TITLE: what is a spinor structure?
QUESTION [18 upvotes]: There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$,

a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$
or an almost complex structure is a section $J$ of $\textrm{End}(TM)$ which is everywhere an anti-involution (i.e. $J_x^2 = - \mathrm{Id}_{T_x M} $)
or an orientation is a non-vanishing section of $\Lambda^m TM$ 

is 
a spin structure, a section of quadratic forms $Q$ on $TM$ (of type $(s,t)$) and a vector bundle $S$ so that $\textrm{End}(S) \simeq \textrm{C}\ell(TM,Q)$?
...or something of the like? any references where it may be stated in this fashion?

REPLY [15 votes]: A spin structure on a real vector space V equipped with a real quadratic form μ
is an invertible bimodule (i.e., a Morita equivalence)
from Cl(V,μ) to Cl(Rdim(V),ν).
Here ν is the direct sum of dim(V) copies of the canonical
quadratic form on R.
A spinc structure on a complex vector space V equipped with a complex quadratic form μ
is an invertible bimodule
from Cl(V,μ) to Cl(Cdim(V),ν).
Here ν is the direct sum of dim(V) copies of the canonical
quadratic form on C.
Note that spin and spinc structures form
a category instead of a mere set, just as we would expect.
Of course, these definitions immediately extend
to vector bundles, with the obvious requirement
that invertible bimodules form bundles themselves.
A spin or spinc structure on a smooth manifold
is a spin or spinc structure on its real
or complex tangent bundle.