We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the V-category of models of a Lawvere V-theory is equivalent to the V-category of algebras for the corresponding V-monad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where V is Cat, and explain how the correspondence extends to pseudo maps of algebras.

Abstract. This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation— a weak functor—from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail. It has applications in constructing quantum field theories, among others. 1.