Goerss–Hopkins obstruction theory via model ∞-categories by Aaron Mazel-Gee A dissertation ...

540 Pages·2016·2.34 MB·English

Goerss–Hopkins obstruction theory via model ∞-categories
by
Aaron Mazel-Gee
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Mathematics
in the
Graduate Division
of the
University of California, Berkeley
Committee in charge:
Peter Teichner, Chair
Ian Agol
Denis Auroux
Alan H.Nelson
Spring 2016
last updated: April 8, 2019
Goerss–Hopkins obstruction theory via model ∞-categories
Copyright 2016
by
Aaron Mazel-Gee
1
Abstract
Goerss–Hopkins obstruction theory via model ∞-categories
by
Aaron Mazel-Gee
Doctor of Philosophy in Mathematics
University of California, Berkeley
Peter Teichner, Chair
We develop a theory of model ∞-categories – that is, of model structures on ∞-
categories – which provides a robust theory of resolutions entirely native to the
∞-categorical context. Using model ∞-categories, we generalize Goerss–Hopkins
obstruction theory from spectra to an arbitrary (presentably symmetric monoidal
stable) ∞-category. We give a sample application of this generalized obstruction
theory in the setting of motivic homotopy theory, where we construct E structures
∞
on the motivic Morava E-theories and compute their automorphism spaces (as E
∞
algebras).
i
This thesis is dedicated to my grandparents.
ii
Contents
0 Introduction 1
0.1 A brief history of derived categories, nonabelian derived categories,
and abstract homotopy theory . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Model ∞-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
0.3 Goerss–Hopkins obstruction theory . . . . . . . . . . . . . . . . . . . 36
0.4 Conventions on ∞-categories and model-independence . . . . . . . . 74
0.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
1 Model ∞-categories I: some pleasant properties of the ∞-category
of simplicial spaces 77
1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
1.1 Model ∞-categories: definitions . . . . . . . . . . . . . . . . . . . . . 84
1.2 Model ∞-categories: examples . . . . . . . . . . . . . . . . . . . . . . 89
1.3 Cofibrantly generated model ∞-categories . . . . . . . . . . . . . . . 101
1.4 The definition of the Kan–Quillen model structure . . . . . . . . . . . 105
1.5 Auxiliary results on spaces and simplicial spaces . . . . . . . . . . . . 108
1.6 Fibrancy, fibrations, and the Ex∞ functor . . . . . . . . . . . . . . . . 111
1.7 The proof of the Kan–Quillen model structure . . . . . . . . . . . . . 127
1.8 The proof of Lemma 1.5.4 . . . . . . . . . . . . . . . . . . . . . . . . 134
2 The universality of the Rezk nerve 144
2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
2.1 Relative ∞-categories and their localizations . . . . . . . . . . . . . . 148
2.2 Complete Segal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 158
2.3 The Rezk nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
2.4 The proof of Theorem 2.3.8 . . . . . . . . . . . . . . . . . . . . . . . 172
3 On the Grothendieck construction for ∞-categories 185
3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
iii
3.1 The Grothendieck construction . . . . . . . . . . . . . . . . . . . . . 188
3.2 Op/lax natural transformations and the global co/limit functor . . . 197
3.3 The Grothendieck construction and colimits of spaces . . . . . . . . . 207
3.4 Homotopy pullbacks in (Cat ) , finality, and Theorems A, B , and C 213
∞ Th n n
3.5 The Bousfield–Kan colimit formula . . . . . . . . . . . . . . . . . . . 227
3.6 The Thomason model structure on the ∞-category of ∞-categories . 240
4 Hammocks and fractions in relative ∞-categories 248
4.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
4.1 Segal spaces, Segal simplicial spaces, and sS-enriched ∞-categories . . 251
4.2 Zigzags and hammocks in relative ∞-categories . . . . . . . . . . . . 266
4.3 Homotopical three-arrow calculi in relative ∞-categories . . . . . . . 288
4.4 Hammock localizations of relative ∞-categories . . . . . . . . . . . . 300
4.5 From fractions to complete Segal spaces, redux . . . . . . . . . . . . . 312
5 Model ∞-categories II: Quillen adjunctions 317
5.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1 Quillen adjunctions, homotopy co/limits, and Reedy model structures 320
5.2 Relative co/cartesian fibrations and bicartesian fibrations . . . . . . . 332
5.3 The proofs of Theorem 5.1.1 and Corollary 5.1.3 . . . . . . . . . . . . 337
5.4 Two-variable Quillen adjunctions . . . . . . . . . . . . . . . . . . . . 346
5.5 Monoidal and symmetric monoidal model ∞-categories . . . . . . . . 357
5.6 Enriched model ∞-categories . . . . . . . . . . . . . . . . . . . . . . 360
6 Model ∞-categories III: the fundamental theorem 362
6.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
6.1 The fundamental theorem of model ∞-categories . . . . . . . . . . . 366
l (cid:13) (cid:13)
6.2 The equivalence hom∼M(x,y) (cid:39) (cid:13)homlMw(cyl•(x),path•(y))(cid:13) . . . . . . . 375
6.3 Reduction to the special case . . . . . . . . . . . . . . . . . . . . . . 378
6.4 Model diagrams and left homotopies . . . . . . . . . . . . . . . . . . 380
6.5 The equivalence (cid:13)(cid:13)homlMw(σcyl•(x),σpath•(y))(cid:13)(cid:13) (cid:39) 3˜(x,y)gpd . . . . . . 394
6.6 The equivalence 3˜(x,y)gpd (cid:39) 3(x,y)gpd . . . . . . . . . . . . . . . . . 404
6.7 The equivalence 3(x,y)gpd (cid:39) 7(x,y)gpd . . . . . . . . . . . . . . . . . 405
6.8 Localization of model ∞-categories . . . . . . . . . . . . . . . . . . . 408
6.9 The equivalence 7(x,y)gpd (cid:39) hom (x,y) . . . . . . . . . . . . . 409
M W−1
6.10 Localization of model ∞-categories(cid:74), red(cid:75)ux . . . . . . . . . . . . . . . 413
7 Goerss–Hopkins obstruction theory for ∞-categories 415
7.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
iv
7.1 The resolution model structure . . . . . . . . . . . . . . . . . . . . . 416
7.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
7.3 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
7.4 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
7.5 Homotopical algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
7.6 Homotopical topology . . . . . . . . . . . . . . . . . . . . . . . . . . 466
7.7 Decomposition of moduli spaces . . . . . . . . . . . . . . . . . . . . . 471
8 E automorphisms of motivic Morava E-theories 480
∞
8.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
8.1 E automorphisms of motivic Morava E-theories . . . . . . . . . . . 483
∞
A Notation, terminology, and conventions 488
A.1 On ∞-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
A.2 Conventions regarding ∞-categories . . . . . . . . . . . . . . . . . . . 495
A.3 On model categories as presentations of ∞-categories . . . . . . . . . 511
A.4 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
B Index of notation 521
Bibliography 526
1
Chapter 0
Introduction
The bulk of this introductory chapter is split into three sections.
In §0.1, we provide an expository overview of abstract homotopy theory. In
the interest of accessibility to a broad mathematical audience, we begin with the
motivation provided by abelian categories. In order to set the stage for the following
section, we place particular emphasis on model categories and ∞-categories.
Next, in§0.2, wedescribethetheoryofmodel ∞-categories whichisintroducedin
this thesis. This provides a robust theory of resolutions entirely native to the world
of ∞-categories. The context provided by §0.1 makes this a fairly straightforward
endeavor. We also describe a number of auxiliary results that we establish in ∞-
category theory that serve as input to the theory of model ∞-categories.
Then, in§0.3wedescribeourgeneralizationofGoerss–Hopkins obstruction theory
from spectra (in the sense of stable homotopy theory) to an arbitrary sufficiently nice
∞-category. This is a powerful tool for constructing “highly structured” objects (e.g.
E algebras) out of purely algebraic data.
∞
The original obstruction theory is based in a point-set model category of spec-
tra satisfying a host of technical assumptions, which makes its direct generalization
rather difficult. Thus, our generalization relieves the original construction of unnec-
essary point-set technicalities. However, as it turns out, relieving the construction of
point-set technicalities is not the same thing as relieving it of model structures: as
we will see, the obstruction theory relies crucially on the notion of a resolution, and
so our generalization necessitates the use of the full strength of the theory of model
∞-categories.
This section begins with an introduction to spectra (as the “nonabelian derived
∞-category of sets”) and to stable ∞-categories more generally. It then proceeds
to give an impressionistic survey of derived algebraic geometry and chromatic ho-
motopy theory, which provide context for some of the most important applications
2
of Goerss–Hopkins obstruction theory to date, most notably the construction of the
cohomology theory tmf of topological modular forms. After describing the obstruc-
tion theory itself (in two passes), it closes by describing our sample application. This
comes from motivic homotopy theory, which is a homotopical context for studying
algebraic varieties and their various cohomology theories. Our application concerns
the motivic Morava E-theories, which are certain “higher chromatic analogs” of (mo-
tivic) algebraic K-theory.
Finally, in §0.4 we say a few words regarding our conventions (which are spelled
out in full detail in §A), and in §0.5 we express our acknowledgments.
0.1 A brief history of derived categories,
nonabelian derived categories, and abstract
homotopy theory
In this expository section, we provide a broad overview of abstract homotopy the-
ory. In the interest of accessibility to a wide mathematical audience, we center our
discussion around the theme of (derived) functors between abelian categories. We
place particular emphasis on the theories of model categories and of ∞-categories,
since the intuition surrounding them will play a prominent role in the remainder of
this introduction (especially in §0.2).
0.1.1 Derived categories, derived functors, and resolutions
In studying abelian categories, one immediately encounters the inescapable fact that
not every functor
F : A → B
among them is exact: some are only left-exact (i.e. preserve kernels), some are only
right-exact (i.e. preserve cokernels), and some are neither left- nor right-exact. For
example, if we take A = B = Mod for a commutative ring R, then for an arbitrary
R
R-module M the functor
M ⊗ − : Mod → Mod
R R R
will always be right-exact but will not generally be left-exact.
In his groundbreaking “Toˆhoku paper” [Gro57], Grothendieck introduced an or-
ganizationalframeworkforunderstandingandquantifyingthesefailuresofexactness,
basedonthecategoryCh(A)ofchaincomplexesinA. Thiscategoryprovidesahome
3
for resolutions of objects of A: these are objects which are “weakly equivalent” to
our original objects of A, but which are better behaved with respect to our given
functor of interest (in a sense to be described shortly). One would now like to define
the derived functor of F to be the value of the induced functor
Ch(F) : Ch(A) → Ch(B)
on an appropriately chosen resolution.
However, such resolutions – and thence their values under the functor Ch(F) –
are generally only well-defined up to weak equivalence (a/k/a “quasi-isomorphism”).
There are two ways of remedying this situation.
• One may take homology of these values in Ch(B) to obtain well-defined objects
of B. For example, this technique leads to the definition of TorR(M,−) as the
∗
derived functor of M ⊗ −.
R
• Alternatively,writingW ⊂ Ch(B)forthesubcategoryofquasi-isomorphisms,
q.i.
one can consider the derived functor of F as taking values in the derived cate-
gory of B, i.e. the localization D(B) = Ch(B)[W−1].
q.i.
In fact, the first approach can always be recovered from the second: by the definition
of quasi-isomorphism, homology descends along the canonical localization functor
Ch(B) → D(B).
Of course, a derived functor should in particular be a functor, but it is not
immediately obvious that the process we have described defines one. In fact, our
desired functoriality will be a consequence of our definition of “resolution”. The
appropriate notion will vary from one application to another, but in any case the
crucial property will be that the restriction
Ch(A)res (cid:44)→ Ch(A) −C−h−(F→) Ch(B)
to the full subcategory of “resolutions” preserves weak equivalences. For example,
≈
givenanyR-moduleN,anyweakequivalenceP → Q betweenprojectiveresolutions
• •
of N induces a weak equivalence
≈
M ⊗ P → M ⊗ Q
R • R •
upon tensoring with M.1 Moreover, every object should admit a resolution: indeed,
in many cases (such as with model categories, as we will see in §0.1.2), the inclusion
1On the other hand, these objects are not generally weakly equivalent to M ⊗ N: this is the
R
entire point of resolving N in the first place.