DOCSLIB.ORG

Kan Complexes As a Univalent Model of Type Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

ALGANT Master's Thesis Kan Complexes as a Univalent Model of Type Theory Gabriele Lobbia Supervisor: Prof. Denis-Charles Cisinski Academic Year 2017-2018 Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. Albert Einstein A mathematician is a device for turning coffee into theorems. Therefore a comathematician is a device for turning cotheorems into ffee. Acknowledgements First of all I would like to express my full gratitude to my supervisor Prof. Denis-Charles Cisisnki, for accepting me as his students, for beeing always available for everything I needed, for helping me in the search for my future academic occupation and for teaching me so many aspects of mathematics. Every meeting has been enlightening math-wise and enjoyable on a personal level. I also have to thanks all the ALGaNT Consortium for the opportunity they gave me. In these two years I was able to meet fantastic people that helped me growing as a mathematician and as a person, and the credit is firstly theirs. The involment in this program has been crucial in the search for a PhD, and I am really grateful for this too. Speaking of people I have met, a special thanks goes to every friend that helped making this experience unique and amazing. Thanks to all my "mathematics" friends that shared with me this incredible passion of mine and adviced me whenever I was struggling. Thanks to all my friends from Milan. They have always been there for me and I am sure they will always be, even though I might be away from home for sometime. Thanks to my "Erasmus" friends, that beared me during all the writing of this thesis and helped me live the experience abroad in all his different parts. Last but not least, grazie alla mia famiglia. Ai miei genitori, ai miei fratelli, ai miei zii, ai miei nonni, a mia cugina e alla mia zia Lella. Senza nes- suno di loro sarei diventato la persona che sono, anche se in effetti potrebbe essere una colpa e non un merito. Ma sopratutto, senza il loro incondizion- ato supporto e aiuto non sarei mai riuscito a raggiungere nessun obbiettivo, quale Laurea o altro. ii Contents Introduction 1 1 Kan Complexes 2 1.1 Preliminary Definitions . 2 1.2 Nerve of a Category . 8 1.3 Fibrations and Complexes . 12 1.4 Homotopy Theory . 18 1.5 Minimal Complexes . 23 2 Type Theory 28 2.1 λ-Calculus . 28 2.2 Simple Type Theory . 33 2.3 Dependent Type Theory . 39 2.4 Homotopy Type Theory . 44 3 A Univalent Model 55 3.1 Categorical Setting . 55 3.2 The Simplicial Model . 58 3.3 The Univalence Axiom . 68 3.4 Simplicial Univalence . 73 A Yoneda Lemma 78 A.1 The Statement . 78 A.2 Important Consequences . 79 B Model Categories 84 B.1 Main Definitions . 84 B.2 Some Properties . 86 iii Introduction The aim of this Thesis is to illustrate an example of a model of Martin-L¨of Dependent Type Theory where Voedvosky's Univalence Axiom holds. The main results are explained in [KL16]. The first chapter will handle the construction of a model category in the enviroment of simplicial sets (sSet). In order to do that we will investigate the definitions of Kan complexes and Kan fibrations. Secondly we will give a rough overview of type theory in general. In particular, we will start giving an idea of what Simple and Dependent Type Theory mean. Then, finally, we will se a natural model of Homotopy Type Theory in category theory, thanks to [Awo16]. The last chapter will deal with the Univalence Axiom itself and the theo- rems that explain why Kan complexes can be seen as a model of type theory where it holds. 1 Kan Complexes 1.1 Preliminary Definitions We denote ∆ the category with objects the sets [n] = f0; :::; ng for any n 2 N, and morphism the non decreasing functions between them. Definition 1.1.1. We call simplicial set (or complex) a controvariant op functor of the kind X : ∆ ! Set. We write Xn for the set X([n]) and we call its elements the n-simplices of X. Usually the image of a function f :[q] ! [p] through the complex X is denoted by f ∗. From now on we will write sSet for the category whose objects are the simplicial sets and morphisms the natural trasformations between them. A subcomplex will be just a suboject in this category (i.e. a subfunctor). Definition 1.1.2. We write ∆n for the complex defined by the contravariant functor represented by [n], more precisely [q] / ∆([q]; [n]) O f −◦f [p] / ∆([p]; [n]) We call it the standard n-simplex. Definition 1.1.3. Let X be a simplicial set, and x 2 Xm an m-simplex of X. We say that x is degenerate if there exists an epimorphism s :[m] [n] 1 with n < m and a n-simplex y 2 Xn such that x = X(s)(y). Proposition 1.1.4. Let X be a simplicial set. Any morphism a : ∆n ! X has a unique factorization a ∆n / X = b " " ∆m where the first map is an epimorphism and b is not degenerate, i.e. the 2 1 correspondent element in Xm (by the Yoneda Lemma ) is not degenerate. 1[GZ67] Chapter II, x3.1. 2Lemma A.1.1 2 Definition 1.1.5. Let X be a complex. We define SknX the n-skeleton of X to be the subcomplex of X defined as follows 3 n (Sk X)m := fx 2 Xm j x is degenerated from a q-simplex with q ≤ ng Remark 1.1.6. The epimorphisms p :[m] [n] in ∆ are simply the non decreasing surjections. Moreover any epimorphism has a section, i.e. there is a morphism s :[n] ! [m] such that p ◦ s = Id[n]. Similarly the monomorphisms s :[n] [m] are the non decresing injections 4 and they have a retraction p :[m] ! [n] in ∆, i.e. p ◦ s = Id[n]. Definition 1.1.7. We denote with ∆_ n ≡ @∆n := Skn−1∆n the boundary of the standard n-simplex. We can see that any complex X is the union of its skeletons. Moreover there is a filtration ; =: SK−1X ⊆ SK0X ⊆ ::: ⊆ SKnX ⊆ ::: ⊆ X But simplicial sets have even a more important property. In fact we can re- cover SKnX from SKn−1X thanks to a pushout diagram, exactly analogously to the construction of CW-complexes in Topology. We have the more general result: Theorem 1.1.8. Let X,! Y be an inclusion of simplicial sets. Let Σn be the set of non degenerate elements of Yn which do not belong in Xn. Let us denote Y i := SkiY [ X. Then the following diagram is a pushout: ` n n−1 s2Σ @∆ / Y n _ _ ` ∆n / Y n s2Σn where the bottom and top morphism are defined componentwise as the mor- phism ∆n ! Y correspondet to s 2 Y . Moreover we have that Y ∼ lim Y i.5 n = −! Example 1.1.9. Let us define the simplicial set S1 as the pushout of the following diagram @∆1 / ∆1 π ∆0 / S1 3[GZ67] Chapter II, x3.5. 4[GZ67] Chapter II, x2.3. 5Generalization of [GZ67] Chapter II, Proposition 3.8. 3 1 We can see that π is both not injective and not degenerate. In fact #S0 = 1 1 1 0 1 and π corresponds to a different element of S1 than the map ∆ ! ∆ ! S . Remark 1.1.10. (/Alternative definition) We can see that actually [ @∆n = Im(∆q ! ∆n) q<n In fact, we can prove the more general statement: [ SknX = Im(∆q ! X) q≤n That is true because by Yoneda any map ∆q ! X correspond to a unique element of Xq. The condition regarding s to be an epimorphism is satisfied because we take the images of these maps. We recall now an important property of morphisms in ∆, which will be useful to make the definition above easier. First of all we have to give the n definition of the i-th coface map δi :[n − 1] ! [n] ( x if x < i δn(x) = i x + 1 if not Another important class of morphisms is formed by the so called i-th code- n generacy maps σi :[n + 1] ! [n] ( x if x ≤ i σn(x) = i x − 1 if not i n ∗ From now on, for any simplicial set X, we will denote with dn := (δi ) : Xn ! i n ∗ Xn−1 and sn := (σi ) : Xn ! Xn+1 the maps induced on the simplices. By straight forward calculation we can verify that these two classes of maps verify the following properties, which are called simplicial relations6: n+1 n n+1 n 1. δj δi = δi δj−1 for any i < j n n+1 n n+1 2. σj σi = σi σj+1 for any i ≤ j 8 n−1 n−2 >δi σj−1 if i < j n−1 n < 3. σj δi = Id[n−1] if i = j or i = j + 1 > n−1 n−2 :δi−1 σj if i > j + 1 6[GZ67] Chapter II, x2.1. 4 Dually we have that: i j j−1 i 1. dndn+1 = dn dn+1 for any i < j i j j+1 i 2. sn+1sn = sn+1sn for any i ≤ j 8 j−1 i >sn−2dn−1 if i < j i j < 3. dnsn−1 = IdX[n−1] if i = j or i = j + 1 > j i−1 :sn−2dn−1 if i > j + 1 Lemma 1.1.11.
Recommended publications
  • Lecture Notes on Simplicial Homotopy Theory

    Lecture Notes on Simplicial Homotopy Theory

    Lectures on Homotopy Theory The links below are to pdf files, which comprise my lecture notes for a first course on Homotopy Theory. I last gave this course at the University of Western Ontario during the Winter term of 2018. The course material is widely applicable, in fields including Topology, Geometry, Number Theory, Mathematical Pysics, and some forms of data analysis. This collection of files is the basic source material for the course, and this page is an outline of the course contents. In practice, some of this is elective - I usually don't get much beyond proving the Hurewicz Theorem in classroom lectures. Also, despite the titles, each of the files covers much more material than one can usually present in a single lecture. More detail on topics covered here can be found in the Goerss-Jardine book Simplicial Homotopy Theory, which appears in the References. It would be quite helpful for a student to have a background in basic Algebraic Topology and/or Homological Algebra prior to working through this course. J.F. Jardine Office: Middlesex College 118 Phone: 519-661-2111 x86512 E-mail: [email protected] Homotopy theories Lecture 01: Homological algebra Section 1: Chain complexes Section 2: Ordinary chain complexes Section 3: Closed model categories Lecture 02: Spaces Section 4: Spaces and homotopy groups Section 5: Serre fibrations and a model structure for spaces Lecture 03: Homotopical algebra Section 6: Example: Chain homotopy Section 7: Homotopical algebra Section 8: The homotopy category Lecture 04: Simplicial sets Section 9:
  • Homotopical Realizations of Infinity Groupoids

    Homotopical Realizations of Infinity Groupoids

    GRAU DE MATEMÀTIQUES Treball final de grau HOMOTOPICAL REALIZATIONS OF INFINITY GROUPOIDS Autor: Jan McGarry Furriol Director: Dr. Carles Casacuberta Vergés Realitzat a: Departament de Matemàtiques i Informàtica Barcelona, 21 de juny de 2020 Abstract Grothendieck’s homotopy hypothesis asserts that the study of homotopy types of topological spaces is equivalent to the study of ¥-groupoids, illustrating how important ideas in higher category theory stem from basic homotopical concepts. In practice there are distinct models for ¥-groupoids, and providing a proof of the homotopy hypothesis is a test for the suitability of any such model. In this thesis, we give a proof of the homotopy hypothesis using topological categories (i.e., categories enriched over topological spaces) as models for ¥-groupoids. In the same context, we propose a manageable model for the fundamental ¥-groupoid of a topological space. 2010 Mathematics Subject Classification: 55U40, 55P15, 18B40 Contents Introduction ii 1 Foundations 1 1.1 Adjoint Functors . .1 1.2 Equivalence of Categories . .5 1.3 Simplicial Sets . .6 1.4 Model Categories . 10 2 The Homotopy Hypothesis 17 2.1 Topological Spaces and Simplicial Sets . 17 2.2 Topological Categories . 22 2.3 Simplicial Categories . 25 2.3.1 From Simplicial Sets to Simplicial Categories . 27 2.3.2 The Homotopy Coherent Nerve . 31 2.4 Comparing Model Structures on Simplicial Sets . 32 2.5 Simplicial Sets and Topological Categories . 34 2.6 Kan Complexes and ¥-Groupoids, back and forth . 34 2.7 Proof of the Homotopy Hypothesis . 36 3 The Fundamental ¥-Groupoid as a Topological Category 39 3.1 Moore Paths .
  • The Theory of Quasi-Categories and Its Applications

    The Theory of Quasi-Categories and Its Applications

    Volume II The Theory of Quasi-Categories and its Applications Andr´eJoyal Contents Introduction 153 Perspective 157 I Twelve lectures 207 1 Elementary aspects 209 2 Three classes of fibrations 221 3 Join and slices 241 4 Quasi-categories and Kan complexes 259 5 Pseudo-fibrations and function spaces 273 6 The model structure for quasi-categories 293 7 The model structure for cylinders 309 8 The contravariant model structure 323 9 Minimal fibrations 339 10 Base changes 353 11 Proper and smooth maps 363 12 Higher quasi-categories 369 151 152 Contents II Appendices 377 A Accessible categories 379 B Simplicial sets 381 C Factorisation systems 393 D Weak factorisation systems 403 E Model categories 427 F Homotopy factorisation systems 447 G Reedy theory 463 H Open boxes and prisms 467 Bibliography 487 Indices 489 Introduction The notion of quasi-category was introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures [BV]. A Kan complex and the nerve of a category are examples. The goal of our work is to extend category theory to quasi-categories and to develop applications to homotopy theory, higher category theory and (higher) topos theory. Quasi-category are examples of (∞, 1)-categories in the sense of Baez and Dolan. Other examples are simplicial categories, Segal categories and complete Segal spaces (here called Rezk categories). To each example is associated a model category and the model categories are connected by a network of Quillen equiva- lences. Simplicial categories were introduced by Dwyer and Kan in their work on simplicial localisation. Segal categories were introduced by Hirschowitz and Simp- son in their work on higher stacks in algebraic geometry.
  • Arxiv:1808.09834V3 [Math.CT] 3 Oct 2019 83,55U10

    Arxiv:1808.09834V3 [Math.CT] 3 Oct 2019 83,55U10

    RECOGNIZING QUASI-CATEGORICAL LIMITS AND COLIMITS IN HOMOTOPY COHERENT NERVES EMILY RIEHL AND DOMINIC VERITY Abstract. In this paper we prove that various quasi-categories whose objects are ∞- categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit cones, whose shape is governed by the weight for pseudo limits over a homotopy coherent diagram but with the defining universal property up to equivalence, rather than isomorphism, of mapping spaces. Our applications follow from the fact that the (∞, 1)-categorical core of an ∞-cosmos admits weighted homotopy limits for all flexible weights, which includes in particular the weight for pseudo cones. Contents 1. Introduction 2 1.1. Size conventions 5 1.2. Acknowledgements 6 2. ∞-cosmoi 6 2.1. ∞-cosmoi and their homotopy 2-categories 6 2.2. Groupoidal objects and the (∞, 1)-core of an ∞-cosmos 13 3. Limits and colimits in an ∞-category 15 3.1. Arrow and comma constructions 16 3.2. Limits and colimits in an ∞-category 17 4. Flexible homotopy limits in an ∞-cosmos and its (∞, 1)-categoricalcore 19 arXiv:1808.09834v3 [math.CT] 3 Oct 2019 4.1.
  • A Survey of Simplicial Sets

    A Survey of Simplicial Sets

    School of Mathematics and Statistics MSc. Project A Survey of Simplicial Sets Author: Supervisor: Gyan Singh Gregory Stevenson 2328552 Declaration of Originality I confirm that this assignment is my own work and that I have: • Read and understood the guidance on plagiarism in the Undergraduate Handbook, including the Uni- versity of Glasgow Statement on Plagiarism • Clearly referenced, in both the text and the bibliography or references, all sources used in the work • Fully referenced (including page numbers) and used inverted commas for all text quoted from books, journals, web etc. • Provided the sources for all tables, figures, data etc. that are not my own work. • Not made use of the work of any other student(s) past or present without acknowledgement. This includes any of my own work, that has been previously, or concurrently, submitted for assessment, either at this or any other educational institution, including school. • Not sought or used the services of any professional agencies to produce this work • In addition, I understand that any false claim in respect of this work will result in action under the University regulations for Student Conduct • I am aware of and understand the University's policy on plagiarism and I certify that this assignment is my own work, except where indicated by referencing, and that I have followed the good academic practices noted above. I also agree that this project can be used by the School of Mathematics and Statistics at University of Glasgow for teaching, recruitment and other aspects of its work. Gyan Pratap Singh September 7, 2018 Abstract A simplicial set is a presheaf over the simplex category ∆, of finite, non-empty, ordered sets, with weakly order preserving functions between them.
  • Notes on Quasi-Categories

    Notes on Quasi-Categories

    NOTES ON QUASI-CATEGORIES ANDRE´ JOYAL To the memory of Jon Beck Contents Introduction 2 1. Elementary aspects 8 2. The model structure for quategories 12 3. Equivalence with simplicial categories 15 4. Equivalence with Rezk categories 16 5. Equivalence with Segal categories 18 6. Minimal quategories 19 7. Discrete fibrations and covering maps 21 8. Left and right fibrations 23 9. Join and slice 26 10. Initial and terminal objects 31 11. Homotopy factorisation systems 34 12. The covariant and contravariant model structures 39 13. Base changes 42 14. Cylinders, correspondances, distributors and spans 46 15. Yoneda lemmas 61 16. Morita equivalences 65 17. Adjoint maps 68 18. Quasi-localisations 70 19. Limits and colimits 72 20. Grothendieck fibrations 80 21. Proper and smooth maps 84 22. Kan extensions 85 23. The quategory K 92 24. Factorisation systems in quategories 96 25. n-objects 101 26. Truncated quategories 102 27. Accessible quategories and directed colimits 104 28. Limit sketches and arenas 109 29. Duality for prestacks and null-pointed prestacks 118 30. Cartesian theories 120 31. Sifted colimits 131 32. Algebraic theories and theaters 135 33. Fiber sequences 150 34. Additive quategories 153 Date: June 22 2008. 1 2 ANDRE´ JOYAL 35. Dold-Kan correspondance and finite differences calculus 162 36. Stabilisation 164 37. Perfect quategories and descent 168 38. Stable quategories 173 39. Para-varieties 178 40. Homotopoi (∞-topoi) 180 41. Meta-stable quasi-categories 183 42. Higher categories 184 43. Higher monoidal categories 186 44. Disks and duality 188 45. Higher quasi-categories 197 46. Appendix on category theory 200 47.
  • Notes on Simplicial Homotopy Theory

    Notes on Simplicial Homotopy Theory

    Notes on simplicial homotopy theory Andr´eJoyal and Myles Tierney The notes contained in this booklet were printed directly from files supplied by the authors before the course. Contents Introduction 1 1 Simplicial sets 3 1.1 Definitions and examples . 3 1.2 The skeleton of a simplicial set . 5 1.3 Geometric realization and the fundamental category . 9 1.4 The nerve of a partially ordered set . 14 2 Quillen homotopy structures 17 2.1 Homotopy structures . 17 2.2 The Quillen structure on groupoids . 20 2.3 Cofibration and fibration structures . 24 2.4 The fundamental groupoid and the Van Kampen theorem . 30 2.5 Coverings . 32 3 The homotopy theory of simplicial sets 37 3.1 Anodyne extensions and fibrations . 37 3.2 Homotopy . 44 3.3 Minimal complexes . 55 3.4 The Quillen homotopy structure . 59 4 Homotopy groups and Milnor's Theorem 65 4.1 Pointed simplicial sets . 65 4.2 Homotopy groups . 69 4.3 A particular weak equivalence . 72 4.4 The long exact sequence and Whitehead's Theorem . 75 4.5 Milnor's Theorem . 77 4.6 Some remarks on weak equivalences . 79 4.7 The Quillen model structure on T opc . 81 iii iv Contents A Quillen Model Structures 85 A.1 Preliminaries . 85 A.2 Weak factorisation systems . 87 A.3 Quillen model structures . 90 A.4 Examples . 92 A.5 The homotopy category . 95 A.6 Quillen functors and equivalences . 102 A.7 Monodial model categories . 108 References 115 Introduction These notes were used by the second author in a course on simplicial homotopy theory given at the CRM in February 2008 in preparation for the advanced courses on simplicial methods in higher categories that followed.
  • Spaces As Infinity-Groupoids

    Spaces As Infinity-Groupoids

    Spaces as infinity-groupoids Timothy Porter, Emeritus Professor, Bangor, Wales, U.K. [email protected] Contents 1 Introduction 2 2 The beginnings: recollections of Poincar´e'sfundamental groupoid. 3 2.1 Homotopy classes of paths . .4 2.2 Covering spaces . .6 2.3 Complexes as `presentations' of spaces and of groupoids . .7 3 Whitehead's algebraic and combinatorial homotopy 14 4 Higher homotopy groups, weak homotopy types, truncation and connectedness 15 4.1 Higher homotopy groups, and weak homotopy types . 15 4.2 n-connectedness . 17 4.3 n-truncation . 19 4.3.1 1-types and groupoids: . 20 4.3.2 2-types, crossed modules and 2-group(oid)s: . 22 4.3.3 2-groupoids, crossed modules and the Mac Lane-Whitehead theorem. 24 4.3.4 n-types for n ≥ 3....................... 25 4.4 Beyond 2-types towards infinity groupoids, from a classical/strict viewpoint . 26 4.5 From `2- ' heading to ‘infinity', from Grothendieck's viewpoint. 29 4.6 Appendix: more technical comments. 31 5 Simplicial sets, higher combinatorics and 1-groupoids 34 5.1 Kan complexes and 1-groupoids . 34 5.2 Simplicially enriched categories and groupoids . 36 5.3 Chain complexes, globular and simplicial models . 40 5.4 Appendix: Some Moore complex properties . 41 5.5 Higher dimensional combinatorial homotopy . 43 5.6 1-categories? . 44 6 Higher Galois theory and locally constant stacks. 45 6.1 `Stacks' . 45 6.2 ... and their pursuit . 47 1 7 Concluding discussion 49 1 Introduction As the title of this chapter suggests, there are aspects of `spaces' mirrored by things called `1-groupoids'.
  • FIBRATIONS in ∞-CATEGORY THEORY Contents 1. Left, Right, and Kan Fibrations 2 2. Inner Fibrations and Isofibrations 5 3. Coca

    FIBRATIONS in ∞-CATEGORY THEORY Contents 1. Left, Right, and Kan Fibrations 2 2. Inner Fibrations and Isofibrations 5 3. Coca

    FIBRATIONS IN ∞-CATEGORY THEORY CLARK BARWICK AND JAY SHAH Abstract. In this short expository note, we discuss, with plenty of examples, the bestiary of fibrations in quasicategory theory. We underscore the simplicity and clarity of the con- structions these fibrations make available to end-users of higher category theory. Contents 1. Left, right, and Kan fibrations 2 2. Inner fibrations and isofibrations 5 3. Cocartesian and cartesian fibrations 7 4. Flat inner fibrations 10 5. Categorical patterns 12 6. Constructing fibrations via additional functoriality 13 References 18 The theory of ∞-categories – as formalized in the model of quasicategories – offers two ways of specifying homotopy theories and functors between them. First, we may describe a homotopy theory via a homotopy-coherent universal property; this is a widely appreciated advantage, and it’s a feature that any sufficiently well-developed model of ∞-categories would have. For example, the ∞-category Top of spaces is the free ∞-category generated under (homotopy) colimits by a single object [7, Th. 5.1.5.6]. The second way of specifying ∞-categories seems to be less well-loved: this is the ability to perform completely explicit constructions with excellent formal properties. This allows one to avoid the intricate workarounds that many of us beleaguered homotopy theorists have been forced to deploy in order solve infinite hierarchies of homotopy coherence problems. This feature seems to be peculiar to the model of quasicategories, and the main instrument that makes these explicit constructions possible is the theory of fibrations of various sorts. In this étude, we study eight sorts of fibrations of quasicategories in use today – left, right, Kan, inner, iso (aka categorical), cocartesian, cartesian, and flat – and we discuss the beautifully explicit constructions they provide.
  • Arxiv:1007.2925V2 [Math.AT] 21 Jan 2015 .Introduction 0

    Arxiv:1007.2925V2 [Math.AT] 21 Jan 2015 .Introduction 0

    A SHORT COURSE ON ∞-CATEGORIES MORITZ GROTH Abstract. In this short survey we give a non-technical introduction to some main ideas of the theory of ∞-categories, hopefully facilitating the digestion of the foundational work of Joyal and Lurie. Besides the basic ∞-categorical notions leading to presentable ∞-categories, we mention the Joyal and Bergner model structures organizing two approaches to a theory of (∞, 1)-categories. We also discuss monoidal ∞-categories and algebra objects, as well as stable ∞-categories. These notions come together in Lurie’s treatment of the smash product on spectra, yielding a convenient framework for the study of A∞-ring spectra, E∞-ring spectra, and Derived Algebraic Geometry. Contents 0. Introduction 2 1. Two models for (∞, 1)-categories 3 1.1. Basics on ∞-categories 3 1.2. Simplicial categories and the relation to ∞-categories 13 2. Categorical constructions with ∞-categories 22 2.1. Functors 23 2.2. Join construction 27 2.3. Slice construction 29 2.4. Final and initial objects 31 2.5. Limits and colimits 32 3. Presentable ∞-categories and the relation to model categories 35 3.1. Locally presentable categories 35 3.2. Combinatorial model categories 39 3.3. Presentable ∞-categories 41 arXiv:1007.2925v2 [math.AT] 21 Jan 2015 4. Monoidal and symmetric monoidal ∞-categories 44 4.1. Monoidal categories via Grothendieck opfibrations 45 4.2. Monoidal ∞-categories via coCartesian fibrations 49 4.3. Algebra objects and monoidal functors 52 4.4. Symmetric monoidal ∞-categories 55 5. Stable ∞-categories and the universal property of spectra 59 5.1. Stable ∞-categories 59 5.2. Stabilization and the universality of spectra 64 5.3.
  • Fibrations and Geometric Realizations

    Fibrations and Geometric Realizations

    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 5, September 1978 FIBRATIONS AND GEOMETRIC REALIZATIONS BY D. W. ANDERSON There is a folk theorem associated to the construction of classifying spaces for topological groups which says that a map of simplicial spaces which is a fibration in every degree has a fibration as its geometric realization. Peter May [GILS] has given a useful form of this which involves quasifibrations. This result has seemed to me a very interesting one, as it relates two rather opposite types of operations. On the one hand, it involves the geometric realization, which is defined by mapping into other things, and on the other hand, it involves fibrations, which are characterized by the properties of maps of other things into them. Any theorem which mixes "left" and "right" mapping properties should be expected to be difficult to prove. Since what is really wanted in applications is a homotopy theoretic result, the possibilities for complication are almost infinite. It seemed to me that a proof should be found which allowed for a homotopy invariant statement of the theorem, and which had the property that the ad hoc part of the argument was isolated in a reasonably small, isolated computation. After some effort, I managed to find such a proof, which I will outline in this paper. The virtue of this proof is that it involves a number of areas of topology which are known only to experts and which deserve a wider audience. Thus the proof of the theorem about the geometric realization of fibration has become the occasion for an exposition of simplicial methods, axiomatic homotopy theory, and homotopy limits and colimits.