Abstract

A calculus of fractions for the homotopy category of a Brown cofibration category

Sebastian Thomas

Dissertation, RWTH Aachen University, 2012

We study the structure of homotopy categories of Brown cofibration categories. Examples are the homotopy category of simplicial sets or the derived category over a module category. Topological spaces yield a Brown fibration category, that is, they fulfil the dual axioms.

In the first part of the thesis, a calculus for these homotopy categories is derived. By definition, the homotopy category of a Brown cofibration category is the localisation at its weak equivalences, that is, certain morphisms are formally inverted. Analogously to the localisation theory of commutative rings, the morphisms of the homotopy category of a Brown cofibration category are, by a theorem of K. Brown, fractions consisting of one numerator and one denominator – they are represented by a so-called 2-arrow. Testing equality of two such fractions is more difficult than in commutative algebra: While two numerator-denominator pairs in commutative rings represent the same fraction in the localisation if and only if they have a common expansion, for 2-arrows in a Brown cofibration category it suffices to have homotopic expansions in order to represent the same fraction.

We show that Brown's homotopy 2-arrow calculus may be replaced by a strict calculus if one restricts to certain representatives for the fractions: Every morphism in the homotopy category is in fact represented by a so-called Z-2-arrow, which is a 2-arrow with an additional property. Two such Z-2-arrows represent the same morphism in the localisation if and only if they have a common expansion. The needed properties of Z-2-arrows are based on an interpretation as generalised, relative cylinders.

In the second part, the Z-2-arrow calculus is used to obtain an additional structure on the homotopy category. Schwede has shown that the homotopy category of a stable Brown cofibration category carries the structure of a triangulated category in the sense of Verdier – a generalisation of Hovey's theorem for stable Quillen model categories. One property of Verdier triangles is the octahedral axiom, which relates triangles and composition. We show that the diagrams constructed in the verification of the octahedral axiom may be seen as a generalisation of Verdier triangles, and that they have analogous properties. Somewhat more generally, we construct an unstable variant of n-triangles on the homotopy category of a (not necessarily stable) Brown cofibration category - Verdier triangles and the particular Verdier octahedra constructed in the verification of the octahedral axiom are the cases n = 2 resp. n = 3. Moreover, it is proved that unstable n-triangles are compatible with (generalised) simplicial operations and have prolongation properties analogously to those of Verdier triangles. This generalises a result of Künzer for Frobenius categories.