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Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E \right).$$

The wedge product of a $E_1$-valued $p$-form $\omega_1 \in \Omega^p(M; E_1)$ with a $E_2$-valued $q$-form $\omega_2 \in \Omega^q(M;E_2)$ is defined in R. W. Sharpe's Cartan geometry text to be the $E_1 \otimes E_2$-valued $p+q$-form $\omega_1 \wedge \omega_2 \in \Omega^{p+q}(M; E_1 \otimes E_2)$ given by $$(\omega_1 \wedge \omega_2) (v_1, \ldots v_{p+q})=\sum_{\text{$(p,q)$ shuffles $\sigma$}} (-1)^{\text{sgn}(\sigma)} \omega_1(v_{\sigma(1)}, \ldots, v_{\sigma(p)})\otimes \omega_2(v_{\sigma(p+1)}, \ldots, v_{\sigma(p+q)}).$$

I am wondering if this multiplication induces a universal object of some sort?

For example, the wedge product of $\mathbb{R}$-valued forms gives you the exterior algebra $\Omega^{\bullet}(M)$, which is the free graded-commutative algebra on $\Omega^1(M)$.

This is more generally true for the exterior algebra $\bigwedge^\bullet V$ for any module $V$ over a commutative ring. Likewise, the tensor algebra $\bigotimes^\bullet V$ is the free algebra on $V$, and the symmetric algebra $\bigodot^\bullet V$ is the free commutative algebra on $V$.

Does the wedge product of bundle-valued forms induce a universal object of some sort as well?

Also, I would really appreciate a reference on bundle-valued forms, especially one from a categorical viewpoint. I haven't been able to find a comprehensive treatment yet.

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    $\begingroup$ I recall Morita's book The Geometry of Differential Forms is a reference for bundle-valued forms, but it's been a while and I don't think it includes a categorical perspective. $\endgroup$ – Neal May 11 '15 at 12:34
  • $\begingroup$ General remark: Don't just look at global sections, look at the whole sheaf. $\endgroup$ – Martin Brandenburg May 12 '15 at 13:14
  • $\begingroup$ @Martin Brandenburg Could you explain how looking at the whole sheaf helps here? $\endgroup$ – ಠ_ಠ May 12 '15 at 13:28

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