Homotopical Algebra
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Edited by A. Dold, Heidelberg and B. Eckmann, Zurich
43
Daniel G. Quillen Massachusetts Institute of Technology Cambridge, Mass.
Homotopical Algebra 1967
Springer-Verlag· Berlin· Heidelberg· New York
AU rights, e5p~ially that of trm~ltion into forelgJllangullges. reserved. It is aho forbidden to roprodlltC thU book, either whole or in part, hy photomechankal means (photostat, microfilm and/or mlcrocord) or by other proadure w:Ilhou~
written permiuloll from Springer Verlag. @ by Springer.Verlag Betllll . HeIdelberg 1967. Library of Congress CilalQjl wd Number 67- 29:no, Title No, 7363
Homotopical Alsebr& Daniel G. Quillen1 Homotopical algebra or non-linear homological algebra is the generalization of homological algebra to arb1trary categories which results by considering a simplicial object as being a general1zation ot a chain complex.
The first step in the theory wa&
presented in [5], [6], where the derived functors of a nonadditive functor from an abelian category t1ves to another category
~
generalizes to the case where
~
with enough projec-
were constructed. ~
This construction
1s a category closed under f1n1te
limits hav1ng sufficiently many projective objects, and these derived functors can be used to give a uniform definition of cohomology for universal algebras.
In order to compute this cohomo-
logy for commutative rings, the author was led to consider the
simplicial Objects over A as forming the objects of a homotopy theory analogous to the homotopy theory of algebraic topology, then using the analogy as a source of intuition for simplicial objects.
'Ibis was suggested by the theorem of Kan [10] that the
homotopy theory of simplicial groups is equivalent to the homotopy theory of connected pointed spaces and by the derived category ([9], [19]) of
an
abelian category.
The analogy turned out
to be very fruitful; but there were a large number ot arguments lSUP20rted in part by the NatiQnal Science Foundation under grant GP 616?
which were formally similar to weJ.l-known ones in algebraic topologYJ so it was decided to define the notion of a homotopy theory in sufficient generality to cover in a uniform way the different homotopy theories encountered.
This is what is done in the pre-
sent paperj applications are reselyed for the future.
The following is a brief outline of the contents of this 'h4~~\~ l"ItrcttUy
B'
where i
-Ml.
is a cof1bratlon,
p 1s a fibration, and either 1 or
p 15 also a weak equivalence, there exists a dotted arrow such that the total diagram is commutative. factored
f ~ pi
where p, pi
and
f
= p'!'
where
H2.
Any map f may be
1, i'
are t!brations, and where p and
weak equivalences.
are colibrations i'
are also
It should be noticed that we do not assume
the existence of a path or cylinder functor; in fact the homotopy relation for maps may be recovered as follows: cotlbrant if the map
~ ~
Call an object X
X is a cofibration {hence in the cate-
gory of simpliCial groups the cotlbrant objects are the free
o.~
simp11cial groups) and li
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