| Introduction | p. 1 |
| Aims and applications | p. 1 |
| Some examples | p. 2 |
| Directed spaces and other directed structures | p. 3 |
| Formal foundations for directed algebraic topology | p. 5 |
| Interactions with category theory | p. 6 |
| Interactions with non-commutative geometry | p. 7 |
| From directed to weighted algebraic topology | p. 7 |
| Terminology and notation | p. 8 |
| Acknowledgements | p. 9 |
| First-order directed homotopy and homology | p. 11 |
| Directed structures and first-order homotopy properties | p. 13 |
| From classical homotopy to the directed case | p. 14 |
| The basic structure of the directed cylinder and cocylinder | p. 28 |
| First-order homotopy theory by the cylinder functor, I | p. 40 |
| Topological spaces with distinguished paths | p. 50 |
| The basic homotopy structure of d-spaces | p. 61 |
| Cubical sets | p. 65 |
| First-order homotopy theory by the cylinder functor, II | p. 79 |
| First-order homotopy theory by the path functor | p. 89 |
| Other topological settings | p. 97 |
| Directed homology and non-commutative geometry | p. 105 |
| Directed homology of cubical sets | p. 106 |
| Properties of the directed homology of cubical sets | p. 114 |
| Pointed homotopy and homology of cubical sets | p. 120 |
| Group actions on cubical sets | p. 127 |
| Interactions with non-commutative geometry | p. 130 |
| Directed homology theories | p. 140 |
| Modelling the fundamental category | p. 145 |
| Higher properties of homotopies of d-spaces | p. 146 |
| The fundamental category of a d-space | p. 153 |
| Future and past equivalences of categories | p. 165 |
| Bilateral directed equivalences of categories | p. 177 |
| Injective and projective models of categories | p. 185 |
| Minimal models of a category | p. 193 |
| Future invariant properties | p. 199 |
| Spectra and pf-equivalence of categories | p. 206 |
| A gallery of spectra and models | p. 214 |
| Higher directed homotopy theory | p. 227 |
| Settings for higher order homotopy | p. 229 |
| Preserving homotopies and transposition | p. 230 |
| A strong setting for directed homotopy | p. 239 |
| Examples, I | p. 250 |
| Examples, II. Chain complexes | p. 254 |
| Double homotopies and the fundamental category | p. 262 |
| Higher properties of h-pushouts and cofibrations | p. 269 |
| Higher properties of cones and Puppe sequences | p. 277 |
| The cone monad | p. 283 |
| The reversible case | p. 290 |
| Categories of functors and algebras, relative settings | p. 296 |
| Directed homotopy of diagrams and sheaves | p. 297 |
| Directed homotopy in slice categories | p. 301 |
| Algebras for a monad and the path functor | p. 309 |
| Applications to d-spaces and small categories | p. 319 |
| The path functor of differential graded algebras | p. 327 |
| Higher structure and cylinder of dg-algebras | p. 334 |
| Cochain algebras as internal semigroups | p. 342 |
| Relative settings based on forgetful functors | p. 345 |
| Elements of weighted algebraic topology | p. 351 |
| Generalised metric spaces | p. 352 |
| Elementary and extended homotopies | p. 362 |
| The fundamental weighted category | p. 366 |
| Minimal models | p. 373 |
| Spaces with weighted paths | p. 376 |
| Linear and metrisable w-spaces | p. 387 |
| Weighted non-commutative tori | p. 391 |
| Tentative formal settings for the weighted case | p. 394 |
| Some points of category theory | p. 397 |
| Basic notions | p. 397 |
| Limits and colimits | p. 405 |
| Adjoint functors | p. 407 |
| Monoidal categories, monads, additive categories | p. 410 |
| Two-dimensional categories and mates | p. 414 |
| References | p. 418 |
| Glossary of symbols | p. 424 |
| Index | p. 427 |
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