Homotopy in functor categories

Author:
Alex Heller

Journal:
Trans. Amer. Math. Soc. **272** (1982), 185-202

MSC:
Primary 55U35; Secondary 18A25, 18G55

Erratum:
Trans. Amer. Math. Soc. **279** (1983), 429.

MathSciNet review:
656485

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If is a small category enriched over topological spaces the category of continuous functors from into topological spaces admits a family of homotopy theories associated with closed subcategories of . The categories , for various , are connected to one another by a functor calculus analogous to the , Hom calculus for modules over rings. The functor calculus and the several homotopy theories may be articulated in such a way as to define an analogous functor calculus on the homotopy categories. Among the functors so described are homotopy limits and colimits and, more generally, homotopy Kan extensions. A by-product of the method is a generalization to functor categories of E. H. Brown's representability theorem.

**[1]**D. W. Anderson,*Axiomatic homotopy theory*, Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978) Lecture Notes in Math., vol. 741, Springer, Berlin, 1979, pp. 520–547. MR**557184****[2]**G. Baumslag, E. Dyer, and A. Heller,*The topology of discrete groups*, J. Pure Appl. Algebra**16**(1980), no. 1, 1–47. MR**549702**, 10.1016/0022-4049(80)90040-7**[3]**A. K. Bousfield and D. M. Kan,*Homotopy limits, completions and localizations*, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573****[4]**Alex Heller,*Abstract homotopy in categories of fibrations and the spectral sequence of Eilenberg and Moore*, Illinois J. Math.**16**(1972), 454–474. MR**0310890****[5]**Alex Heller,*Adjoint functors and bar constructions*, Advances in Math.**12**(1974), 8–31. MR**0334266****[6]**Alex Heller,*On the homotopy theory of topogenic groups and groupoids*, Illinois J. Math.**24**(1980), no. 4, 576–605. MR**586797****[7]**Alex Heller,*On the representability of homotopy functors*, J. London Math. Soc. (2)**23**(1981), no. 3, 551–562. MR**616562**, 10.1112/jlms/s2-23.3.551**[8]**Saunders MacLane,*Categories for the working mathematician*, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR**0354798****[9]**Dusa McDuff,*On the classifying spaces of discrete monoids*, Topology**18**(1979), no. 4, 313–320. MR**551013**, 10.1016/0040-9383(79)90022-3**[10]**Daniel G. Quillen,*Homotopical algebra*, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR**0223432****[11]**Graeme Segal,*Classifying spaces and spectral sequences*, Inst. Hautes Études Sci. Publ. Math.**34**(1968), 105–112. MR**0232393****[12]**N. E. Steenrod,*Milgram’s classifying space of a topological group*, Topology**7**(1968), 349–368. MR**0233353****[13]**Rainer M. Vogt,*Homotopy limits and colimits*, Math. Z.**134**(1973), 11–52. MR**0331376****[14]**Rainer M. Vogt,*Commuting homotopy limits*, Math. Z.**153**(1977), no. 1, 59–82. MR**0451237**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55U35,
18A25,
18G55

Retrieve articles in all journals with MSC: 55U35, 18A25, 18G55

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0656485-2

Keywords:
Abstract homotopy,
functor category,
homotopy limit,
homotopy Kan extension,
half-exact functor

Article copyright:
© Copyright 1982
American Mathematical Society