Cofibration

In mathematics, in particular homotopy theory, a continuous mapping between topological spaces

:<math>i: A \to X</math>

is a cofibration if it has the homotopy extension property with respect to all topological spaces <math>S</math>. That is, <math>i</math> is a cofibration if for each topological space <math>S</math>, and for any continuous maps <math>f, f': A\to S</math> and <math>g:X\to S</math> with <math>g\circ i=f</math>, for any homotopy <math>h : A\times I\to S</math> from <math>f</math> to <math>f'</math>, there is a continuous map <math>g':X \to S</math> and a homotopy <math>h': X\times I \to S</math> from <math>g</math> to <math>g'</math> such that <math>h'(i(a),t)=h(a,t)</math> for all <math>a\in A</math> and <math>t\in I</math>. (Here, <math>I</math> denotes the unit interval <math>[0,1]</math>.)

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.

Definition

Homotopy theory

In what follows, let <math>I = [0,1]</math> denote the unit interval.

A map <math>i\colon A \to X</math> of topological spaces is called a cofibration^{pg 51} if for any map <math>f:A \to S</math> such that there is an extension to <math>X</math> (meaning: there is a map <math>f':X \to S</math> such that <math>f'\circ i = f</math>), we can extend a homotopy of maps <math>H:A\times I \to S</math> to a homotopy of maps <math>H': X\times I \to S</math>, where<blockquote><math>\begin{align}

H(a,0) &= f(a) \\

H'(x,0) &= f'(x)

\end{align}</math></blockquote>We can encode this condition in the following commutative diagram<blockquote>frameless</blockquote>where <math>S^I</math> is the path space of <math>S</math> equipped with the compact-open topology.

For the notion of a cofibration in a model category, see model category.

Examples

In topology

Topologists have long studied notions of "good subspace embedding", many of which imply that the map is a cofibration, or the converse, or have similar formal properties with regards to homology. In 1937, Borsuk proved that if <math>X</math> is a binormal space (<math>X</math> is normal, and its product with the unit interval <math>X\times I</math> is normal) then every closed subspace of <math>X</math> has the homotopy extension property with respect to any absolute neighborhood retract. Likewise, if <math>A</math> is a closed subspace of <math>X</math> and the subspace inclusion <math>A\times I \cup X\times {1}\subset X\times I</math> is an absolute neighborhood retract, then the inclusion of <math>A</math> into <math>X</math> is a cofibration.

Hatcher's introductory textbook Algebraic Topology uses a technical notion of good pair which has the same long exact sequence in singular homology associated to a cofibration, but it is not equivalent. The notion of cofibration is distinguished from these because its homotopy-theoretic definition is more amenable to formal analysis and generalization.

If <math>f:X \to Y</math> is a continuous map between topological spaces, there is an associated topological space <math>Mf</math> called the mapping cylinder of <math>f</math>. There is a canonical subspace embedding <math>i: X\to Mf</math> and a projection map <math>r: Mf\to Y</math> such that <math>r\circ i = f</math> as pictured in the commutative diagram below. Moreover, <math>i</math> is a cofibration and <math>r</math> is a homotopy equivalence. This result can be summarized by saying that "every map is equivalent in the homotopy category to a cofibration."

:frameless|108x108px

Arne Strøm has proved a strengthening of this result, that every map <math>f:X \to Y</math> factors as the composition of a cofibration and a homotopy equivalence which is also a fibration.

A topological space <math>X</math> with distinguished basepoint <math>x</math> is said to be well-pointed if the inclusion map <math>{x}\to X</math> is a cofibration.

The inclusion map <math>S^{n-1} \to D^n</math> of the boundary sphere of a solid disk is a cofibration for every <math>n</math>.

A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if <math>(X, A)</math> is a CW pair, then <math>A \to X</math> is a cofibration). This follows from the previous fact and the fact that cofibrations are stable under pushout, because pushouts are the gluing maps to the <math>n-1 </math> skeleton.

In chain complexes

Let <math>\mathcal{A}</math> be an Abelian category with enough projectives.

If we let <math>C_+(\mathcal{A})</math> be the category of chain complexes which are <math>0</math> in degrees <math>q << 0</math>, then there is a model category structure^{pg 1.2} where the weak equivalences are the quasi-isomorphisms, the fibrations are the epimorphisms, and the cofibrations are maps<blockquote><math>i:C_\bullet \to D_\bullet</math></blockquote>which are degreewise monic and the cokernel complex <math>\text{Coker}(i)_\bullet</math> is a complex of projective objects in <math>\mathcal{A}</math>. It follows that the cofibrant objects are the complexes whose objects are all projective.

Simplicial sets

The category <math>\textbf{SSet}</math> of simplicial sets