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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 $F_{*p} \circ i_N = i_N \circ F$?

ItsNotObvious
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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
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    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
  • @JesseMadnick If you want to put your comment as an answer, I'll upvote/mark as answered. – ItsNotObvious Jun 07 '12 at 20:45
  • The fact that the (global) pushforward is the tangent functor applied to morphisms in the category of smooth manifolds explains the fact that some authors use the notation $T$ for the pushforward, as they are being consistent with category theory. – setnoset Mar 17 '19 at 18:42

1 Answers1

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(Promoted from Jesse Madnick's comment. Please upvote his comment instead!)

There is indeed a tangent functor $T$, and it maps $M$ to the entire tangent bundle $TM$ (which is the discriminated union of all tangent spaces $T_p M$, as a vector bundle on M).

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 $TM→M$, and to each smooth map $F:M→N$ it assigns the pushforward $F_∗:TM→TN$."

w123
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