What are the algebras over $\Omega^k\Sigma^k$ ? - MathOverflow

What are the algebras over $\Omega^k\Sigma^k$ ?

2011-03-02T18:28:25Z

Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair $$ \Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k, $$ where $\Sigma^k$ is the $k$-th supension functor and $\Omega^k$ is the $k$-fold loop space functor ($k\ge 1)$. This adjunction has as associated monad the functor $\Omega^k\Sigma^k$. My question is what are the algebras over this monad (a precise reference to this fact would be welcome).

If we denote by $Ho^s$ the homotopy category of spectra, then there is another adjunction $$ \Sigma^{\infty} \colon Ho(Spc) \leftrightarrows Ho^s\colon \Omega^{\infty}, $$ where $\Sigma^{\infty}$ is the suspension spectrum functor. My question is again what are the algebras over the monad $\Omega^{\infty}\Sigma^{\infty}$.

Answer by Tyler Lawson for What are the algebras over $\Omega^k\Sigma^k$ ?

2011-03-02T18:41:14Z

(Everything I say here is up to homotopy equivalence.)

Algebras over $\Omega^k \Sigma^k$ are spaces $X$ equivalent to a $k$-fold loop space $\Omega^k Y$. Algebras over $\Omega^\infty \Sigma^\infty$ are infinite loop spaces; this is a little harder to say, but it is essentially that there is a sequence of spaces $Y_n$ with $Y_0 = X$ and equivalences $Y_n \simeq \Omega Y_{n+1}$ .

The original and still canonical reference, which covers all of this in detail, is J.P. May's book "The geometry of iterated loop spaces," Lectures Notes in Mathematics 271.

EDIT: As Neil points out, I've misread the question. The statements above are for spaces, not for objects in the homotopy category of spaces.

Answer by Neil Strickland for What are the algebras over $\Omega^k\Sigma^k$ ?

2011-03-02T19:42:35Z

I know interesting answers to two questions that are not the same as the one asked, but are related.

Consider the monad $T=\Omega^\infty L_{K(n)}\Sigma^\infty$ on the homotopy category of spaces. It is straightforward to construct a functor $$\Omega^\infty:Ho(\{K(n)-\text{local spectra}\})\to \{T-\text{algebras}\}.$$ One can show using the Bousfield-Kuhn functor and related ideas that this is actually an equivalence.
Consider the monad $Q=\Omega^\infty\Sigma^\infty$ on the category of based spaces (not up to homotopy). If we use spectra in the sense of Lewis and May, there is an evident functor $\Omega^\infty:\{\text{spectra}\}\to\{Q-\text{algebras}\}$. This is actually full and faithful (even on spectra whose homotopy groups are concentrated in negative degrees), which means that the point-set level $Q$-action carries a lot more information than you might naively guess. The key point in the proof is that we can use a trick with the Hopf map and space-filling curves to express $S^2$ as the coequaliser of two based maps from $S^3$ to $S^3$. This gives a natural way to express $\Omega^2X$ as the equaliser of two maps from $\Omega^3X$ to $\Omega^3X$, which allows us to do a bunch of things by induction.