# How is topological André-Quillen homology (TAQ) a "stabilization", exactly?

Let $$S\to A\to B$$ be cofibrations of commutative $$S$$-algebras. Then the topological André-Quillen $$B$$-module $$TAQ(B|A)$$ can be computed as a stabilization. Precisely, I think it means the following: let $$I$$ be the augmentation ideal functor from augmented commutative $$B$$-algebras to $$B$$-modules; it is right Quillen. These categories are enriched and tensored over based spaces: let $$\wedge$$ denote the tensor of $$B$$-modules and $$\odot$$ denote the tensor of augmented commutative $$B$$-algebras.

There are canonical maps $$X\wedge IC\to I(X\odot C)$$ for any augmented commutative $$B$$-algebra $$C$$ and based space $$X$$. In particular, there are maps $$S^1\wedge I(S^n\odot C) \to I(S^{n+1}\odot C)$$. Taking $$n$$-th loops (in $$B$$-modules) of the adjoint gives maps $$\Omega^nI(S^n\odot C)\to \Omega^{n+1} I(S^{n+1} \odot C).$$ The claim is that if we take $$C=B\wedge_A B$$ (a $$B$$-algebra on the first variable and the augmentation is the multiplication map), then the homotopy colimit in $$B$$-modules of this tower is weakly equivalent to $$TAQ(B|A)$$. Am I saying it right? In symbols,

Is $$TAQ(B|A)\simeq \operatorname{hocolim} (\Omega^nI(S^n \odot (B\wedge_AB)))$$?

It seems the answer can be deduced from the results in Basterra-Mandell, Homology and cohomology of $$E_\infty$$-ring spectra, but it's not obvious to me. If it's the case, then how?

Another presentation of the "$$TAQ$$ as stabilization" statement I've seen (up to my own misinterpretations) is the following: the category of commutative $$A$$-algebras (no augmentation) is tensored over unbased spaces: call this tensor $$\otimes_A$$. One can take the tensors $$S^n\otimes_A B$$, and consider the inclusion of the point $$*\to S^n$$ which gives maps $$B\to S^n\otimes_A B$$ of which we can take the cofiber in $$B$$-modules. Denote this cofiber by $$S^n\overline{\otimes}_AB$$. These also form a direct system by passing to cofibers some maps gotten similarly as above.

Is $$TAQ(B|A)\simeq \operatorname{hocolim}(\Omega^n(S^n\overline\otimes_AB))$$?

Note: There is yet another formulation which might be useful and which is equivalent to the first one. The functor $$I$$ factors via the category of non-unital commutative $$B$$-algebras (nucas), as a functor $$I_0$$ followed by the forgetful functor $$U$$ (both right Quillen). The category of nucas is also tensored over pointed spaces; call the tensor $$\tilde\otimes$$. Instead of considering $$I(S^n\odot C)$$ in the directed system above, we could consider $$U(S^n\tilde\otimes I_0C)$$. We get a directed system similarly as above, and

$$\operatorname{hocolim} (\Omega^nI(S^n \odot C )) \simeq \operatorname{hocolim}(\Omega^nU(S^n\tilde\otimes I_0 C))$$.

This follows from the fact that $$I_0$$ is a right Quillen equivalence, so it preserves tensors (properly interpreted; see Does the right adjoint of a Quillen equivalence preserve homotopy colimits?).

These stabilization formulas do indeed follow from the paper of Basterra-Mandell. Fix a commutative $$S$$-algebra $$A$$. Then Basterra and Mandell prove the following:
1) [Theorem 3] Given a commutative $$A$$-algebra $$B$$, the $$(\infty-)$$category of $$\Omega$$-spectrum objects in augmented $$B$$-algebras is equivalent to the $$(\infty-)$$category of $$B$$-modules via the functor that takes an $$\Omega$$-spectrum object $$\{X_n\}$$ in augmented $$B$$-algebras to the augmentation ideal of $$X_0$$.
2) [Theorem 4] Under the identification of (1), the $$B$$-module $$TAQ(B|A)$$ corresponds to (the $$\Omega$$-spectrum replacement of) the suspension spectrum of the augmented $$B$$-algebra $$B \wedge_A B$$.
Combining (1) and (2) we get your first stabilization formula. Indeed, the suspension spectrum of $$B \wedge_A B$$ is given in degree $$n$$ by what you denote by $$S^n \odot (B \wedge_A B)$$. The $$\Omega$$-spectrum replacement of this suspension spectrum then has in degree $$0$$ the homotopy colimit $${\rm hocolim}_n \Omega^nS^n \odot (B \wedge_A B)$$, where the loop operation $$\Omega$$ is performed in augmented $$B$$-algebras. Applying the equivalence of (1) and using the fact that the augmentation ideal functor $$I$$ commutes with loops we get that the $$B$$-module corresponding to the suspension spectrum of $$B \wedge_A B$$ is given by the formula $${\rm hocolim}_n \Omega^n I(S^n \odot (B \wedge_A B))$$. By (2) this is also $$TAQ(B|A)$$.
To get the second stabilization formula from the first note that $$B \wedge_A (-)$$ is left Quillen from $$A$$-algebras to $$B$$-algberas and that homotopy colimits of augmented $$B$$-algebras are computed in $$B$$-algebras. This means that $$S^n \otimes_A B \simeq S^n \odot (B \wedge_A B)$$. Then use the fact that for an augmented $$B$$-algebra $$C$$ there is a natural equivalence between $$I(C)$$ and the cofiber of $$B \to C$$. The third stabilization formula is also equivalent to the first, as you mention.
• Thank you for your answer! I'm confused by your last formula, $S^n\otimes_A B \simeq S^n \odot (B\wedge_A B)$, I don't see how it follows from what you wrote. Nov 29 '18 at 14:49
• Tensoring with $S^n$ is a form of homotopy colimit. The functor $B \wedge_A (-)$ from $A$-algebras to $B$-algebras preserves homotopy colimits and hence preserves tensoring with $S^n$. The same is also true if we consider $B \wedge_A (-)$ as a functor from $A$-algebras over $B$ to $B$-algebras over $B$, since colimits (or homotopy colimits) in over categories are computed in the ambient category. Nov 29 '18 at 20:02