Note on monoidal monads

Abstract

The representation theory of categories is used to embed each promonoidal monad in a monoidal biclosed monad. The existence of a promonoidal structure on the ordinary EilenbergMoore category generated by a promonoidal monad is examined. Several results by previous authors (notably A. Kock and F. E. J. Linton) are reproved and extended.

Promonoidal functor categories

“…The problem of the existence of the Eilenberg-Moore construction in Pmon was considered in a previous note Day (1977). This problem is related to the present one in that both constructions are, in some sense, inverse limit constructions in Pmon.…”

“…This definition, together with the definitions of promonoidal functor and promonoidal natural transformation are described in Day (1970b) and Day (1977). Thus we obtain the 2-category Pmon whose cocompleteness follows from the fact that "convolution" with V (see Day (1970a)) transforms the 2-colimit of a diagram in Pmon into the corresponding limit of monoidal biclosed categories of the form [A, V], at least if V is cartesian closed (see §4).…”

“…For convenience we also recall the strong trace conditions from Day (1977) §2. Given a full embedding M : A -»B of a category A into a monoidal category B (more generally, into a promonoidal category) we ask for conditions under which B induces a trace promonoidal structure on A.…”

“…For this note we have assumed some knowledge of the elementary definitions and theory (as presented, for example, in Eilenberg and Kelly (1966), Day and Kelly (1969) and Day (1977)). Moreover the calculations may be transferred easily into the "several-objects" version thus yielding analogous results for bicategories of functor categories and probicategories (cf.…”

On the abstract characterization of quasi-varieties

“…In fact, if B(P , − • P ) preserves filtered colimits, then P is abstractly finite since a set S is the filtered union of its finite subsets. Conversely, it is known that a functor T : SET → SET preserves filtered colimits iff it preserves filtered colimits of diagrams of monos, and this holds for T = B(P , − • P ) since P is abstractly finite (see [8] REMARK. The passage from one-sorted varieties to many-sorted varieties corresponds to the passage from monads over SET to monads over a power of SET.…”