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

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• Author: Sebastian Thomas

Homotopical algebra may be thought of as the study of homotopy categories in the following sense. We consider
a category C that is equipped with a set (1
) of morphisms that we want to call weak equivalences. We would like
to consider the objects in C that are connected by weak equivalences as essentially equal, although a given weak
equivalence in C is not an isomorphism in general. To make this mathematically precise, we have to pass to the
homotopy category Ho C of C, which is defined to be the localisation of C with respect to the weak equivalences.
Here localisation is a purely category theoretical device that produces the universal category in which the weak
equivalences become isomorphisms – the idea being borrowed from localisation of rings.
The archetypical example is given by the category of topological spaces, with the weak equivalences being
continuous maps that induce isomorphisms on all homotopy groups. Similarly, we may consider the category
of simplicial sets, with the weak equivalences being simplicial maps that induce, after topological realisation,
isomorphisms on all homotopy groups. An additive example is given by the category of complexes with entries
in an abelian category, with weak equivalences being the quasi-isomorphisms, that is, the complex morphisms
that induce isomorphisms on all (co)homology objects. A further example, which is somehow degenerate from
our point of view, is given by an abelian category, with the weak equivalences being those morphisms having
kernel and cokernel in a chosen thick subcategory.

Table of contents:

I Localisations of categories 1
1 Categories with denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Localisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Saturatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
II Z-2-arrow calculus 19
1 Categories with denominators and S-denominators . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 S-2-arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 S-Ore completions and the classical S-Ore localisation . . . . . . . . . . . . . . . . . . . . . . . 27
4 Z-2-arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Z-fractionable categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6 The S-Ore localisation of a Z-prefractionable category . . . . . . . . . . . . . . . . . . . . . . . 50
7 The Z-Ore localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8 Maltsiniotis’ 3-arrow calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
III Cofibration categories 81
1 Categories with weak equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2 Categories with cofibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3 Categories with cofibrations and weak equivalences . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Cofibration categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Coreedian rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 Some structures on diagram categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7 Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8 The gluing lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9 The homotopy category of a Brown cofibration category . . . . . . . . . . . . . . . . . . . . . . 128
IV Combinatorics for unstable triangulations 137
1 Objects with shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2 Diagram categories on categories with shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3 Semiquasicyclic types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4 (Co)semiquasicyclic objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5 Semistrip types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 Cosemistrips and cosemicomplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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