# Function complexes in homotopical algebra

@article{Dwyer1980FunctionCI, title={Function complexes in homotopical algebra}, author={William G. Dwyer and Daniel M. Kan}, journal={Topology}, year={1980}, volume={19}, pages={427-440} }

1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4 (iv))) as a general framework for “doing homotopy theory”. To each model category M there is associated a homotopy category. If W C M denotes the subcategory of the weak equivalences, then this homotopy category is just the localization M[W-‘I, i.e. the category obtained from M by formally inverting… Expand

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#### References

SHOWING 1-10 OF 15 REFERENCES

Calculating simplicial localizations

- Mathematics
- 1980

1.1. Summary. This paper is essentially a continuation of [3], where we introduced a (standard) simplicial localization functor, which assigned to every category C and subcategory W c C, a simpficiaf… Expand

Pseudo-isotopy and invariant theory—I

- Mathematics
- 1980

such that f(M X 0 = id; the space of all such pseudo-isotopies (with the C” Whitney topology) is denoted P(M). This paper looks at the problem of computing the rational homotopy groups qP( M>@0 for i… Expand

Realizing simply connected 4-manifolds by blowing down

- Mathematics
- 1980

A USUAL STARTING point in studying a smooth, compact, simply connected 4-manifold W4 is to describe W4 as a handlebody. In the case where W4 can be obtained by attaching 2-handles to B4, Kirby’s… Expand

Planes triply tangent to curves with nonvanishing torsion

- Mathematics
- 1980

EXPERIMENTATION with a closed loop of wire and a desk top quickly leads to the conclusion that except for certain special configurations, only a finite number of planes are tangent to a given curve… Expand