We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups uq(sl2).

In a pointed category with kernels and cokernels, we characterize torsion–free classes in terms of their closure under extensions. They are also described as indexed reflections. We obtain a corresponding characterization of torsion classes by formal categorical dualization.

for Scientific Research, and the "Badische Landesverband" for cancer research, and in part by a grant G-23739 of the National Science Foundation. Part of this work was submitted as a thesis to the Faculty of Medicine of the University of Freiburg (Br.) in behalf of B. Langer. t Research Career Awardee of the USPHS; on sabbatical leave from the University of California between June and October, ...

We describe a certain category of Ind-towers into which the Pro category of towers of simplicial sets embeds, and in which all colimits (rather than just the finite ones) may be constructed explicitly.

The following discusses a generalization of Baire category in a continuous set. The objective is to provide a meaningful classification of subsets of a continuous set as “large” or “small” sets in linearly ordered continuous sets. In particular, for cardinal number κ, the continuous ordered set 2∗ a subset of the set of dyadic sequences of length κ is discussed. We establish that this space, an...

The notion of a subtractive category recently introduced by the author, is a pointed categorical counterpart of the notion of a subtractive variety of universal algebras in the sense of A. Ursini (recall that a variety is subtractive if its theory contains a constant 0 and a binary term s satisfying s(x, x) = 0 and s(x, 0) = x). Let us call a pointed regular category C normal if every regular e...

Today, Nicky spoke on a few approaches to higher K-theory. Let C be a pointed ∞-category with finite colimits (as in Lurie’s approach) or a category with cofibrations and weak equivalences satisfying certain axioms (as in Waldhausen’s approach). Recall that K0(C) was defined to be the free abelian group on isomorphism classes of objects of C modulo [X] = [X′] + [X′′] whenever we have a pushout ...

In previous work we defined a Quillen model structure, determined by cycles, on the category Gph of directed graphs. In this paper we give a complete description of the homotopy category of graphs associated to our model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category H...