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First, I apologize if this question is poorly-worded or otherwise vague, I'll try to be as clear as possible.

If $F:N\rightarrow M$ is a smooth map between smooth manifolds $N$ and $M$, then at each point $p \in N$ the map $F$ induces the derivation $F_{*p}:T_pN \rightarrow T_{F(p)}M$ between tangent spaces, called the differential, that is determined by $F_{*p}(X_p)f = X_p(f \circ F)$ for all smooth real-valued functions $f$ on $M$.

To me, this seems like a covariant functor from the category of smooth manifolds to the category (?) of tangent spaces. My understanding though if it is to be a functor it must also assign, for example, a manifold $M$ to a tangent space $T_pM$. Are there additional aspects of defining the differential that would facilitate this?

Is there a way, perhaps, that this can be achieved with the inclusion maps $i_N$ and $i_M$ of $N$ and $M$ into $T_pN$ and $T_pN$ since the differential, satisfies $F_{*p} \circ i_N = i_N \circ F$?

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If this is going to work, I think you want to look at the whole tangent space $TM$, as a manifold and possibly a vector bundle. Also, it seems like you're saying that to specify a functor you need a functor going the opposite way, which I don't think is the case. –  Dylan Moreland Jan 23 '12 at 21:11
    
@DylanMoreland Sorry, I meant to say "covariant" in the body of my question; I'll fix that. –  ItsNotObvious Jan 23 '12 at 21:15
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Quoting John Lee's text: "The tangent functor is a covariant functor from the category of smooth manifolds to the category of smooth vector bundles. To each smooth manifold $M$ it assigns the tangent bundle $TM \to M$, and to each smooth map $F\colon M \to N$ it assigns the pushforward $F_*\colon TM\to TN$." –  Jesse Madnick Jan 23 '12 at 21:17
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If you don't have a copy of John Lee's "Introduction to Smooth Manifolds," by the way, you should really look into getting one. It's an excellent text. –  Jesse Madnick Jan 23 '12 at 21:22
    
The Lee reference is perfect, exactly what I was looking for. And yes, I have Lee's text - Just haven't read it far enough to know that he discusses what I now know is called a "tangent functor"! –  ItsNotObvious Jan 23 '12 at 21:24

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