# Theory of Homotopy – How Topological Homology André-Quillen (TAQ) Is it a "Stabilization", Exactly?

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 $$E_ infty$$ghostsbut 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?).