*Semantics Club 23 02 2001*

# Two-dimensional linear algebra

## John Power

(joint work with Martin Hyland)### Abstract

We introduce two-dimensional linear algebra. For ordinary linear algebra,
the fundamental object of study is the category Ab of Abelian groups,
and the fundamental fact about Ab is that it is monadic over Set for
a commutative monad. That implies that Ab is a symmetric monoidal
closed category, with the forgetful functor to Set forming part of
a symmetric monoidal closed adjunction. For two-dimensional linear
algebra, the category SymMon of small symmetric monoidal categories
plays the role of Ab, and Cat plays the role of Set. However, one
does not have a commutative monad here, because a diagram that must
commute does not in fact do so. So we need to develop a notion of
pseudo-commutative 2-monad, then a notion of pseudo-closed 2-category,
then prove that if T is a pseudo-commutative 2-monad on Cat, it follows
that the 2-category of T-algebras is pseudo-closed. We also need
to provide some independent indication that our definitions are
robust. This work provides a unified framework for modelling various sorts of
simply typed context such as linear contexts, ordinary contexts, and
combinations of them, and provides a theoretical basis for some
of the choices in axiomatic domain theory, action calculi, and the like.