Cofibration
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In mathematics, in particular homotopy theory, a continuous mapping between topological spaces
,
is a cofibration if it has the homotopy extension property with respect to all topological spaces . That is,
is a cofibration if for each topological space
, and for any continuous maps
and
with
, for any homotopy
from
to
, there is a continuous map
and a homotopy
from
to
such that
for all
and
. (Here,
denotes the unit interval
.)
This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology.
Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a model category is endowed with three distinguished classes of morphisms called fibrations, cofibrations and weak equivalences satisfying certain lifting and factorization axioms.