nLab m-cofibrant space

Definition

An m-cofibrant space is a topological space that is homotopy equivalent to a CW complex.

Idea

The importance of m-cofibrant spaces comes from the fact any weak homotopy equivalence between m-cofibrant spaces is a homotopy equivalence. It is a classical theorem – the Whitehead theorem – that this is true for CW complexes, and it is easy to see that it is a property preserved under homotopy equivalence. In fact, it is easy to see that m-cofibrant spaces are the largest subcategory $C$ of $Top$ which includes all CW complexes and such that any weak homotopy equivalence between objects of $C$ is a homotopy equivalence.

Moreover, since any space is weakly equivalent to a CW complex, any space is a fortiori weakly equivalent to an m-cofibrant one. It follows that the homotopy category of topological spaces in which the weak homotopy equivalences are inverted is equivalent to the homotopy category of m-cofibrant spaces in which homotopic maps are identified.

Therefore, m-cofibrant spaces are a natural place to do homotopy theory if one is interested in spaces up to weak homotopy type, but doesn’t want to have to worry about the distinction between weak homotopy equivalence and homotopy equivalence. For instance, when higher category theorists say that an $\infty$-groupoid “is” a topological space, they generally mean an m-cofibrant space, so that $\infty$-functors, natural transformations, and so on can be identified with continuous maps, homotopies, etc.

Milnor, in particular, has argued that the category of m-cofibrant spaces is a convenient category of spaces for algebraic topology. It is not cartesian closed, but Milnor proved that if $Y$ is m-cofibrant and $X$ is compact Hausdorff, then $Y^X$ is also m-cofibrant. Hence many desirable constructions are still available, in particular finite and infinite loop spaces.

More recently it has been discovered that there is a model structure on Top, called the mixed model structure, in which the weak equivalences are the weak homotopy equivalences, the cofibrant objects are precisely the m-cofibrant spaces, and the fibrations are the Hurewicz fibrations. This is where the name “m-cofibrant” comes from; “m-” is for the “mixed” model structure, as contrasted with the “q-” or Quillen model structure and the “h-” or Hurewicz/Strøm model structure (see model structure on topological spaces).