Let $ S to A to B $ to be commutative cofibrations $ S $-algebras. Then the topology André-Quillen $ B $-module $ TAQ (B | A) $ can be calculated as a *stabilization*. Precisely, I think that means the following: $ I $ to be the ideal functor of increased commutative augmented $ B $Algebra to $ B $-modules; it's true Quillen. These categories are getting richer and tending to basic spaces: let $ wedge $ denote the tensor of $ B $-modules and $ odot $ denote the augmentative commutative tensor $ B $-algebras.

There are canonical cards $$ X wedge IC to I (X odot C) $$ for any increased commutative $ B $-algebra $ C $ and space based $ X $. In particular, there are maps $ S ^ 1 coin I (S ^ n odot C) to I (S ^ {n + 1} odot C) $. taking $ n $-the loops (in $ B $-modules) of the deputy gives cards

$$ Omega ^ nI (S ^ n odot C) to Omega ^ {n + 1} I (S ^ {n + 1} odot C). $$

The assertion is that if we take $ C = B wedge_A B $ (a $ B $-algebra on the first variable and the increase is the multiplication card), then the homotopy colimit in $ B $-modules of this tower is weakly equivalent to $ TAQ (B | A) $. Am I saying that? In symbols,

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

It seems that the answer can be deduced from the results of Basterra-Mandell, *Homology and cohomology of $ E_ infty $ghosts*but it's not obvious to me. If so, how?

Another presentation of the "$ TAQ $ "Stabilization" that I saw (up to my own misinterpretations) is: the commutative category $ A $-algebras (no increase) is stretched over non-based spaces: call this tensor $ otimes_A $. We can take tensors $ S ^ n otimes_A B $, and consider the inclusion of the point $ * to S ^ n $ who gives cards $ B to S ^ n otimes_A B $ which we can take the cofiber $ B $-modules. Rate this cofiber by $ S ^ n overline { otimes} _AB $. These also form a direct system by transmitting to the fibers cards obtained in the same manner as above.

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

**Note**: There is yet another formulation that could be useful and 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 $ (Quillen right). The category of nucas is also stretched over pointed spaces; call the tensor $ tilde otimes $. Instead of considering $ I (S ^ n odot C) $ in the system headed above we could consider $ U (S ^ n tilde otimes I_0C) $. We get a directed system similar to the one 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 Quillen right *equivalence*, so it preserves the tensors (correctly interpreted, see Does the Quillen equivalence right guard preserve the homotopy colimits?).