suspension object



Homotopy theory

Stable homotopy theory



In a (∞,1)-category CC admitting a final object *{*}, for any object XX its suspension object ΣX\Sigma X is the homotopy pushout

X * * ΣX, \array{ X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \Sigma X } \,,

This is the mapping cone of the terminal map X*X \to {*}. See there for more details.

This concept is dual to that of loop space object.

Suspension functor

As an (infinity,1)-functor

Let CC be a pointed (infinity,1)-category. Write M ΣM^\Sigma for the (infinity,1)-category of cocartesian squares of the form

X * * Y, \array{ X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& Y } \,,

where XX and YY are objects of CC. Supposing that CC admits cofibres of all morphisms, then one sees that the functor M ΣCM^\Sigma \to C given by evaluation at the initial vertex (XX) is a trivial fibration. Hence it admits a section s:CM Σs : C \to M^\Sigma. Then the suspension functor Σ C:CC\Sigma_C : C \to C is the composite of ss with the functor M ΣCM^\Sigma \to C given by evaluating at the final vertex (YY).

Σ C\Sigma_C is left adjoint to the loop space functor Ω C\Omega_C.

As an ordinary functor

Let CC be a category admitting small colimits. Let Φ\Phi be a graded monoid in the category of groups and F:CCF : C \to C a Φ\Phi-symmetric endofunctor of CC that commutes with small colimits. Let Spect F Φ(C)Spect_F^{\Phi}(C) denote the category of Φ\Phi-symmetric FF-spectrum objects in CC.

Following Ayoub, the evaluation functor

Ev n:Spect F Φ(C)C, Ev^n : Spect_F^{\Phi}(C) \to C,

which “evaluates” a symmetric spectrum at its nnth component, admits under these assumptions a left adjoint

Sus n:CSpect F Φ(C) Sus^n : C \to \Spect_F^\Phi(C)

called the nnth suspension functor, more commonly denoted Σ C n\Sigma_C^{\infty-n}.

When CC is symmetric monoidal, and in the case Φ=Σ\Phi = \Sigma and F=TF = T \otimes - for some object TT, there is an induced symmetric monoidal structure on Spect T Σ(C)Spect^\Sigma_T(C) as described at symmetric monoidal structure on spectrum objects.

Proposition. One has

Sus T p(X)Sus T q(Y)Sus T p+q(XY) Sus^p_T(X) \otimes Sus^q_T(Y) \simeq Sus^{p+q}_T(X \otimes Y)

for all X,YCX,Y \in C. In particular, Sus=Sus 0:CSpect T Σ(C)Sus = Sus^0 : C \to \Spect^\Sigma_T(C) 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)

Revised on February 7, 2014 00:49:35 by Adeel Khan (