Some remarks on Cartan–Eilenberg categories

Abstract

Abstract. In this note we collect some remarks and examples on Cartan-Eilenberg categories.

Homotopy theory of mixed Hodge complexes

“…For r = 0 we recover the notions of filtered quasi-isomorphism and filtered derived category studied by Illusie in [Ill71] (see also [Kel90] and [Pas12] for an account in the frameworks of exact categories and Cartan-Eilenberg categories respectively). There is a chain of functors…”

Cotorsion Pairs and Cartan–Eilenberg Categories

“…In [19], as a counterexample, C + (Qco(P 1 k )), where P 1 k is the projective line, was proved not to be a left Cartan-Eilenberg category with the usual homotopy and weak equivalences. In the next example, we show that under some conditions on the scheme, vector bundles may substitute for projective objects.…”

“…The motivation of this work comes from two examples on C + (A) given in [19]: the category C + (A) of bounded below complexes with the usual homotopy equivalences S and quasi-isomorphisms W is a left Cartan-Eilenberg category if A has enough projectives. In contrary, the other one is a counterexample in which A has no enough projectives.…”

“…To solve whether (A, S, W ) is a left Cartan-Eilenberg category is the same as proving that the subcategory A cof consisting of (S, W )-cofibrant objects is a coreflective subcategory of A[S−1 ]. For more detailed treatment, we refer to[12,19].Section 4 is devoted to exhibiting our claims. Essentially, we find new examples of left Cartan-Eilenberg categories on complexes.…”