Paths and cylinders
Stable homotopy theory
In a (∞,1)-category admitting a final object , for any object its suspension object is the homotopy pushout
This is the mapping cone of the terminal map . See there for more details.
This concept is dual to that of loop space object.
As an (infinity,1)-functor
Let be a pointed (infinity,1)-category. Write for the (infinity,1)-category of cocartesian squares of the form
where and are objects of . Supposing that admits cofibres of all morphisms, then one sees that the functor given by evaluation at the initial vertex () is a trivial fibration. Hence it admits a section . Then the suspension functor is the composite of with the functor given by evaluating at the final vertex ().
is left adjoint to the loop space functor .
As an ordinary functor
Let be a category admitting small colimits. Let be a graded monoid in the category of groups and a -symmetric endofunctor of that commutes with small colimits. Let denote the category of -symmetric -spectrum objects in .
Following Ayoub, the evaluation functor
which “evaluates” a symmetric spectrum at its th component, admits under these assumptions a left adjoint
called the th suspension functor, more commonly denoted .
When is symmetric monoidal, and in the case and for some object , there is an induced symmetric monoidal structure on as described at symmetric monoidal structure on spectrum objects.
Proposition. One has
for all . In particular, is a symmetric monoidal functor.
A detailed treatment of the 1-categorical case is in the last chapter of
- Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I. Astérisque, Vol. 314 (2008). Société Mathématique de France. (pdf)