Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2018 The 1-Type of Algebraic K-Theory as a Multifunctor Yaineli Valdes Follow this and additional works at the DigiNole: FSU's Digital Repository. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES THE 1-TYPE OF ALGEBRAIC K-THEORY AS A MULTIFUNCTOR By YAINELI VALDES A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2018 Copyright © 2018 Yaineli Valdes. All Rights Reserved. Yaineli Valdes defended this dissertation on April 10, 2018. The members of the supervisory committee were: Ettore Aldrovandi Professor Directing Dissertation John Rawling University Representative Amod Agashe Committee Member Paolo Aluffi Committee Member Kate Petersen Committee Member Mark van Hoeij Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii ACKNOWLEDGMENTS The completion of this research was only made possible through the assistance and support from so many throughout my educational career. I am very thankful to my committee members for their dedication as they not only served in my supervisory committee for my dissertation, but as well for my candidacy exam. I also had the immense pleasure of taking some of their courses as a graduate student and I owe much of my research achievements to their great teaching. I am especially thankful to my major professor Dr. Aldrovandi who has been my insight, professor, and guide throughout this research project, and has eloquently taught me so much of what I know today. This would definitely have not been possible if it were not for his support throughout these last several years. I am proud of the work Dr. Aldrovandi and I have completed together and am so grateful to him and everyone else that invested even a minute of their life to help me with this mission. I cannot leave out my family who have always been there for me unconditionally, especially my husband Alejandro who has supported and encouraged me to help me realize my dream. Finally, and most importantly, to my Lord and Savior, the One I owe everything to, for His unconditional love. I am beyond blessed and very hopeful for what the future holds in store! iii TABLE OF CONTENTS Abstract . vi 1 Introduction 1 2 Preliminaries 3 3 (Closed, Symmetric) Multicategories 6 3.1 Definitions . .6 3.2 Examples and Properties . .8 4 Waldhausen Categories 9 4.1 Definitions and Examples . .9 4.2 The closed structure on Wald ............................... 11 4.3 The S∗ Construction . 12 4.4 K-theory for a Waldhausen Category . 13 4.5 K-theory Simplices from Total Simplicial Set . 15 4.6 Classical K-theory of a Ring . 18 5 Picard Groupoids 19 5.1 Definitions and Examples . 19 5.2 The closed structure on Pic ................................ 23 6 Stable Quadratic Modules 25 6.1 Definitions and Properties . 25 6.2 An Algebraic Model for the 1-type of K(W) as a Stable Quadratic Module . 26 7 Crossed Modules 29 7.1 Definitions and Examples . 29 7.2 Properties . 30 8 Determinant Functor 32 8.1 Definitions and Examples . 32 8.2 Universal Determinant Functor for a Waldhausen Category . 34 9 P1K as a Multifunctor 36 9.1 Proof of Main Theorem . 40 iv Appendix A Computations for Proofs 44 A.1 ! defines a natural transformation on morphisms . 44 A.2 ! is a well-defined symmetric, monoidal functor . 50 Bibliography . 52 Biographical Sketch . 54 v ABSTRACT It is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic K-theory is a functor from the category of Waldhausen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1-type of its K-theory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1-types are classified by Picard groupoids. We prove that this functor is a multifunctor to a corresponding multicategory of Picard groupoids. vi CHAPTER 1 INTRODUCTION Given any Waldhausen category, W, we can construct its K-theory spectrum, K(W), and for any exact functor of Waldhausen categories W!V, we get a morphism of spectra K(W) ! K(V). It is well known that given an infinite loop space X its 1-type can be realized as a Picard groupoid, P1(X) [5]. Therefore, realizing the 1-type of the K-theory of a Waldhausen category as a Picard groupoid gives us a functor from the category of Waldhausen categories to the category of Picard groupoids: Wald −−−!P1K Pic W 7! P1(K(W)) We also know that although the category of Waldhausen categories does not have a symmetric monoidal structure, it does have a (closed) symmetric multicategorical structure. (Blumberg and Mandell gave a proof of this in [4], as well as Zakharevich in [2].) We call the multifunctors of Waldhausen categories multiexact functors. Similarly, we can show that Picard groupoids also form a (closed) symmetric multicategory when equipped with multimonoidal functors. Therefore, since P1K is a functor between categories underlying multicategories, a natural question to ask is if it extends to a multifunctor. Our main result answers this question in the affirmative, namely we prove the following theorem: Theorem 9.1. Wald −−−!P1K Pic extends to a multifunctor. To be clear, if a map between multicategories is a just a functor between their underlying categories, then we say it \extends to a multifunctor" if there is a multifunctor between the multi- categories such that when seen as a functor between the underlying categories, it agrees with the original one. This is defined in Definition 3.3. This theorem is based on two main ingredients: (1) the existence of a universal determinant functor detW : wW! P1K(W) 1 that satisfies certain properties, described in detail in section 8.1, where wW is the subcategory of weak equivalences of W; and (2) the closed multicategorical structures on Wald and Pic. The former is known thanks to [2], [4] and we give some attention to the latter. With these premises, our key ingredient is that the functor P1K \respects the closed structures" in the following sense: P1K(Wald(W; V)) ! Pic(P1K(W);P1K(V)) which allows us to prove the theorem. An application of this result is to use this multifunctor to study the algebraic structures on the crossed modules, which model Picard groupoids, coming from multiexact maps of Waldhausen categories. Aldrovandi shows there is an equivalence between bimonoidal functors and biexten- sions of crossed modules, which in fact extends to multimonoidal functors and multiextensions [3]. Therefore, an application of our results is to study the multiextensions corresponding to the multimonoidal functors of Picard groupoids under the multifunctor, PD·. ∗ ∗ ∗ This thesis is organized as follows: we start with reminding the reader of relevant algebraic structures such as Waldhausen categories, Picard groupoids, and stable quadratic modules. We cover universal determinant functors as they play a major role in proving the main result. We also recall the K-theory construction for a Waldhausen category by forming the S-construction. Finally, we end with stating and proving two propositions that allow us to prove the main theorem. Certain cumbersome but necessary computations are in the appendix. 2 CHAPTER 2 PRELIMINARIES Definition 2.1. Let [n] denote the finite, nonempty, totally ordered set f0 < 1 < ··· < ng considered as a category. This means the objects are 0; 1; : : : ; n and 8 ; j > k <> Hom[n](j; k) = idj j = k :>j ! k j < k Definition 2.2. For any two categories C and D, we can form another category Fun(C; D) called the functor category whose objects are functors F : C!D and whose morphism are natural transformations between these functors. Definition 2.3. The arrow category of some category C is Ar C := Fun([1]; C). This is truly the category of arrows in C because any object in Ar C is a functor from [1] !C and such a functor just chooses two objects in C and a morphism between them since there is only two objects in [1] and one non-identity morphism. One particular arrow category used in this thesis is Ar[n]. Definition 2.4. The ordinal category ∆ is the category whose objects are the ordered sets [n] and whose morphisms are order preserving maps. Specifically, f :[m] ! [n] is a morphism in ∆ if whenever i < j in [m], then f(i) ≤ f(j) in [n]. There are special maps in ∆ called the coface and codegeneracy maps. For n ≥ 0, the n + 1 coface maps are the injections di :[n − 1] ! [n], which send j 7! j for all j < i and j 7! j + 1 for j ≥ i. For n ≥ 0, the n + 1 codegeneracy maps are the surjections si :[n + 1] ! [n], which send j 7! j for all j ≤ i and j 7! j − 1 for j > i. These morphisms satisfy several relations and it can be verified that every morphism in ∆ is a composite of coface and codegeneracy maps. A nice reference on this is in [16]. 3 Definition 2.5. A simplicial set is a contravariant functor from the ordinal category to the category of sets. More generally, for any category C, a simplicial object in C, X, is a contravariant functor from the ordinal category to C X : ∆op !C So the data of a simplicial object in C comprises of an object in C for every natural number and face and degeneracy maps, denoted di and si respectively, which are the images of the coface and codegeneracy maps in ∆.