The map \alpha:V_1\times \cdots \times V_n\rightarrow
V_1\otimes\cdots \otimes V_n is initial in the (slice) category whose objects are the F-multilinear maps
f:V_1\times \cdots \times V_n\rightarrow W from
V_1\times \cdots \times V_n to vector space W over F,
and whose morphisms are and F-linear maps "between" such multilinear maps--meaning a commuting triangle as follows.
Given f:V_1\times \cdots \times V_n\rightarrow W
and g:V_1\times \cdots \times V_n\rightarrow X,
a morphism \beta:f\rightarrow g "is" (which means "is given by") an F-linear map \beta:W\rightarrow X such that
g = \beta f. Now of course, one has to specify the category in which the equation holds...which would be the category from which the slice category is obtained.
This is all elementary. I hope I haven't deployed the "fake it until you make it" affectation of academic authority that escape_goat (a cute nick) finds deplorable. I do too, though were I to write about it, I would raise the suspicion that I am a disaffected outsider who deserved exile. (I seem to have just done that.)
Nelson's text on tensor analysis is another route to tensorial enlightenment..
I could also add my own obscure blog to the long tail of wannabe math \cap programming blogs: http://publicsphere.org.
Given f:V_1\times \cdots \times V_n\rightarrow W and g:V_1\times \cdots \times V_n\rightarrow X, a morphism \beta:f\rightarrow g "is" (which means "is given by") an F-linear map \beta:W\rightarrow X such that g = \beta f. Now of course, one has to specify the category in which the equation holds...which would be the category from which the slice category is obtained.
This is all elementary. I hope I haven't deployed the "fake it until you make it" affectation of academic authority that escape_goat (a cute nick) finds deplorable. I do too, though were I to write about it, I would raise the suspicion that I am a disaffected outsider who deserved exile. (I seem to have just done that.)
Nelson's text on tensor analysis is another route to tensorial enlightenment..
I could also add my own obscure blog to the long tail of wannabe math \cap programming blogs: http://publicsphere.org.