# Cobordism invariance and “reverse-extended TQFTs”

According to Atiyah, an $n$-dimensional TQFT is a functor from the bordism category of $(n-1)$-dimensional manifolds into $\mathrm{Hilb}$. That is, for every $(n-1)$-dimensional manifold it assigns a vector space, and for every $n$-dimensional bordism a linear map.

One can also generalize this to the notion of an extended TQFT where one also assigns values to lower-dimensional manifolds. However, here I am interested in the other direction, which I will call a "reverse-extended TQFT". That is, suppose we have an $(n+1)$-dimensional "bordism with corners" between $n$-dimensional bordisms. A reverse-extended TQFT should assign a value to such objects. We can treat $\mathrm{Hilb}$ as an 2-category where the only 2-morphisms are the identity morphisms. So from this point of view, a reverse-extended TQFT is equivalent to an ordinary TQFT for which the linear maps assigned to any two $n$-dimensional cobordisms which are themselves cobordant must be equal.

So my questions are: is this a natural condition to impose on a TQFT? Is there a standard terminology for it? Does it follow from any other axioms?

Related (I think): http://arxiv.org/abs/1403.1467

• – AHusain Sep 26 '16 at 19:25