# What's a good reference for the following obstruction theory yoga?

Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I will let $Y$ denote any algebra, and I will let $X$ denote an algebra such that if you forget the differential, then $X$ is free on a well-ordered set of generators, and such that for each generator, its differential is a composition of strictly earlier (for the well-ordering) generators.

Here are some facts I know how to prove, but my proofs are in places long-winded. They are not in any way due to me — they are "classical results", where "classical" is defined as "something the author learned in graduate school". My question is: what is a good reference that proves these results?

1. One may try to build an algebra homomorphism $\eta: X \to Y$ inductively. When trying to $\eta(x)$ for a generator $x$, the only condition is that $\partial \eta(x) = \eta(\partial x)$, the latter having already been defined. The induction can continue iff $\eta(\partial x)$ is exact (it is already closed), and this is measured by the class of $\eta(\partial x)$ in $\mathrm H_{\deg x - 1}(Y)$. Here $x$ has homological degree $x$, and I'm using homological conventions, so that $\deg(\partial x) = \deg x - 1$. I'm also being a bit sloppy with notation — really I mean $\mathrm H_\bullet($the part of $Y$ with the appropriate colors for $\eta(x)$ to be valued there$)$.

2. Suppose that $\eta(\partial x)$ is exact. Different choices for $\eta(x)$ might affect whether later steps of the induction succeed. Changing $\eta(x)$ by something exact will not affect the success or failure of later steps. So the "true" set of choice for $\eta(x)$ is a torsor for $\mathrm H_{\deg x}(Y)$.

3. Recall Sullivan's simplicial dg commutative algebra $\Omega(\Delta_\bullet)$, whose $k$-simplices are the dgca $\mathbb K[t_0,\dots,t_k,\partial t_0,\dots,\partial t_k] / (\sum t_i = 1, \sum \partial t_i = 0)$. By definition, the space $\hom_\bullet(X,Y)$ of homomorphisms $X \to Y$ is the simplicial set whose $k$-simplices are $\hom_{\text{algebras}}(X,Y \otimes \Omega(\Delta_k))$. This simplicial set satisfies the Kan horn filling condition.

4. Suppose we have chosen a homomorphism $\eta : X \to Y$. What is the homotopy type of the connected component of $\eta$ in $\hom_\bullet(X,Y)$? For $k\geq 1$, the $k$th homotopy group $\pi_k(\hom_\bullet(X,Y);\eta)$ is an extension of abelian groups, one for each generator $x$ of $X$, such that the $x$th group describes $\pi_k($space of choices for $\eta(x))$. The $x$th group in the extension is precisely $\mathrm{H}_{\deg x + k}(Y)$.

• One correction: I think my proof for 4 depends on earlier $\pi_k$s vanishing. I don't know whether 4 is true as stated. – Theo Johnson-Freyd Feb 1 '14 at 20:44

Theo, there is a nice, abstractly developed, obstruction theory in Baues's "Combinatorial Foundation of Homology and Homotopy":

Your context can be fitted into the general framework of this book with little effort. This would give you a much more structured obstruction theory than what you suggest in 1 and 2. Computations are possible, and Baues himself has successfully applied it in many papers (and other books).

In 3, that's going to be a Kan complex whenever $X$ is cofibrant, your conditions may not be enough, but probably close.

As for 4, and to some extent for a different answer to 1 and 2, you can take a look at Bousfield's "Homotopy spectral sequences and obstructions"

• Great. For the colored operads I care about, the category of dg algebras admits a model structure in which fibrations are surjections and acyclics are quasiisomorphisms. In this case, everything is fibrant, and the conditions on $X$ imply that $X$ is cofibrant. I've always assumed that this is true for any colored operad, but I realize the proofs I've seen tend to take specific ones. – Theo Johnson-Freyd Feb 1 '14 at 20:47