The notion of a fibration has a nice geometric intuition of one topological space (a fiber) being parametrized by another topological space (the base) -- this is taken from the Wikipedia entry on Fibration.

Now, I would like to know if there is an analogous geometric picture for the situation of cofibrations. Like that of a parametrized inclusion of a space into another space (apologies if this is nonsense).


1 Answer 1


Yes, there are precisely the dual diagrams defining a cofibration. The way to think about them -- the way I think about them, at least -- is just that they're sufficiently nice inclusions, such that the subspace has a bit of "wiggle room". This is made precise in the definition of a "neighborhood deformation retract (NDR) pair", which you can read about in May's Concise Course in Algebraic Topology. Maybe the best example to keep in mind is when you turn an arbitrary map into a cofibration with the "mapping cylinder" construction; then it's obvious that the subspace has plenty of wiggle room.

  • $\begingroup$ From this perspective, the important aspect of a fibration is not that it has a well-defined fiber (as a homotopy type), but rather that it behaves well with respect to lifting maps through the morphism. Dually, the important aspect of a cofibration is that it behaves well with respect to extending maps along the morphism. $\endgroup$ Nov 13, 2012 at 22:47
  • $\begingroup$ Thanks for the answer. So for cofibration there is not a "dual" geometric intuition to that of the fibration? I mean, I understand the algebraic duality between fibration and cofibration, but cannot grasp the geometric idea (if any) so easily. $\endgroup$
    – camilo
    Nov 14, 2012 at 15:02
  • $\begingroup$ (cont.) An instance of the nice intuition for fibration is for fiber bundles. Here one may have non-trivial bundles, as is always exemplified by the Moebius band. Is there a "dual" geometric picture to this situation? that is, where you define a cofibration "locally" which is nevertheless globally nontrivial? $\endgroup$
    – camilo
    Nov 14, 2012 at 15:06
  • $\begingroup$ @camilo: I'm not sure; there might be some connection that I just don't see though. However, here is another duality-type statement. A fibration sits in a fiber sequence, which loops backwards indefinitely (look up the "Puppe sequence", or maybe Dold-Puppe) and induces a lexseq for homotopy classes of maps in (e.g. from spheres -- this is the lexseq in homotopy). On the other hand, a cofibration sits in a cofiber sequence, which suspends forward indefinitely, and this induces a lexseq for homotopy classes of maps out (e.g. to Eilenberg-MacLanes -- this is the lexseq in cohomology). $\endgroup$ Nov 15, 2012 at 1:25
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    $\begingroup$ I'll let you know if I come across anything, but I think I've read enough at this point that I would've seen something if it existed (beyond what I've already said). The point of both of these notions -- from a model category point of view, where one converts (up to weak equivalence) any morphism to a fibration or a cofibration willy-nilly -- is that they give us different measurements the "difference" of a morphism, i.e. its failure to be a homotopy equivalence. [cont] $\endgroup$ Nov 15, 2012 at 22:56

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