Two-dimensional linear algebra

We introduce previous termtwonext term-previous termdimensionalnext termprevious termlinearnext term algebra, by which we do not mean previous termtwonext term-previous termdimensionalnext term vector spaces but rather the systematic replacement in previous termlinearnext term algebra of sets by categories. This entails the study of categories that are simultaneously categories of algebras for a monad and categories of coalgebras for comonad on a category such as SymMons, the category of small symmetric monoidal categories. We outline relevant notions such as that of pseudo-closed previous term2next term-category, symmetric monoidal Lawvere theory, and commutativity of a symmetric monoidal Lawvere theory, and we explain the role of coalgebra, explaining its precedence over algebra in this setting. We outline salient results and perspectives given by the dual approach of algebra and coalgebra, extending to previous termtwonext term dimensions the study of previous termlinearnext term algebra.