PARAMETRIZED HIGHER CATEGORY THEORY AND HIGHER ALGEBRA: A GENERAL INTRODUCTION CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH Let 푘 denote a field, and let 퐸 ⊇ 푘 be a finite Galois extension thereof with Galois group 퐺. The algebraic 퐾-groups 퐾푛(푘) and 퐾푛(퐸), as defined by Quillen, together exhibit some interesting structure. Since these groups are defined in terms of the categories of finite-dimensional vector spaces (along with their additive structure), the forgetful functor Vect(퐸) Vect(푘) and the functor Vect(푘) Vect(퐸) given by 푋 푋 ⊗푘 퐸 give rise to homomorphisms 푉∶ 퐾푛(퐸) 퐾푛(푘) and 퐹∶ 퐾푛(푘) 퐾푛(퐸). Ordinary Galois theory shows that the composite functor Vect(퐸) Vect(퐸) given by 푌 푌 ⊗푘 퐸 can be described as the direct sum ⨁ 푔∶ Vect(퐸) Vect(퐸), 푔∈퐺 where 퐺 acts in the obvious manner. Accordingly, we have an action of 퐺 on 퐾푛(퐸) for which both 푉 and 퐹 are equivariant, and a formula 퐹푉 = ∑ 푔. 푔∈퐺 Note that the equivariance of 푉 implies that it factors through the orbits 퐾푛(퐸)퐺, and the 퐺 equivariance of 퐹 implies that it factors through the fixed points 퐾푛(퐸) , but these maps do not typically identify 퐾푛(푘) with either the orbits or the fixed points. The data of 퐾푛(푘) is an added piece of structure; that is, 퐾푛(푘) cannot in general be recovered from 퐾푛(퐸) as a 퐺-module. But the problem is even deeper than this. Even if one considers all the 퐾-groups together as a single entity (by thinking of these groups as the homotopy groups of a space or spectrum), one can construct a descent spectral sequence 2 −푝 퐸푝,푞 = 퐻 (퐺, 퐾푞(퐸)), but this will not, as a rule, converge to the groups 퐾푝+푞(푘). In other words, the space or spectrum 퐾(푘) is not the homotopy fixed point space/spectrum of the action of 퐺 on the space/spectrum 퐾(퐸). Consequently, even knowing the homotopy type 퐾(퐸) with its action of 퐺 is insufficient to recover the groups 퐾푛(푘). This is the descent problem in algebraic 퐾-theory. There is, of course, no need to consider the 퐾-theories of 퐸 and 푘 in isolation. One can also include the information of the 퐾-groups of all the various subextensions 퐸 ⊇ 퐿 ⊇ 푘. In 퐻 other words, for any subgroup 퐻 ≤ 퐺, one can contemplate the 퐾-groups 퐾푛(퐸 ) of the fixed field 퐸퐻. These abelian groups each have conjugation homomorphisms 퐻 푔퐻푔−1 푐푔 ∶ 퐾푛(퐸 ) 퐾푛(퐸 ) 1 2 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH for any 푔 ∈ 퐺. Additionally, for subgroups 퐾, 퐿 ≤ 퐻 ≤ 퐺, one again has the forgetful functor 퐾 퐻 퐻 퐿 퐿 Vect(퐸 ) Vect(퐸 ) and the functor Vect(퐸 ) Vect(퐸 ) given by 푌 푌 ⊗퐸퐻 퐸 , so again one has homomorphisms 퐻 퐾 퐻 퐻 퐻 퐿 푉퐾 ∶ 퐾푛(퐸 ) 퐾푛(퐸 ) and 퐹퐿 ∶ 퐾푛(퐸 ) 퐾푛(퐸 ). Again, a small amount of Galois theory reveals that these two homomorphisms compose in the following manner: 퐻 퐻 퐿 퐾 퐾 퐿 퐹퐿 푉퐾 = ∑ 푉퐿∩(푥퐾푥−1)푐푥퐹(푥−1퐿푥)∩퐾 ∶ 퐾푛(퐸 ) 퐾푛(퐸 ). 푥∈퐿\퐻/퐾 퐻 And again, of course, the groups 퐾푛(퐸 ) cannot be recovered from the 퐺-module 퐾푛(퐸) or the homotopy type 퐾(퐸) with its action of 퐺. 퐻 Combined, this structure on the assignment 퐻 퐾푛(퐸 ) makes up what is called a Mackey functor for 퐺. As we see, this is strictly more structure than a 퐺-module. Similarly, the assignment 퐻 퐾(퐸퐻) is a spectral Mackey functor for 퐺 in the sense of the first author [2]. This is strictly more structure than a spectrum with a 퐺-action. We call this object the 퐺-equivariant 퐾-theory of 퐸 over 푘. In this monograph, we tease out the kind of structure on the categories Vect(퐸퐻) that provides their 퐾-theory with the structure of a spectral Mackey functor for 퐺. As a first approximation, we note that, because the category of subextensions of 퐸 is equivalent to the 퐿 category of transitive 퐺-sets, the functors 푌 푌 ⊗퐸퐻 퐸 together define what we call a 퐺-category – a diagram of categories indexed on the opposite of the orbit category O퐺 of 퐺. Let us write Vect퐸⊇푘 for this 퐺-category. Of course, the 퐺-category Vect퐸⊇푘 is relatively simple: after all, if one thinks of the action of 퐺 on Vect(퐸), then Vect(퐸퐻) is the category of 퐸-vector spaces equipped with a semilinear action of 퐻. In other words, Vect(퐸퐻) is simply the homotopy fixed point category for the action of 퐻 on Vect(퐸). So we might at first contemplate Vect(퐸) with its 퐺-action. However, the adjoints to the functors in this 퐺-category – the forgetful functors – contain extra information that compels us to contemplate entire 퐺-category structure. For example, the forgetful functor Vect(퐸) Vect(푘) is a kind of generalized product of vector spaces: we regard it as indexed, not over a mere set, but over the 퐺-set 퐺/푒. To see why this is appropriate, first note that by the normal basis theorem, if 푌 is an 퐸-vector space with basis {푣푖}1≤푖≤푛, then there is an element 휃 ∈ 퐸 such that 푌 has basis {푔휃푣푖}1≤푖≤푛,푔∈퐺 over 푘. But without choosing this element, we would still be entitled to write ∏ 푌 훼∈퐺/푒 for this 푘-vector space. In the same manner, the presence of all the other right adjoints Vect(퐸퐻) Vect(퐸퐾) in this diagram of categories can be regarded as the existence of various indexed products ∏ 푍 훼∈퐾/퐻 on this 퐺-category. At the same time, since our field extensions are separable, these right adjoints are all also left adjoints, and so we even think of this as endowing our 퐺-category with indexed direct sums ⨁ 푍. 훼∈퐾/퐻 The point here is that the transfer structure on the equivariant algebraic 퐾-groups arises from the additional structure of indexed products or coproducts on the 퐺-category Vect퐸⊇푘. And this example refects a general principle: to get the full structure of a Mackey functor A GENERAL INTRODUCTION 3 on equivariant algebraic 퐾-theory of 퐸 over 푘, one must work not only with the diagram of op categories indexed by O퐺 , but also the 퐺-direct sums thereupon. The 퐺-category Vect퐸⊇푘 also carries a sophisticated multiplicative structure. Of course, the tensor product over 푘 provides an external product 퐾 퐿 fg 퐾 퐿 (푥−1퐿푥)∩퐾 Vect(퐸 ) × Vect(퐸 ) Proj (퐸 ⊗푘 퐸 ) ≃ ∏ Vect(퐸 ). 푥∈퐿\퐺/퐾 In [9], we demonstrated that the external products provide the equivariant algebraic 퐾- groups with the structure of a graded Green functor, and, even better, they provide the equivariant algebraic 퐾-theory spectra with the structure of a spectral Green functor. However, there is a still richer multiplicative structure, whose impact on equivariant 퐾-theory is studied here for the first time. Just as the usual norm of an element of 퐸 is automatically Galois-invariant, we see that for any finite-dimensional 퐸-vector space 푉, the tensor power 푉⊗퐺 comes with canonical descent data. We call the resulting 푘-vector space 푘 푘 푁퐸(푉) the multiplicative norm from 퐸 to 푘. Quite simply, 푁퐸(푉) is the 푘-vector space (of #퐺 푘 dimension (dim 푉) ) such that the set Hom푘(푁퐸(푉), 푊) is in bijection with the set of ×퐺 norm forms 푉 푊 ⊗푘 퐸 for 퐸/푘 – i.e., 푘-multilinear maps ×퐺 훷∶ 푉 푊 ⊗푘 퐸 ×퐺 such that for any element (푣ℎ)ℎ∈퐺 ∈ 푉 , any element 푔 ∈ 퐺, and any element 휆 ∈ 퐸, ′ 훷((푣ℎ)ℎ∈퐺) = (푔휆)훷((푣ℎ)ℎ∈퐺), where ′ 휆푣푔 if ℎ = 푔; 푣ℎ = { 푣ℎ if ℎ ≠ 푔, and 푔훷((푣ℎ)ℎ∈퐺) = 훷((푣푔ℎ)ℎ∈퐺. 푘 ×퐺 So, 푁퐸(푉) is the dual of the 푘-vector space of norm forms 푉 퐸 for 퐸/푘. In particular, R when 푘 = R and 퐸 = C, then 푁C(푉) is precisely the dual space of the R-vector space of hermitian forms on 푉. More generally, there are multiplicative norms for any subgroups 퐾 ≤ 퐿 ≤ 퐺. Together with the external products, these multiplicative norms furnish Vect퐸⊇푘 with a 퐺-symmetric monoidal structure. In effect, this provides tensor products indexed over any finite 퐺-set 푈 = ∐푖∈퐼(퐺/퐻푖), which amount to functors ⨂∶ ∏ Vect(퐸퐻푖 ) Vect(푘), 푢∈푈 푖∈퐼 which are suitably associative and commutative. This additional structure on Vect퐸⊇푘 descends to an analogous structure on the equivari- ant algebraic 퐾-theory of 퐸 over 푘. These provide the equivariant algebraic 퐾-theory of 퐸 over 푘 with the full structure of a 퐺-퐸∞-algebra. Hill’s program To tell this story, we pursue here the general theory of 퐺-∞-categories. But we are by no means the first to contemplate this possibility. In their landmark solution of the Kervaire Invariant Problem [25], Mike Hill, Mike Hop- kins, and Doug Ravenel developed a perspective on equivariant stable homotopy thery that centered on the study of indexed products, indexed coproducts, and indexed symmetric 4 CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH monoidal structures (incorporating their multiplicative norms).