I don't understand a statement in Randal-Williams Characteristic classes and $K$-theory, at the beginning of page 29. The book says:
Tensor product (of vector bundles) gives a homomorphism of abelian monoids: $$vec(X)\times vec(X)\to vec(X)$$ $$([E],[F])\mapsto [E\otimes F].$$
Here $X$ is a (compact Hausdorff) space and $vec(X)$ is the set of isomorphism classes of complex vector bundles over $X$, which is a monoid with the trivial $0$-dimensional bundle as identity and the Whitney sum as oepration.
I guess $vec(X)\times vec(X)$ is a monoid with its operation defined coordinate-wise. Hence if the tensor product preserves the monoid structure, for $([E],[F])$, $([E'],[F'])$, we have $$[(E\otimes E')\oplus (F\otimes F')]=[(E\oplus E')\otimes (F\oplus F')],$$ i.e. $$(E\otimes E')\oplus (F\otimes F')\cong(E\oplus E')\otimes (F\oplus F').$$ However this fact isn't true for vector spaces, and doesn't seem true for vector bundles either to me. Do you understand the issue / the meaning of the text? Thanks
Indeed, this statement is incorrect. The tensor product is not a homomorphism; instead it is a bilinear map (i.e. if you fix a value for one input then you get a homomorphism in the other input). To get a homomorphism you would instead have to take the induced map $vec(X)\otimes vec(X)\to vec(X)$ (where $vec(X)\otimes vec(X)$ is the tensor product of abelian monoids, defined analogously to the tensor product of abelian groups).