I have difficulties assimilating the notion of cofibration. I seem to get lost in the diagrams and technicalities (e.g. Hatcher/Bredon/Spanier). I'd be grateful if someone helped me out with that.

Please correct me if I'm wrong.

Suppose $i: A \hookrightarrow X$ is a cofibration (for simplicity, let's say it is an inclusion). Now let $Y$ be some topological space and $g: X \rightarrow Y$ any map.

In words, the cofibration seems to tell me that if we have a map $f: A \rightarrow Y$ which homotopy commutes with $g \circ i$ via some homotopy $G$, then there exists another homotopy $F: X \times [0,1] \rightarrow Y$ such that [...]. This is where I'm stuck. I understand the technical definition but I don't "feel" what it really means.

As a consequence, I fail to see how it can be used. Could someone please give me a specific illustration of a cofibration at work and fill in the blank with a non-technical explanation?



2 Answers 2


... such that the restriction of $F$ to $A\times [0,1]$ is $G$. All you need to say is that "we can extend the homotopy to $X$".

In algebraic topology, whenever you say "inclusion" you almost always mean "cofibration", though this is always true e.g. for CW-complexes, which is very possibly why the distinction is often blurred. A nice condition is that when your spaces are Hausdorff, a cofibration is a closed inclusion.

  • $\begingroup$ After applying put-it-away/digest/try-again method, I think I finally got it with your simple but clear explanation. Thanks! $\endgroup$
    – johnny
    Aug 5, 2011 at 13:29

The point of the notion of cofibration that it is, well, dual to the notion of fibration: fibrations are maps adapted to taking fibers (“kernels”) and cofibrations are maps adapted to taking cofibers aka quotients (“cokernels”); this duality is especially clear in the definition of a model category.

In some sense, this is the same duality that links (co)homology and homotopy groups — e.g. there is a long exact sequence of homotopy groups for a fibration and a long exact sequence of (co)homology groups for a cofibration. The later long exact sequence is the example of «cofibrations at work» in algebraic topology, I believe.


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