# Contents

## Idea

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

## Spherical objects

Let $𝒞$ be a pointed model category.

###### Definition

A spherical object for $𝒞$ is a cofibrant homotopy cogroup in $𝒞$.

## Examples

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

2. In the category of path connected pointed spaces with action of a discrete group, $\mathrm{Gr}.{\mathrm{Top}}_{0}^{*}$ and space of form ${S}_{G}^{n}={\bigvee }_{G}{S}^{n}$ 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 $ϵ:E\prime {T}^{0}$ be a finite-to-one function. By ${S}_{ϵ}^{n}$ 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 ${ϵ}^{-1}\left(v\right)$. This space ${S}_{ϵ}^{n}$ is a spherical object in the proper category, ${\mathrm{Proper}}_{\infty }^{T}$, 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 $T={ℝ}_{\ge 0}$, made up of an infinite number of closed unit intervals (end-to-end), then ${S}_{ϵ}^{n}$ will be the infinite string of spheres considered in the entry on the Brown-Grossmann homotopy groups? if we take $ϵ$ to be the identity function on ${T}^{0}$.

###### Definition

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

## The theory ${\Pi }_{𝒜}$

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

### Example

For $𝒜=\left\{{S}^{n}{\right\}}_{n=1}^{\infty }$ in $\mathrm{Top}$, ${\Pi }_{𝒜}=\Pi$, the theory of Pi-algebras.

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

## ${\Pi }_{𝒜}$-algebras

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

### Properties

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

### Example

If $X$ is in $𝒞$, define ${\pi }_{𝒜}\left(X\right):=\left[-,X{\right]}_{\mathrm{Ho}\left(𝒞\right)}:{\Pi }_{𝒜}^{\mathrm{op}}\to {\mathrm{Set}}_{*}$. This is the homotopy ${\Pi }_{𝒜}$-algebra of $X$. As with $\Pi$-algebras, there is a realisablity problem, i.e., given $\Lambda$, find a $X$ and an isomorphism, ${\pi }_{𝒜}\left(X\right)\cong \Lambda$. The realisability problem is discussed in Baues-Blanc (2010) (see below).

## References

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 (64.89.59.121)