/sci/ - Science & Math

>>9481473
I liked the one where Yukari and Yuyuko raped this one village boy.

>>9381744
Sorry for the slow response, my phone battery died as I was at the gym.
To see why the functor maps the empty set to the ground ring R, note that [math]\mathscr{T}[/math] maps the manifold [math]\overline{M}[/math] with the reverse orientation of [math]{M}[/math] to the dual [math]\mathscr{T}(M)^*[/math] of the R-module [math]\mathscr{T}(M)[/math], so since [math]\overline{\emptyset} = \emptyset[/math], we have [math]\mathscr{T}(\emptyset)^* = \mathscr{T}(\emptyset)[/math] and hence by the non-degeneracy (prove this!) of the pairing [math]\mathscr{T}(\emptyset) \times \mathscr{T}(\emptyset)^* = \mathscr{T}(\emptyset) \times \mathscr{T}(\emptyset) \rightarrow R[/math] we see that there is an isomorphism [math] \mathscr{T}(\emptyset) \cong R[/math], since the pairing would just be a multiplication by an element in R. Hence for any cobordism [math]W[/math] that ends/starts with the empty set, we can pre/post-compose the R-linear homomorphism [math]\mathscr{T}(W)[/math] with the isomorphism [math] \mathscr{T}(\emptyset) \cong R[/math] such that the new [math]\mathscr{T}(W)[/math] maps whatever space to the empty set (or vice versa). This isomorphism is unique up to isotopy so this doesn't change the categorical structure of R-modules.
This "shifting" of morphisms can be considered as a "normalization".