homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Spherical objects in a general pointed model category play the role of the spheres in $Top$.
Let $\mathcal{C}$ be a pointed model category.
The spheres form the obvious examples of spherical objects in the category $Top$, but the rational spheres give other examples.
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).
Any rational sphere is a sphere object (in a suitable category for rational homotopy theory).
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 $T^0$.
By a family of spherical objects for $\mathcal{C}$ is meant a collection of spherical objects in $\mathcal{C}$ closed under suspension.
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 $\mathcal{A}$.
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:
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 $\Pi_\mathcal{A}-Alg$.
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
The example from proper homotopy theory is discussed in