Ross Street

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This is an expanded, revised and corrected version of the rst author's preprint [1]. The discussion of one-dimensional cohomology H 1 in a fairly general category E involves passing to the (2-)category Cat(E) of categories in E. In particular, the coecient object is a category B in E and the torsors that H 1 classies are particular functors in E. We only(More)
Strong promonoidal functors are defined. Left Kan extension (also called " existential quantification ") along a strong promonoidal functor is shown to be a strong monoidal functor. A construction for the free monoidal category on a promonoidal category is provided. A Fourier-like transform of presheaves is defined and shown to take convolution product to(More)
The purpose is to give a simple proof that a category is equivalent t o a small category if and only if both it and its presheaf category are locally small. In one of his lectures (University of New South Wales, 1971) on Yoneda structures SW], the second author conjectured that a category A is essentially small if and only if both A and the presheaf(More)
Given a horizontal monoid M in a duoidal category F , we examine the relationship between bimonoid structures on M and monoidal structures on the category F * M of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the so-called Tannaka adjunction; that is, an adjunction inducing the equivalence which(More)
We show that every algebraically{central extension in a Mal'tsev variety | that is, every surjective homomorphism f : A?! B whose kernel{congruence is contained in the centre of A, as deened using the theory of commutators | is also a central extension in the sense of categorical Galois theory; this was previously known only for varieties of-groups, while(More)