In this talk I will articulate and contextualize the following sequence of results. - The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group. - Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories. - In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.