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,



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



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



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> $\operatorname{hocolim} (\Omega^nI(S^n \odot C )) \simeq \operatorname{hocolim}(\Omega^nU(S^n\tilde\otimes I\_0 C))$.
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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?](https://mathoverflow.net/questions/82813/does-the-right-adjoint-of-a-quillen-equivalence-preserve-homotopy-colimits)).



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


