spherical object



Spherical objects in a general pointed model category play the role of the spheres in TopTop.

Spherical objects

Let 𝒞\mathcal{C} be a pointed model category.


A spherical object for 𝒞\mathcal{C} is a cofibrant homotopy cogroup in 𝒞\mathcal{C}.


  1. The spheres form the obvious examples of spherical objects in the category TopTop, but the rational spheres give other examples.

  2. In the category of path connected pointed spaces with action of a discrete group, Gr.Top 0 *Gr.Top^*_0 and space of form S G n= GS nS^n_G= \bigvee_G S^n is a spherical object.(see Baues, 1991, ref. below, p.273).

  3. Any rational sphere is a sphere object (in a suitable category for rational homotopy theory).

  4. Let TT be a contractible locally finite 1-dimensional simplicial complex, with T 0T^0 its 0-skeleton. Let ϵ:ET 0\epsilon : E'T^0 be a finite-to-one function. By S ϵ nS^n_\epsilon we mean the space obtained by attaching an nn-sphere to the vertices of TT with at vertex vv, the spheres attached to vv being indexed by ϵ 1(v)\epsilon^{-1}(v). This space S ϵ nS^n_\epsilon is a spherical object in the proper category, Proper TProper_\infinity^T, of TT-based spaces. (In this context TT is acting as the analogue of the base point. It gives a base tree within the spaces. This is explored a bit more in proper homotopy theory.)

For instance, take T= 0T = \mathbb{R}_{\geq 0}, made up of an infinite number of closed unit intervals (end-to-end), then S ϵ nS^n_\epsilon will be the infinite string of spheres considered in the entry on the Brown-Grossmann homotopy groups? if we take ϵ\epsilon to be the identity function on T 0T^0.


By a family of spherical objects for 𝒞\mathcal{C} is meant a collection of spherical objects in 𝒞\mathcal{C} closed under suspension.

The theory Π 𝒜\Pi_\mathcal{A}

Let 𝒜\mathcal{A} be such a family of spherical objects. Let Π 𝒜\Pi_\mathcal{A} denote the full subcategory of Ho(𝒞)Ho(\mathcal{C}), whose objects are the finite coproducts of objects from 𝒜\mathcal{A}.


For 𝒜={S n} n=1 \mathcal{A} = \{S^n\}^\infty_{n=1} in TopTop, Π 𝒜=Π\Pi_\mathcal{A} = \Pi, the theory of Pi-algebras.

Of course, Π 𝒜\Pi_\mathcal{A} is a finite product theory in the sense of algebraic theories, and the corresponding models/algebras/modules are called:

Π 𝒜\Pi_\mathcal{A}-algebras

We thus have that these are the product preserving functors Λ:Π 𝒜 opSet *\Lambda : \Pi_\mathcal{A}^{op}\to Set_*. Morphisms of Π 𝒜\Pi_\mathcal{A}-algebras are simply the natural transformations. This gives a category Π 𝒜Alg\Pi_\mathcal{A}-Alg.


  • Such a Π 𝒜\Pi_\mathcal{A}-algebra, Λ\Lambda, is determined by its values Λ(A)Set *\Lambda(A)\in Set_* for AA in 𝒜\mathcal{A}, together with, for every ξ:A iIA i\xi\colon A \to \bigsqcup_{i\in I}A_i in Π 𝒜\Pi_\mathcal{A}, a map
ξ *:Λ(A i)Λ(A).\xi^*\colon \prod \Lambda(A_i)\to \Lambda(A).
  • The object AA being a (homotopy) cogroup, Λ(A)\Lambda(A) is a group (but beware the ξ *\xi^* need not be group homomorphisms).


If XX is in 𝒞\mathcal{C}, define π 𝒜(X):=[,X] Ho(𝒞):Π 𝒜 opSet *\pi_\mathcal{A}(X):= [-,X]_{Ho(\mathcal{C})} : \Pi_{\mathcal{A}}^{op}\to Set_*. This is the homotopy Π 𝒜\Pi_{\mathcal{A}}-algebra of XX. As with Π\Pi-algebras, there is a realisablity problem, i.e., given Λ\Lambda, find a XX and an isomorphism, π 𝒜(X)Λ\pi_\mathcal{A}(X)\cong \Lambda. The realisability problem is discussed in Baues-Blanc (2010) (see below).


Spherical objects are considered in

Examples are given in earlier work by Baues and by Blanc.

The group action case is in

  • Hans-Joachim Baues, Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).

The example from proper homotopy theory is discussed in

  • H.-J. Baues and Antonio Quintero, Infinite Homotopy Theory, K-monographs in mathematics, Volume 6, Kluwer, 2001.

Revised on August 18, 2011 00:23:06 by Toby Bartels (