Proof: We define the multiplication factorwise as (C∧AB)∧A(C∧AB) ∼= C∧AC∧AB∧AB→C∧AB. Here the isomorphism in the middle is the one from lemma 2.7. It is clear that the maps fromCandB into the smash product
are maps ofS-algebras. 2
We will now specialize to structured ring and module spectra. More precisely let (MS,∧, S) now be the monoidal category ofS-modules as introduced in [36].
In particularS denotes the sphere spectrum. The categories ofS-modules and S-algebras are tensored and cotensored over the category of unbased topological spaces. Cotensors in the category ofS-algebras are created in the category of S-modules. For an unbased spaceX and anS-moduleK the cotensor is denoted byF(X+, K). Here are some more examples of uncas.
(c) IfR→T is a map of associative rings, then the induced map of Eilenberg-MacLane spectraHR →HT givesHT the structure of an HR-unca. The spectrumHT is an HR-algebra if and only if R→T is central.
(d) Let K be an associative S-algebra and X → Y a map of spaces. Then the function spectumF(X+, K) is anF(Y+, K)-unca but in general not an F(Y+, K)-algebra.
(e) LetRbe an even commutativeS-algebra, i.e. a commutativeS-algebra with homotopy groups concentrated in even degrees. LetI be a regular sequence in R∗. One can construct an associative S-algebra R/I that realizes the homotopy groups ofR∗/I, see chapter 7 for details of this construction. Let I =J1+J2be a decomposition ofI into regular sequencesJ1andJ2. Then R/I=R/J1∧RR/J2 and the canonical mapsR/Ji →R/I make R/I into anR/Ji-unca.
2.4 Model structures
Proposition 2.9. Let A be an associativeS-algebra and assume that the cat-egory of associativeS-algebras is a model category. Then so is the category of A-uncas where a map is a weak-equivalence, fibration or cofibration if it is so as a map in the model category AS of S-algebras. In particular this is the case when specifying to the category ofS-modules from [36].
Proof: This follows as the under-categoryA↓ AS inherits the model structure fromAS with fibrations, cofibrations and weak equivalences as in the
proposi-tion, see [35, 3.10 p.15] 2
We will again specialize to the category ofS-modules from [36] for the rest of this chapter. In this case the category of associativeS-algebras AS is a model category. Hence by the last proposition so is the category of A-uncas A↓ AS. Recall that for a commutativeS-algebraAalso the categoryCAof commutative A-algebras is a model category. Moreover the model structures on CA, AA and A↓ AS are such that weak equivalences and fibrations are created in the category of spectra or likewise in the category of S-modules. This property is
special to the categories introduced in [36] and is in general not satisfied, see [79, Rem. 4.5.].
Lemma 2.10 (Full-subcategory lemma). Let A be an associative S-algebra in the sense of [36]. Then the inclusion
AA⊂A↓ AS (2.3)
is an inclusion of a full subcategory. If A is a commutative S-algebra then
CA⊂ AA (2.4)
is an inclusion of a full subcategory.
Proof: Maps inCAandAA are maps ofS-algebras underA, compare also [36,
1.3.]. 2
The following is a formal consequence.
Lemma 2.11. Let A be a commutativeS-algebra. If α is a map inAA that is a cofibration as a map in A↓ AS, then α is a cofibration in AA.
If αis a map inCAthat is a cofibration as a map inAA, thenαis a cofibration in CA.
Proof: For the first case, note that every acyclic fibration in AA is also an acyclic fibration inA↓ AS. Henceαhas the left lifting property (LLP) with re-spect to every acyclic fibration inAAproducing a lift inA↓ AS. By lemma 2.10 this lift is automatically inAA. The proof for the second statement is similar. 2
Chapter 3
Model structures for uncas
The definition and examination of Galois extensions in stable homotopy theory was made possible by the construction of symmetric monoidal model categories of spectra. For several decades, such categories had been thought impossible to exist. When the goal was finally achieved in the mid-nineties, it was a real breakthrough. One of our main references, [36], gives such a construction and also investigates categories of module- and algebra-spectra. The categories of S-algebras, S-modules, R-algebras and R-modules constructed in [36] share a lot of good properties whence we decided to work with them as far as possible.
However, our needs go beyond this volume as we will now explain. In short, we can say that [36] is written “under acommutative S-algebra”. Many results are only stated for the category ofR-algebras, whereRis acommutativeS-algebra.
The case of associative S-algebras under an associative S-algebra A, i.e. the category of objects we calledA-uncas before, is not dealt with. However, as the previous chapter suggests and as will become evident in chapter 4, we need to consider this more general situation. In this chapter we generalize most of the statements of [36, chapter VII] to the category ofS-algebras underA, whereA is a not necessarily commutativeS-algebra.
The aim of this chapter is to show that the smash product of cofibrant algebras is homotopically meaningful. This is made precise and proved at the end of this chapter in theorem 3.16. In order to achieve this, we have to gain better con-trol over the cofibrations in the various categories under consideration. For this purpose, we establish the model structure on the category ofA-uncas along the lines of [36], even though we already obtained the model structure almost for free in proposition 2.9. Philosophically the quintessence is that the categories of uncas share most of the good properties of the various categories introduced in [36]. This is not surprising, but we have to go through the necessary techni-calities.
Again a comment on terminology seems appropriate: In this thesis, the term
“cofibration” will always be used in the model category theoretic sense as is the usual usage in most publications. In [36] these maps were called “q-cofibrations”. Moreover, “cofibration” in [36] denotes a map which has the homotopy extension property (HEP), see definition 3.6. We will instead always
19
keep the term HEP whenever we are talking about maps with the homotopy extension property.
Let MS = (MS,∧, S) be the categoriy of S-modules constructed in [36], so in particular S from now on denotes the sphere spectrum. We will also work with the categories of S-algebras AS and more generally with R-modules and R-algebras without further comments.
For the whole chapter, let A be a cofibrant associativeS-algebra.
3.1 Basic properties of the category A ↓ A
SAs mentioned at the beginning of the chapter, we want to establish model structures and hence have to check several properties of the category of S-algebras underA in order to make the machinery of [36] work. We will do this in this and the next section. We collect formal properties and the first is the following.
The category A↓ AS is enriched over unbased spaces. (3.1) This is rather trivial as the category of S-algebrasAS is topologically enriched and the morphism sets in A↓ AS are subsets of the morphism sets inAS. For the other properties to be verified another description ofA↓ AS is useful.
3.1.1 An operadic description of uncas
A very helpful description of the category A↓ AS is via monads. In fact all the model structures in [36] are obtained as follows: One starts with a model category C and wants to lift this structure to a category of algebras C[T] over a monad T:C → C. Assumptions under which this is possible are given in [36]
and [79].
Recall from chapter 2 that an object in A↓ AS is just a monoid in MAe
(proposition 2.2). In other words it is an algebra over the monad given by the free algebra functor TA:MAe → MAe. On objects this functor is given by TA(M) := W
j≥0M∧Aj. Another convenient notation for the category of A-uncas hence is MAe[TA]. Remember that MAe itself has an analogous de-scription asMAe =MS[FAe] with the monadFAe:MS → MAe sendingM to A∧M ∧A. By [36, II.6.1] there is an equality MS[FAe][TA] = MS[TA◦FAe] and this is our category of A-uncas as can be seen by the adjunction
A↓ AS(TAM, B)∼=MAe(M, B).
It is clear that a bimodule map M → B into an algebraB ∈A↓ AS defines a map TAM → B of uncas. Vice versa, given a map of uncas TAM → B this provides a bimodule map M → B by lemma 2.5. These mappings are inverse to each other. Hence TAis left adjoint to the forgetful functor
TA:A↓ AS oo //MAe :U .