KZ-monadic categories and their logic

Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed. We present a sound logic for Kan-injectivity consequences and prove that in "reasonable" categories (such as Pos or Top_0) it is also complete for every set H of squares

KZ-monadic categories and their logic

Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed. We present a sound logic for Kan-injectivity consequences and prove that in "reasonable" categories (such as Pos or Top_0) it is also complete for every set H of squares

Enriched weakness

The basic notions of category theory, such as limit, adjunction, and orthogonality, all involve assertions of the existence and uniqueness of certain arrows. Weak notions arise when one drops the uniqueness requirement and asks only for existence. The enriched versions of the usual notions involve certain morphisms between hom-objects being invertible; here we introduce enriched versions of the weak notions by asking that the morphisms between hom-objects belong to a chosen class of "surjections". We study in particular injectivity (weak orthogonality) in the enriched context, and illustrate how it can be used to describe homotopy coherent structures.Comment: 25 pages; v2 minor changes, to appear in JPA

Order-preserving reflectors and injectivity

AbstractWe investigate a Galois connection in poset enriched categories between subcategories and classes of morphisms, given by means of the concept of right-Kan injectivity, and, specially, we study its relationship with a certain kind of subcategories, the KZ-reflective subcategories. A number of well-known properties concerning orthogonality and full reflectivity can be seen as a particular case of the ones of right-Kan injectivity and KZ-reflectivity. On the other hand, many examples of injectivity in poset enriched categories encountered in the literature are closely related to the above connection. We give several examples and show that some known subcategories of the category of T0-topological spaces are right-Kan injective hulls of a finite subcategory

Algebraic K-theory of group rings and the cyclotomic trace map

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of B\"okstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of $K_n(\mathbb{Z}[G])\otimes_{\mathbb{Z}}\mathbb{Q}$ in terms of group homology. In many cases the number theoretic conjectures are true, so we obtain rational injectivity results about assembly maps, in particular for Whitehead groups, under homological finiteness assumptions on the group only. The proof uses the cyclotomic trace map to topological cyclic homology, B\"okstedt-Hsiang-Madsen's functor C, and new general isomorphism and injectivity results about the assembly maps for topological Hochschild homology and C.Comment: To appear in Advances in Mathematics. 77 page

Homotopy locally presentable enriched categories

We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.Comment: 48 pages. Significant changes in v2, especially in the last sectio