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Enriched Lawvere Theories
"... 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 Vcategory of mod ..."
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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 Vcategory of models of a Lawvere Vtheory is equivalent to the Vcategory of algebras for the corresponding Vmonad. 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.
2VECTOR SPACES AND GROUPOIDS
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"... Abstract. This paper describes a relationship between essentially finite groupoids and 2vector spaces. In particular, we show to construct 2vector spaces of Vectvalued presheaves on such groupoids. We define 2linear maps corresponding to functors between groupoids in both a covariant and contrav ..."
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Abstract. This paper describes a relationship between essentially finite groupoids and 2vector spaces. In particular, we show to construct 2vector spaces of Vectvalued presheaves on such groupoids. We define 2linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representationâ€” a weak functorâ€”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.