Contents

Idea

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

Spherical objects

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

Definition

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

Examples

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

2. In the category of path connected pointed spaces with action of a discrete group, $Gr.Top^*_0$ and space of form $S^n_G= \bigvee_G S^n$ 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 $T$ be a contractible locally finite 1-dimensional simplicial complex, with $T^0$ its 0-skeleton. Let $\epsilon : E'T^0$ be a finite-to-one function. By $S^n_\epsilon$ we mean the space obtained by attaching an $n$-sphere to the vertices of $T$ with at vertex $v$, the spheres attached to $v$ being indexed by $\epsilon^{-1}(v)$. This space $S^n_\epsilon$ is a spherical object in the proper category, $Proper_\infinity^T$, of $T$-based spaces. (In this context $T$ 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 $T = \mathbb{R}_{\geq 0}$, made up of an infinite number of closed unit intervals (end-to-end), then $S^n_\epsilon$ will be the infinite string of spheres considered in the entry on the Brown-Grossmann homotopy groups? if we take $\epsilon$ to be the identity function on $T^0$.

Definition

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

The theory $\Pi_\mathcal{A}$

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

Example

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

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

$\Pi_\mathcal{A}$-algebras

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

Properties

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

Example

If $X$ is in $\mathcal{C}$, define $\pi_\mathcal{A}(X):= [-,X]_{Ho(\mathcal{C})} : \Pi_{\mathcal{A}}^{op}\to Set_*$. This is the homotopy $\Pi_{\mathcal{A}}$-algebra of $X$. As with $\Pi$-algebras, there is a realisablity problem, i.e., given $\Lambda$, find a $X$ and an isomorphism, $\pi_\mathcal{A}(X)\cong \Lambda$. 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.