homotopy theory, (∞,1)-category theory, homotopy type theory
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see also algebraic topology
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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 of 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 $[a,b] = a b a^{-1} b^{-1}$, 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 always 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 the theory of 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.