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I have to solve an Excercise about the following:

Classify all locally constant sheaves of $\mathbb{C}$-vector spaces on the space $T^{2} := S^1 \times S^1$.

I was learning something about the category $LCSH(k_X) := Fct((Op_X)^{op}, Mod(k))$ for some unitary ring $k$.

How do i have to think about this?

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    Locally constant sheaves with stalk $V$ is the same as a representation of the fundamental group $\rho : \pi_1(X,x) \to \rm{GL}(V)$. In particular, since $\pi_1(X) \cong \mathbb Z^2$, rank $r$ local systems are classified by pairs of commuting matrices $A,B \in GL(\mathbb C^r)$, up to conjugaison. – 2017-01-29
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    I'm sorry, i should have said that i am an undergrad. I have read about the fact that local-systems are the same as reps of the fundamental group. But is there a more intuitive way of thinking of this? i would realy like to have a sort of picture in my mind. and what do you mean by rank $r$ – 2017-01-29
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    I did assume that the stalk of your sheaf was a vector space of finite dimensional, and then $r$ is by definition the dimension of the stalk as a vector space. I don't think it is easy to do this without knowing anything about local system. Maybe you can admit the following lemma without proofs : any local system $\mathcal L$ on $[0,1]$ is trivial, and in particular you have canonical isomorphism $\mathcal L_0 \cong \mathcal L_1$. Anyways I think this correspondance is really intuitive, maybe you should try more easy examples first ? – 2017-01-29
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    For example, do you know how to classify coverings map with degree $3$ $\pi : X \to S^1 \times S^1$ ? – 2017-01-29
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    No. The point is, I'm trying to understand the following: 1. The category of locally-constant sheaves with values in $Mod(k)$ is equivalent to the category of representations of the fundamental group $Rep(\pi_1(X,x),Mod(k)) \cong Fkt(\Pi_1(X),Mod(k))$ if $X$ is pathconnected. So I wanted to see an example of a Rep of a Fundamental group and the analogy to the locally-constant sheafes – 2017-01-29
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    I wanted to write an example but since I know it from a mathoverflow thread, this is simpler to give you the link : http://mathoverflow.net/questions/47351/how-to-think-of-monodromy-transformations I also advice the reading of the book "Galois groups and fundamental groups" which underline a lot the geometric idea between local systems, especially chapter 2. – 2017-01-29
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    the example looks very detailed. im going to have a closer look at the book. thanks bunches – 2017-01-29
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    Sure ! I would advice to take your time and work lot of examples, and also playing with covering since somehow coverings could be considered as a "discrete" version of local system. The book is really well made ! – 2017-01-29
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    So i was looking trough the example and the book. The book shows a very detailed walktrough on the prove. But just thinking about the category equivalence, i am wondering if there is sort of a "geometrical" intuition about this. Say, if i am thinking about the Torus, what do the LCSH on X have in common with the reps of the F-Group. I realy realy would like to have a picture in my mind. I guess this would make things much more clearer – 2017-01-30
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    The picture you should have is the same as for covering : you have a fiber, and a path starting at $x$ and finishing at $y$ will identiy canonically the fiber at $x$ and the fiber at $y$. Taking a loop you will get an automorphism of the fiber, and this automorphism fully characterize the local system/covering. This is really geometric. – 2017-01-30
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    so by taking a loop, do you meen a loop from $x \rightarrow y \rightarrow x$, or a loop $x \rightarrow p^{-1}(x) \rightarrow p^{-1}(y) \rightarrow y \rightarrow x$ ? – 2017-01-31
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    A loop $x \to p^{-1}x$ does not make sense since $x$ and $p {-1}(x)$ are not in the same space. I mean a loop $x \to y \to x$. I advice to read more about covering spaces : you will get lot of feeling and intuition by understanding how these spaces work. – 2017-01-31
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    Hello again. so i have a read a bit on coverings etc. and i think i do now understand very well what you where trying to tell me. so for example if i have point $x = 1$ on the circle and i am taking a loop $\gamma: x \rightarrow x$ then i get an automorphism of the fibers $p^{-1}(\gamma) : 1 \rightarrow 2$ (just for an example). But why are the Stalks at a point $x$ and the fibers the same thing? This bothers me a little bit. – 2017-02-12
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    Hi ! I think you need to be more precise again. What covering of the circle do you consider ? What loop do you consider ? You are not looking at $p^{-1}(\gamma)$ but $p^{-1}(1)$. You have a sheaf associated to your covering which is the pullback of the constant sheaf by the covering map. This sheaf has stalk $\mathbb C \oplus \mathbb C$ and if you take the loop $\gamma(t) = e^{2\pi i t}$ for $t \in [0,1]$ you will see that the automorphism you will get is just permutation of the two components : $\rho(\gamma) = \begin{bmatrix}0 & 1 \\1 & 0 \end{bmatrix}$. – 2017-02-12
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    So now fiber and stalk are **not** the same. In my example, the fiber over $1$ is $\{1,-1 \}$. The stalk is $\mathbb C \oplus \mathbb C$. But you can notice in this construction, the stalk will be for a general covering $\mathbb C ^{n}$ where $n$ is the degree of the covering. I suggest you to take a little break and read more about covering spaces, lifting properties and then go back with local system. Did you read the chapter about fundamental group and coverings in the book of Lee ? It can help you to have more intuition. – 2017-02-12
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    Finally : I explained to you how to go from a covering to a local system, but we don't need it in a first time. So you basically forgot what I said before and go back later. For the moment you should understand this : let $p : S^1 \to S^1$ the covering $z \mapsto z^2$, prove there is an action of $\pi_1(S^1)$ on $p^{-1}(1)$. Show that $\gamma \cdot 1 = -1$ and $\gamma \cdot -1 = 1$. Once you will understand it you can try to understand the "local system" version which I did wrote before. – 2017-02-12
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    I think i understood the monodromy-action of the fundamental group on the fiber ($\gamma*1 = -1$ for ex., because that is very imaginable). I am very sorry for not being precise enough, the cover you considered was the one is was thinking about actually! What i meant with $p^{-1}(\gamma)$ was the lifiting of the loop $\hat{\gamma}$ in the covering space. Some remaining questions are: How do i find the Stalk in the first place and more interest. what is an element of the stalk? – 2017-02-12
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    and to be more precise this time :) how do i find the automorphism representation $\rho : \pi_1(\mathbb{S}^1,1) \rightarrow GL(\mathbb{C})$ $\gamma(t) := e^{2\pi i t} \mapsto \rho(\gamma)$ – 2017-02-12
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    I have to go to ski now, but I can give details later. I made a little mistake before, it was the direct image and not the pullback. So consider $F$ to be the constant sheaf with stalk $\mathbb C$ on $S^1$. Try to compute $p_* F$ (its stalk at $1$, and try to deduce an action of $\pi_1(S^1,1)$ on $(p_* F)_1$. This is basically a construction which shows you that in some sense covering is a really particular case of a local system. – 2017-02-12

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