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.

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 ${ϵ}^{-1}(v)$. This space $S_{ϵ}^n$ is a spherical object in the proper category, ${\mathrm{Proper}}_{\mathrm{\infty }}^T$, 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 $T^0$.

The theory ${\Pi }_{𝒜}$

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

Example

For $𝒜=\{S^n{\}}_{n=1}^{\mathrm{\infty }}$ 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 $\Lambda (A)\in {\mathrm{Set}}_{*}$ for $A$ in $𝒜$, together with, for every $\xi :A\to {⨆}_{i\in I}{A}_i$ in ${\Pi }_{𝒜}$, a map

• 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 $𝒞$, define ${\pi }_{𝒜}(X):=[-,X{]}_{\mathrm{Ho}(𝒞)}:{\Pi }_{𝒜}^{\mathrm{op}}\to {\mathrm{Set}}_{*}$. 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, ${\pi }_{𝒜}(X)\cong \Lambda$. The realisability problem is discussed in Baues-Blanc (2010) (see below).

References

Spherical objects are considered in

• Hans-Joachim Baues and David Blanc, Comparing cohomology obstructions, (2010), Arxiv (to appear JPAA).

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.