb{C}:\text{Im}(z)>\frac{1}{2}\}\cup \{z\in\mathbb{C}:\text{Im}(z)<-\frac{1}{2}\}\cup \{z\in\mathbb{C}:\text{Re}(z)>1\}$ and let $V=\{z\in\mathbb{C}:\left|z\right|<1\}$. Note that $U$ and $V$ are path connected open subsets of $\mathbb{C}$ but that the intersection $U\cap V$ is not even connected. – 2011-06-29
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0@Amitesh: Thank you. Please do not send purposeless comments. – 2011-06-29
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0@lion2011 I do not understand why my comment is "purposeless". Georges noted at the end of his answer a fact to the effect that "the intersection of two open path connected sets is not in general path connected". I thought that, for the sake of completeness, it would be good to justify this remark. (It might not be obvious to everyone reading the answer above.) – 2011-06-29
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0@Amitesh: Ok. People usually write circular answers and comments, but yours has a purpose now.+1 – 2011-06-29
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2Dear lion, there is a huge literature concerned with sheaves defined on more general categories than $\mathcal Open(X)$. The buzz words are: étale cohomology, Grothendieck topologies,topos theory, sites. This is a rather sophisticated subject, so depending on your background you can jump in right now or wait a little longer. Here are some excellent online notes by Milne, a masterful expert and expositor, which will help you take your decision. Good luck! http://www.jmilne.org/math/CourseNotes/lec.html – 2011-06-29