Abstract

The main purpose of this paper is, among other things, to study cotriple (co) homology defined on a fibred category, which includes a unified account of introducing products of various derived functors, known or unknown, in a categorical setting. This approach is motivated by an attempt to find a suitable way, in relative homological algebra, of discussing the derived functors of a functor of two variables. In fact, this is done in this paper by considering cotriple (co) homology defined on a fibred product which is a subcategory of a product category. More precisely speaking, we introduce first a category 3e= (2), 31, Q)@,»,P) of fibred functors (T,0): (£, S3, P )->(?), 21, Q), which inherits the fibre wise properties of (2), 21, Q). Since a cotriple on the fibred category (£, S3, P) induces a cotriple on the category £?<? in the usual sense, relative homological algebra can be applied to 39. Consider the situation where a fibred functor (T, 0) is defined on a fibred category (£, S3, P) into an abelian category (2), 21, Q) and a cotriple (G, E, A) is given on (3t, 93, P). Then the cotriple (co) homology Jt^(TG) can be defined as an object in £?$. Moreover, if the fibred categories are both multiplicative and if the functors G, T satisfy certain conditions involved in the multiplicative functors, then an external product can be defined on H*(TG). For applications, T is

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