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I can't seem to wrap my mind around what is going on when I try to visualize $Fct(Op_X, Set)$, as one example. Now I know that a functor is a morphism between categories hence we have a morphism between the category of open sets of $X$ and the category of sets.

What does this mean, intuitively speaking what is really going on here and can anyone help me (and others that are finding some of this category theory quite abstract), visualize this?

Edit: The image below is my ideas of what is going on. Note, ()'s are open sets, |'s are for closed sets. I don't know if denoting the functor as $F^{op}$, but what I imagine is that since U is an "inclusion" in V as $U \subset V$, that there are morphisims $r$ which map elements in $U$ to those same elements in $V$ - so as to get an inclusion i.e. it is these morphisms that create the inclusion. The like idea is the same with $F(V) \rightarrow F(U)$ as although $F(U) \subset F(V)$, we are dealing with the opposite category open sets of $X$. Please let me know if anything in the diagram is wrong.

Thanks in advance,

Brian

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  • $\begingroup$ en.wikipedia.org/wiki/Sheaf_%28mathematics%29#Presheaves (or do you really mean $\mathrm{Op}_X$ and not $\mathrm{Op}_X^{op}$?) $\endgroup$ – Martin Brandenburg Nov 12 '13 at 7:13
  • $\begingroup$ Thanks Martin, yes, I am working my way towards Presheaves and I know that I am just supposed to reverse the arrows as far as $Op_X$ is concerned. As far as I can see, the category of open sets on X ($Op_X$) may have some morphisms between the open sets $U,V \subset X$ Whatever those morphisms are, we map those morphisms to some relationship between sets, say $A,B \in Set$. I don't even know if I am correct in my thinking, but even if I am, what does this mean or do for me? Thanks again. $\endgroup$ – Relative0 Nov 12 '13 at 7:22
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From the comments, I assume you're working on presheaves and so you're interesting in functors ${\mathcal O_X}^\mathrm{op} \to \mathbf{Set}$ rather than ${\mathcal O_X} \to \mathbf{Set}$ ($\mathcal O_X$ being the category of open sets of $X$).

First, just recall that $\mathcal O_X$ is the category with objects the open sets $U$ of the topological space $X$, and with morphisms the inclusions of open sets : it means that considering $U,V$ open sets of $X$, $$ \mathrm{Hom}(U,V) = \left\{ \begin{aligned} \{ \ast \} \quad &\text{if $U \subseteq V$} \\ \emptyset \quad &\text{otherwise.} \end{aligned} \right. $$

So a presheaf $F \colon {\mathcal O_X}^\mathrm{op} \to \mathbf{Set}$ is the data of a set $F(U)$ forevery open set $U$ of $X$, and of a set function $F(V) \to F(U)$ for every inclusion $U \subseteq V$ of open sets in $X$. This data is required to satisfy : for open sets $U \subseteq V \subseteq W$ of $X$, the composite set function $F(W) \to F(V) \to F(U)$ is the function $F(W) \to F(U)$, and the function $F(U) \to F(U)$ coming from the trivial inclusion $U \subseteq U$ need to be the identity function.

With the hands, a presheaf is a set that evolves continuously along the topological space $X$. Especially, that set is compatible with zooming in the space $X$.


Mostly, nice visual examples of presheaves are sheaves (or at least separated presheaves). The most popular example is certainly the following : take a topological space $X$ (a topological variety is nice for visualisation), then the functor $U \mapsto \mathcal C(U,\mathbb R)$ of the continuous maps from $X$ to $\mathbb R$ is a presheaf (and even a sheaf) ; the set function $\mathcal C(V, \mathbb R) \to \mathcal C(U, \mathbb R)$ is just the restriction of the continuous maps defined on $V$ to $U \subseteq V$.

An other example : take again a topological variety $X$, the functor $U \mapsto \mathcal C_b(X, \mathbb R)$ of bounded continuous maps from $X$ to $\mathbb R$ is again a presheaf (actually a separated one).

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  • $\begingroup$ "The most popular example is certainly the following:" How do you know that? And what is a topological variety? $\endgroup$ – Martin Brandenburg Nov 12 '13 at 9:13
  • $\begingroup$ @MartinBrandenburg Well, the sheaf on continuous functions on a (topological or differential) variety is often used in textbooks to introduce sheaves and presheaves. As for a topological variety, a compact way to say it is to defined it as a $\mathcal C^0$-differential variety : that is a separated topological space with a countable basis of open sets, admitting a open covering $(U_i)_i$ with $U_i$ homeomorphic to an open set of $\mathbb R^n$ for all $i$. $\endgroup$ – Pece Nov 12 '13 at 9:19
  • $\begingroup$ @MartinBrandenburg Maybe I should have used 'manifold' instead of 'variety' ? If so, I'll edit the post. $\endgroup$ – Pece Nov 12 '13 at 10:34
  • $\begingroup$ Thanks Pece, I have a long way to go on this.. First, in your answer (and I have heard before), "with morphisms the inclusions of open sets". Now to start with, the Functor F (Presheaf) maps the category of open sets to Sets. So if we have an open set $U, V \in Op_x$ and after mapping this to another category (Set) in which $F(U), F(V) \in Set$ This "inclusion", how is it a morphism? I mean if $U \subset V$, then $U$ is included in $V$ hence would be an "inclusion". Wiki says a morphism is a structure preserving mapping, I don't see how morphisms are inclusions. Continued.. $\endgroup$ – Relative0 Nov 13 '13 at 20:37
  • $\begingroup$ since morphisms are structure preserving maps, we would have that structure preserving maps are included in open sets. I suppose that I am not very well versed at maps being in open sets. I can see maps from one set to another as we would have one element in one set being mapped to another element in another set. The best I can imagine is that we have arrows from one open set U to another open set V, then these arrows are operated on by the functor F which in preserving arrows would map $F(U) \rightarrow F(V)$. I suppose then $Op_{X}^{op} \rightarrow Set$ gives $F(V) \rightarrow F(U)$ ... $\endgroup$ – Relative0 Nov 13 '13 at 20:47

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