homotopy theory, (∞,1)-category theory, homotopy type theory
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A $\Pi$-algebra is an algebraic model for the homotopy groups $\pi_*X$ of a pointed topological space, $X$, together with the action of the primary homotopy operations on them, in the same sense that algebras over the Steenrod algebra are models for the cohomology of a space.
Constructions of this type exist in many pointed model categories. It suffices to have a collection of spherical objects.
The category $\Pi$ of homotopy operations has
as objects - pointed CW-complexes with the homotopy type of a finite wedge product of spheres of dimensions $\geq 1$;
as morphisms - homotopy classes of (pointed) continuous functions between them.
$\Pi$ is a pointed category and has finite coproducts (given by the finite wedges), but not products.
There is a functor, smash product $i : \Pi\times \Pi \to \Pi$, which sends an object $(U,V)$ to $U\wedge V = (U\times V)/((U\times *)\vee(*\times V))$, which preserves coproducts in each variable.
This category $\Pi^{op}$ is a finite product theory, in the sense of algebraic theories whose models are:
Let $Set_*$ denote the category of pointed sets.
A $\Pi$-algebra is a functor $A: \Pi^{op}\to Set_*$, which sends coproducts to products.
A morphism of $\Pi$-algebras is a natural transformation between the corresponding functors.
A $\Pi$-algebra $A$ satisfies $A* = *$.
The values of a $\Pi$-algebra $A$ are determined by the values $A_n = A(S^n)$, that it takes on the spheres, $S^n$, $n\geq 1$.
A $\Pi$-algebra can be considered to be a graded group $\{A_n\}_{n=1}^\infty$ with $A_n$ abelian for $n\gt 1$, together with
for $p,q \geq 1$ (the case where they are equal to 1 needs special mention, see below.)
which satisfy the identities that hold for the Whitehead products and composition operations on the higher homotopy groups os a pointed space, and
The Whitehead products include
$[\xi,a] = {}^\xi a - a$, where ${}^\xi a$ is the result of the $A_1$-action of $\xi \in A_1$ on $a\in A_r$, $r\gt 1$; similarly for a right action;
the commutators $[a,b] = aba^{-1}b^{-1}$, for $a,b \in A_1$.
For a pointed space $X$, and $U \in \Pi$, define a $\Pi$-algebra $\pi_* X$ by $\pi_* X(U) = [U,X]_*$, the set of pointed homotopy classes of pointed maps from $U$ to $X$.
This is a $\Pi$-algebra called the homotopy $\Pi$-algebra of $X$.
Suppose $A: \Pi \to sets_*$ is an abstract $\Pi$-algebra, the realisability problem for $A$ is to construct, if possible, a pointed space $X$, such that $A\simeq \pi_* X$. The space $X$ is called a realisation of $A$.
Things can be complicated!
The homotopy type of $X$ is not be determined by $A$ (hence ‘a’ rather than ‘the“ realisation) , so that raises the additional problem of classifying the realisations.
Not all $\Pi$-algebras can be realised, in fact
Given a $\Pi$-algebra, $A$, there is a sequence of higher homotopy operation?s depending only on maps between wedges of spheres, such that $A$ is realisable if and only if the operations vanish coherently.
For $p\neq 2$, a prime and $r\geq 4(p-1)$, $\pi_*S^r \otimes \mathbb{Z}/p$ cannot be realised (and if $p = 2$, one uses $r\geq 6$).
(There is a problem, discussed in Blanc’s 1995 paper, that the composition operations need not be homomorphisms, so tensoring with $\mathbb{Z}/p$ has to be interpreted carefully.)
A $\Pi$-algebra, $A$, is said to be simply connected if $A_1= 0$.
In this case the universal identities for the primary homotopy operations can be described more easily (see Blanc 1993). These include the structural information that the Whitehead products make $A$ into a graded Lie ring (with a shift of indices).
The beginnings of a classification theory for $n$-truncated $\Pi$-algebras can be found in Frankland’s thesis (link given below).
David Blanc has written a lot on these objects. An example is
The realisability problem is discussed in
and further in
There are more recent results on the realisability problem in Martin Frankland’s thesis.