Enriched Locally Generated Categories

Are Banach spaces monadic?

“…Let CAlg be the category of unital C * -algebras and CCAlg the category of commutative unital C * -algebras. The forgetful functor G : CAlg → Ban preserves limits, isometries and directed colimits (see [8] 6.10). Thus it has a left adjoint F .…”

Locally bounded enriched categories

“…We conclude this subsection by considering the relationship between locally α-bounded Vcategories and the M -locally α-generated V -categories of [9]. If (E , M ) is an enriched factorization system on a cocomplete V -category C and α is a regular cardinal, then (E , M ) is said to be αconvenient [9, 4.3] if C is E -cowellpowered and for every α-directed diagram D : I → C 0 of M -morphisms, every colimit cocone for D consists of M -morphisms, and the factorizing morphism from colim D to the vertex of any cocone consisting of M -morphisms itself lies in M .…”

“…In Section 6 we examine the relationship between the local boundedness of a V -category versus that of its underlying ordinary category. In Section 7 we compare locally bounded V -categories with locally presentable V -categories and the enriched locally generated categories of [9]. In Section 8 we show that locally bounded V -categories satisfy particularly useful adjoint functor and representability theorems.…”

Metric monads

“…The resulting enriched Lawvere theories from [27] do not fit our aim to do universal algebra over a V-category K. Recently, Bourke and Garner [11] introduced A-pretheories for every small dense subcategory A of K and related them to A-nervous enriched monads on K. Their pretheories perfectly suit our need and describe λ-ary monads on every locally λpresentable V-category K provided that V is locally λ-presentable as a closed category. Indeed, if A is the full subcategory of λ-presentable objects of K the A-pretheory of [11] is given by (X, Y )-ary operations and equations; here X and Y are λ-presentable objects of K. Over Met, we can apply [13] and relate equational theories whose operations have finite metric spaces as arities to monads preserving directed colimits of isometries.…”

“…In Section 3, we show that λ-ary pretheories of [11] can be presented as equational theories over a category K introduced in [31] (motivated by [23]). We show how it can be used to prove the result from [11] that λ-ary pretheories over a locally λpresentable category K correspond to λ-ary monads on K. Then we apply it to the case when K is not locally finitely presentable but only locally finitely generated in the sense of [13], which is the case of metric spaces. In Section 4, we do the enriched case and show that it yields recent results from [2] and [1].…”