Homotopy coherent category theory

Transactions of the American Mathematical Society

Volumn 349, Issue 1, 1997, Pages 1-54

  Cordier, Jean Marc ^a   Porter, Timothy ^b  

^a UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

^b BANGOR UNIVERSITY   (United Kingdom)

Author keywords

Homotopy coherent ends and coends; Simplicially enriched categories; Yoneda lemma

Indexed keywords

EID: 21444438225     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-97-01752-2     Document Type: Article

References (52)

1. J. F. Adams, A variant of E.H. Brown's representability theorem, Topology, 10, (1971), 185-198. MR 44:1018
2. D. W. anderson, Axiomatic Homotopy Theory, in Algebraic Topology, Waterloo 1978, Lecture Notes in Mathematics, No. 741, Springer-Verlag, Berlin, Heidelberg, New York, 1979. MR 81c:55031
3. M. andré, Méthode Simpliciale en Algèbre Homologique et Algèbre Commutative, Lecture Notes in Mathematics, No. 32, Springer-Verlag, Berlin, Heidelberg, New York, (1967). MR 35:5493
4. M. Artin and B. Mazur, Étale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin, Heidelberg, New York, (1969). MR 39:6883
5. M, Artin and B. Mazur, On the van Kampen Theorem, Topology, 5, (1966), 179-189. MR 33:720
6. M. Barr and J. Beck, Homology and Standard Constructions, in Seminar on Triples and Categorical Homology Theory, (B. Eckmann, ed.), Lecture Notes in Mathematics, No. 80, pp. 245-335, Springer-Verlag, Berlin, Heidelberg, New York, 1969. MR 41:3562
7. M. A. Batanin, Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers Top. Géom. Diff. Cat, 34, (1993), 279-304. MR 95b:18006
8. H. J. Baues, Geometry of loop spaces and the cobar construction, Memoirs Amer. Math. Soc. 25, (1980), Number 230. MR 81m:55010
9. H. J. Baues, Algebraic Homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge Univ. Press, (1989). MR 90i:55016
10. D. Bourn and J.-M. Cordier, A general formulation of homotopy limits, J. Pure Appl. Algebra, 29, (1983), 129-141. MR 85c:55013
11. A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math. No. 304, Springer-Verlag, Berlin, Heidelberg, New York, (1972). MR 51:1825
12. S. Bozapalides, Theorie formelle des bicatégories, thèse, 3-ème cycle, (1976), Paris. MR 57:16380
13. S. Bozapalides, Some remarks on Lax presheafs, Illinois J. Math., 24, (1980), 676-680. MR 81m:18006
14. K. S. Brown, Abstract homotopy theory and generalised sheaf cohomology, Trans. Amer. Math. Soc., 186, (1973), 419-458. MR 49:6220
15. R. Brown and N. D. Gilbert, Algebraic Models for 3-types and automorphism structures for crossed modules, Proc. London Math. Soc., 59, (1989), 51-73. MR 90e:18015
16. J.-M. Cordier, Sur la notion de diagramme homotopiquement cohérent, Cahiers Top. et Géom. Diff., 23, (1982), 93-112, Proc., Sème Coll. sur les Catégories, Amiens, (1980). MR 83g:18018
17. J.-M. Cordier, Extensions de Kan simplicialement cohérentes, Prépublications, Amiens, (1985).
18. J.-M. Cordier, Sur les limites homotopiques de diagrammes homotopiquement cohérents, Comp. Math., 62, (1987), 367-388. MR 88m:55029
19. J.-M. Cordier, Homologie de Steenrod-Sitnikov et limite homotopique algébrique, Manuscripta Math., 59, (1987), 35-52. MR 88m:55003
20. J.-M. Cordier, Comparison de deux catégories d'homotopie de morphismes cohérents, Cahiers Top. Géom. Diff. Cat., 33, (1989), 257-275. MR 91d:55005
21. J.-M. Cordier and T. Porter, Vogt's theorem on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc., 100, (1986), 65-90. MR 87i:55027
22. J.-M. Cordier and T. Porter, Cohérent Kan Extensions, (i). Simplicially Enriched Ends and Coends, U.C.N.W. Pure Maths. Preprint 86.19, (1986).
23. J.-M. Cordier and T. Porter, Fibrant diagrams, rectifications and a construction of Loday, J. Pure Appl. Algebra, 67, (1990), 111-124. MR 92a:55017
24. J.-M. Cordier and T. Porter, Categorical Aspects of Equivariant Homotopy, Applied Categorical Structures (to appear).
25. W. G. Dwyer and D. M. Kan, Realizing diagrams in the homotopy category by means of diagrams of simplicial sets, Proc. Amer. Math. Soc., 91, (1984), 456-460. MR 86c:55010b
26. W. G. Dwyer and D. M. Kan, Function complexes for diagrams of simplicial sets, Proc. Kon. Ned. Akad. Wet., 86, (1983), 139-147. MR 85e:55038
27. D. A. Edwards and H. M. Hastings, Cecil and Steenrod Homotopy Theories with Applications to Geometric Topology, Lecture Notes in Mathematics No. 542, Springer-Verlag, Berlin, Heidelberg, New York, (1976). MR 55:1347
28. A. Elmendorf, Systems of Fixed Point Sets, Trans. Amer. Math. Soc, 277, (1983), 275-284. MR 84f:57029
29. P. E. Gardener, Generalized Derived Functors, Thesis, University of Warwick, 1981.
30. J. M. Gray, Closed categories, lax limits and homotopy limits, J. Pure Appl. Algebra, 19, (1980), 127-158. MR 82f:18007a
31. A. Grothendieck, (1983), Pursuing Stacks, typed manuscript, (c.600 pages).
32. B. Günther, The use of semisimplicial complexes in strong shape theory, Glasnik Mat., 27, (1992), 101-144. MR 94g:55014
33. A. Heller, Homotopy in Functor Categories, Trans. Amer. Math. Soc., 272, (1982), 185-202. MR 84j:55009a
34. A. Heller, Homotopy Theories, Memoirs Amer. Math. Soc. Vol. 71, No. 383, Amer. Math. Soc., Providence, R.I., (1988). MR 89b:55013
35. K. H. Kamps and T. Porter, Abstract homotopy and simple homotopy theory, World Scientific, (1996?).
36. G. M. Kelly, The Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series, 64, Cambridge University Press, Cambridge, (1983). MR 84e:18001
37. J. T. Lisica and S. Mardesic, Coherent prohomotopy and strong shape theory, Glasnik Mat., 19, (1984), 335-399. MR 87h:55005
38. S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math., Vol. 5, Springer-Verlag, Berlin, Heidelberg, New York, (1971). MR 50:7275
39. S. Mardesic and J.Segal, Shape Theory, the Inverse Systems Approach, North-Holland Mathematical Library vol 26, North-Holland, Amsterdam, (1982). MR 84b:55020
40. J. P. May, Classifying spaces and fibrations, Memoirs Amer. Math. Soc., Vol. 155, (1975). MR 51:6806
41. J.-P. Meyer, Bar and cobar constructions, J. Pure Applied Alg., 33, (1984), 163-207. MR 86g:18010
42. J.-P. Meyer, Mappings of bar constructions, Israel J. Math., 48, (1984), 331-339. MR 86a:18011
43. J. E. Roberts, Mathematical aspects of local cohomology, in Proceedings of the Colloquium on Operator Algebras and their Application to Mathematical Physics, Marseille, 1977. MR 81e:18017
44. C. A. Robinson, Torsion Products as Homotopy Groups, J. Pure Applied Algebra, 21, (1981), 167-182. MR 83f:18015
45. D. G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, Berlin, Heidelberg, New York, (1967). MR 36:6480
46. G. Segal, Categories and cohomology theories, Topology, 13, (1974), 293-312. MR 50:5782
47. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, (I960). MR 35:1007
48. R.Street, The algebra of oriented Simplexes, J. Pure Appl. Algebra 49 (1987) 283-335; also, Higher Dimensional Nerves, Notes, McGill 16 and 23 April 1985, and Weak n-categories, Notes of lectures at Bangor, June 1993. MR 89a:18019
49. M. Tierney and W. Vogel, Simplicial Resolutions and Derived Functors, Math. Z. Ill, (1969), 1-14. MR 40:2733
50. F. Ulmer, Kan Extensions, Cotriples and andré (Co)homology, in Category Theory, Homology Theory and their Applications, II, Lecture Notes in Mathematics, No. 92, (1969), 278-308. MR 41:1840b
51. D. Verity, Nerves of n-categories, Notes of lectures at Bangor, 1993.
52. R. M. Vogt, Homotopy Limits and Colimits, Math. Z., 134, (1973), 11-52. MR 48:9709