Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
Paths and cylinders
The mapping cocone of a morphism , with a pointed object, is a particular model category realization of the homotopy fiber of , i.e. of the homotopy pullback
of the point along .
The concept is dual to that of mapping cone.
Mapping co-cones are realized in terms of mapping cocylinders, as described below. These typically play the role of generalized universal bundles.
Under sufficiently nice conditions (…spell out…), the homotopy pullback
may be computed as the ordinary limit over the diagram
where is a path space object of . This, in turn, may be computed as two consecutive ordinary pullbacks. The first one
yields the mapping cocylinder . This is the kind of object discussed at generalized universal bundle. The second pullback then produces the mapping cocone
In the category Top of topological spaces, let be group and its classifying space, and let be a morphism. Then
Note on terminology
Whitehead complained about the term cocone back in the old days, because of the seeming (though false) double dualization, so he used mapping path space. This practice was followed by his school (in most of US for example). But he himself was not confident in that terminology. For example there is a table in his book where he lists the dual notions and at the place where mapping cocone/mapping path space should fit he puts just the symbol for the construction while on the dual side he puts the whole name. Similarily for the mapping cocylinder.
Somebody – maybe Samuel Eilenberg – (– check –) suggested to Whitehead to use ne insteaad of cocone , jokingly cancelling one co against the other, as if both expressed abstract duality.
Postnikov uses the term mapping cocylinder , while for on Whitehead’s complaint he comments:
we do not see a particular criminal in the cocone terminology, but will anyway not use it.