Relative Homological Algebra. Cohomology of Categories, Posets and Coalgebras

Modular Classes of Lie Algebroid Morphisms

“…The differential δ is defined by the usual formula. This definition is simpler than the one which we have found in the literature (see, e.g., [1], [9]), but it defines the same object in the cases studied here, and seems therefore to be good enough for our limited purposes.…”

Resolutions, relation modules and Schur multipliers for categories

“…We first recall the notion of an extension of a category which appears in Hoff [13], bearing in mind that there are also other approaches to category extensions (see [1,8,11,13] and also [19] for an exposition of this basic material). An extension of categories is a pair of functors…”

“…A discussion of background and the uses of representations of categories can also be found in many of the other references we give, including [1][2][3]6,8,11,14,15,18]. In any case, we now give some of the basic definitions.…”

On the cohomology rings of small categories

“…We want to compare the cohomology rings of C and of its various subcategories. In [5,16,12,32] the authors studied the case where D ⊂ C is a full subcategory which has fewer objects, and showed under certain assumptions one can have H * (C) ∼ = H * (D). Here we investigate subcategories D ⊂ C with the same set of objects but with fewer morphisms.…”

“…When [x] only has one or two objects, the construction is trivial. When there are three objects in [x], say x 1 , x 2 , x 3 , we can choose arbitrary α 12 ∈ Hom C (x 1 , x 2 ), α 13 ∈ Hom C (x 1 , x 3 ) and then define α 23 = α 13 α −1 12 . Suppose we have constructed such sets for all isomorphism classes with less than n objects.…”