# Bicategorical Yoneda Lemma

Total Page：16

File Type：pdf, Size：1020Kb

Bicategorical Yoneda lemma Igor Bakovi¶c Rudjer Bo·skovi¶cInstitute Abstract We review the Yoneda lemma for bicategories and its connection to 2-descent and some universal constructions. 1 The 2-category of 2-presheaves De¯nition 1.1. Let B be a bicategory. A 2-presheaf P on a bicategory B is a homomor- phism of bicategories P : Bop ! Cat. Let P : Bop ! Cat be the 2-presheaf. It takes an object B 2 Ob(B) to the category P (B), a 1-morphism f : A ! B to the functor P (f): P (B) ! P (A); and any 2-morphism Ã : f ! g to the natural transformation P (Ã): P (f) ! P (g): We also have natural trans- formations ¹g;f : P (f) ± P (g) ! P (g ± f) and ´A : iP (A) ! P (iA), which satisfy following coherence laws ¹ ±P (h) ¹ (P (f) ± P (g)) ± P (h) g;f / P (g ± f) ± P (h) h;g±f / P (h ± (g ± f)) O P (ah;g;f ) / / P (f) ± (P (g) ± P (h)) P (f) ± P (h ± g) ¹ P ((h ± g) ± f) P (f)±¹h;g h±g;f ¹i ;f ¹f;i P (f) ± P (i ) B / P (i ± f) P (i ) ± P (f) A / P (f ± i ) O B B A O A P (f)±´B P (¸f ) ´A±P (f) P (½f ) ² ² P (f) P (f) P (f) P (f) 1 De¯nition 1.2. Let P; R: Bop ! Cat be two 2-presheaves on B. A (left) pseudo natural transformation σ : P ) R consists of the functors σA : P (A) ! R(A) for each object A 2 B and is given by the square σ P (B) B / R(B) P (f) ÄÄÄ R(f) {¤ Äσf ² ² P (A) / R(A) σA which represents a natural transformation σf : R(f) ± σB ) σA ± P (f) for any 1-morphism f : A ! B in B. These natural transformations satisfy the coherence law given by the diagram P (B) Ä? ?? ÄÄ ?? P (g) ÄÄ ?? P (f) ÄÄ ?? ÄÄ ?? ÄÄ ÂÂ ?? Ä ÂÂ ¹P ? Ä ®¶ g;f Â P (C) / P (A) P (g±f) σB σg ;C ???σf ÄÄÄ # ² σC R(B) σA Ä? ?? ÄÄ ?? R(g) ÄÄ ?? R(f) ÄÄ ÂÂ ?? ÄÄ ¹R ?? ÄÄ ®¶ g;f ?? ² ÄÄ ?Â ² R(C) / R(A) R(g±f) which becomes a commutative diagram R(f)±σg / R(f) ± R(g) ± σC R(f) ± σB ± P (g) R ¹g;f ±σC ² R(g ± f) ± σC σf ±P (g) σg±f ² ² σ ± P (g ± f) o σ ± P (f) ± P (g) A P A σz±¹g;f 2 f g for any composable pair A / B / C of 1-morphisms in B. The second coherence is given by the commutative diagram σA Ä ?? ÄÄ ?? ÄÄ ?? ÄÄ ?? ´R±σ Ä ? σ ±´P A A ÄÄ ?? A A ÄÄ ?? ÄÄ ?? ÄÄ ?? ÄÄ ?? Ä Ä ?Â / R(iA) ± σA σA ± P (iA) σiA of natural transformations. De¯nition 1.3. A modi¯cation ¡: σ ! ¿ consists of the natural transformation ¡A : σA ! ¿A for each object A in B such that the following diagram σB \$ ÂÂ P (B) ¡B R(B) ®¶ : ¿B P (f) R(f) ÂÂ ®¶ ¿f f _ X l σA R ² s K\$ ² ÂÂ P (A) ¡A R(A) ®¶ : ¿A commutes. This means that the above diagram becomes a commutative diagram R(f)±¡B / R(f) ± σB R(f) ± ¿B σf ¿f ² ² / σA ± P (f) ¿A ± P (f) ¡A±P (f) of 2-morphisms in B. 3 Proposition 1.1. Let B be a bicategory. Then 2-preshevaes are the objects of the 2- op category P := HomBicat(B ; Cat) in which 1-morphisms are pseudo natural transforma- tion and 2-morphisms are modi¯cations. σ » Proof. For any two pseudo natural transformations P +3 R +3 S the composition is de¯ned by (» ± σ)A := »A ± σA and (» ± σ)f := (»A ± σf )(»f ± σB) which is just the pasting σ » P (B) B / R(B) B / S(B) P (f) ÄÄÄ R(f) ÄÄÄ S(f) {¤ σf {¤ »f ² ² ² / / P (A) σ R(A) S(A) A »A composite of the above diagram. That this is well de¯ned composition we deduce from the pasting composite P (B) ? ? ÄÄ ?? P (f) Ä ÂÂ ? P (g) ÄÄ ¹P ?? ÄÄ ®¶ g;f ?? ÄÄ ?Â P (C) / P (A) P (k±h) σB σf ;C ???σg ÄÄÄ # ² σC R(B) σA ? ? ÄÄ ?? R(f) Ä ÂÂ ? R(g) ÄÄ ¹R ?? ÄÄ ®¶ g;f ?? ² ÄÄ ?Â ² R(C) / R(A) R(k±h) »B »f ;C ???»g ÄÄÄ # ² »C S(B) »A ? ? ÄÄ ?? S(f) Ä ÂÂ ? S(g) ÄÄ ¹S ?? ÄÄ ®¶ g;f ?? ² ÄÄ ?Â ² S(C) / S(A) S(g±f) which gives a coherence for a pseudo natural transformation. This composition is asso- σ » ! ciative since for any three pseudo natural transformations P +3 R +3 S +3 T we have the associativity [(! ±»)±σ]A = (! ±»)A ±σA = (!A ±»A)±σA = !A ±(»A ±σA) = !A ±(» ±σ)A = [! ±(» ±σ)]A 4 for components indexed by the object B in B, and [(! ± ») ± σ]f = [(! ± »)A ± σf ][(! ± »)f ± σB] = [(!A ± »A) ± σf ][((!A ± »f )(!f ± »B)) ± σB] = = [(!A ± »A) ± σf ][(!A ± »f ) ± σB][(!f ± »B) ± σB] = [!A ± (»A ± σf )][!A ± (»f ± σB)][!f ± (»B ± σB)] = = [(!A ± (»A ± σf )(»f ± σB)][(!f ± (»B ± σB))] = [!A ± (» ± σ)f ][!f ± (» ± σ)B] = [! ± (» ± σ)]f for components indexed by the morphism f : A ! B in B. The vertical composition for any two modi¯cations σ ¡ _*4 ¿ ¦ _*4 ¼ is de¯ned simply by (¦¡)A := ¦A¡A. The horizontal composition for any two modi¯cations σ » ÂÂ ¿ ÂÂ ¿ P ®¶ ¡ BR ®¶ ­ B S ¿ ³ is de¯ned by (­ ± ¡)A := ­B ± ¡B. We need to show that pasting of the diagram σB »B ! ! ÂÂ ÂÂ P (B) ¡B R(B) ­B S(B) ®¶ = ®¶ = ¿ B ³B P (f) R(f) S(f) ÂÂ ÂÂ ¿f ³ h _®¶ V h _®¶ Vf p σA N p » N ² y E! ² y A E! ² ÂÂ ÂÂ P (A) ¡A R(A) ­A S(A) ®¶ = ®¶ = ¿ A ³A 5 gives the coherence diagram for the composite modi¯cation ­ ± ¡: » ± σ ! ³ ± ¿. (»±σ)B ! ÂÂ P (B) (­±¡)B S(B) ®¶ = (³±¿)B P (f) S(f) ÂÂ ÂÂ (³±¿) h _®¶ V f p (»±σ) N ² y A E! ² ÂÂ P (A) (­±¡)A S(A) ®¶ = (³±¿)A The commutativity of the above diagram is deduced from the commutative diagram S(f)±(»B ±¡B ) / S(f)±(­B ±¿B ) / S(f) ± (»B ± σB) S(f) ± (»B ± ¿B) S(f) ± (³B ± ¿B) ® ® ® S(f);»B ,σB S(f);»B ;¿B S(f);³B ;¿B ² ² ² (S(f)±»B )±¡B / (S(f)±­B )±¿B / (S(f) ± »B) ± σB (S(f) ± »B) ± ¿B (S(f) ± ³B) ± ¿B »f ±σB »f ±¿B ³f ±¿B ² ² ² (»A±R(f))±¡B / (­A±R(f))±¿B / (»A ± R(f)) ± σB (»A ± R(f)) ± ¿B (³A ± R(f)) ± ¿B ® ® ® »A;R(f),σB »A;R(f);¿B ³A;R(f);¿B ² ² ² »A±(R(f)±¡B ) / ­A±(R(f)±¿B ) / »A ± (R(f) ± σB) »A ± (R(f) ± ¿B) ³A ± (R(f) ± ¿B) »A±σf »A±¿f ³A±¿f ² ² ² »A±(¡A±P (f)) / ­A±(¿A±P (f)) / »A ± (σA ± P (f)) »A ± (¿A ± P (f)) ³A ± (¿A ± P (f) ®¡1 ®¡1 ®¡1 »A,σA;P (f) »A;¿A;P (f) ³A;¿A;P (f) ² ² ² / / (»A ± σA) ± P (f) (»A ± ¿A) ± P (f) (³A ± ¿A) ± P (f) (»A±¡A)±P (f) (­A±¿A)±P (f) where the ¯rst square in the second row and the second square in the the fourth row commute because they de¯ne horizontal compositions »f ±¡B and ­A ±¿f , respectively. The commutativity of the second square in the second row and the ¯rst square in the fourth row follows from the coherence for modi¯cations ­: » ! ³ and ¡: σ ! ¿, respectively 6 and squares in the ¯rst, third and ¯fth row are associativity coherence. But the pasting composition of the top and front faces of the coherence for the modi¯cation ­ ± ¡ is equal to the composition of the top and right edges of the above diagram, and the pasting composition of the back and bottom faces of the coherence is equal to the composition of the left and bottom edges of the diagram. The associativity and identities for horizontal composition follows immediately from the associativity of horizontal composition of natural transformations in Cat. Also, for any diagram of modi¯cations σ » ÂÂ ÂÂ ®¶ ¡ ¿ ®¶ ­ ¿ / / P ¿ BR B S ÂÂ ³ÂÂ ®¶ ¦ ®¶ ¤ ¼ Â we have Godement interchange law (¤ ± ¦)(­ ± ¡) = (¤­) ± (¦¡) since it holds for natural transformations and the composition were de¯ned componentwise. 7 2 The Yoneda embedding We will now take a more closer look to the representable 2-presheaves. Each object C in op B gives rise to the 2-presheaf YC : B ! Cat on B, de¯ned by YC (B) := B(B; C); for each object B in B, and for any two objects A; B 2 Bop by the functor A;B YC : B(A; B) ! Cat(YC (B); YC (A)); for which we will usually omit indexing objects in the superscript. The value of this functor at any object f 2 B(A; B) is a functor YC (f): YC (B) !YC (A) de¯ned by YC (f)(g) := g ± f; YC (f)(Á) := Ã ± f for any 1-morphism g 2 B(B; C) and any 2-morphism Ã 2 B(B; C). The value of the A;B C C 0 functor YC : B(A; B) ! Cat(Y (B); Y (A)) for any 2-morphism Á: f ! f (viewed as a morphism of the category B(A; B)), is the natural transformation 0 YC (Á): YC (f) !YC (f ) as in the diagram Y (Á) C g / 0 YC (f)(g) YC (f )(g) 0 YC (f)(º) YC (f )(º) ² ² 0 / 0 0 YC (f)(g ) YC (f )(g ) YC (Á)g0 for any 2-morphism º : g ! g0 in B(B; C), whose component for any object g 2 B(B; C) is de¯ned by YC (Á)g := g ± Á.

Recommended publications
• Fibrations and Yoneda's Lemma in An
Journal of Pure and Applied Algebra 221 (2017) 499–564 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa Fibrations and Yoneda’s lemma in an ∞-cosmos Emily Riehl a,∗, Dominic Verity b a Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA b Centre of Australian Category Theory, Macquarie University, NSW 2109, Australia a r t i c l e i n f o a b s t r a c t Article history: We use the terms ∞-categories and ∞-functors to mean the objects and morphisms Received 14 October 2015 in an ∞-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent Received in revised form 13 June of an enriched category of ﬁbrant objects. Quasi-categories, Segal categories, 2016 complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, Available online 29 July 2016 θ -spaces, and ﬁbered versions of each of these are all ∞-categories in this sense. Communicated by J. Adámek n Previous work in this series shows that the basic category theory of ∞-categories and ∞-functors can be developed only in reference to the axioms of an ∞-cosmos; indeed, most of the work is internal to the homotopy 2-category, astrict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi- categories, we recapture precisely the same category theory developed by Joyal and Lurie, although our deﬁnitions are 2-categorical in natural, making no use of the combinatorial details that diﬀerentiate each model. In this paper, we introduce cartesian ﬁbrations, a certain class of ∞-functors, and their groupoidal variants.
[Show full text]
• AN INTRODUCTION to CATEGORY THEORY and the YONEDA LEMMA Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transfo
AN INTRODUCTION TO CATEGORY THEORY AND THE YONEDA LEMMA SHU-NAN JUSTIN CHANG Abstract. We begin this introduction to category theory with deﬁnitions of categories, functors, and natural transformations. We provide many examples of each construct and discuss interesting relations between them. We proceed to prove the Yoneda Lemma, a central concept in category theory, and motivate its signiﬁcance. We conclude with some results and applications of the Yoneda Lemma. Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transformations 6 4. The Yoneda Lemma 9 5. Corollaries and Applications 10 Acknowledgments 12 References 13 Introduction Category theory is an interdisciplinary ﬁeld of mathematics which takes on a new perspective to understanding mathematical phenomena. Unlike most other branches of mathematics, category theory is rather uninterested in the objects be- ing considered themselves. Instead, it focuses on the relations between objects of the same type and objects of diﬀerent types. Its abstract and broad nature allows it to reach into and connect several diﬀerent branches of mathematics: algebra, geometry, topology, analysis, etc. A central theme of category theory is abstraction, understanding objects by gen- eralizing rather than focusing on them individually. Similar to taxonomy, category theory oﬀers a way for mathematical concepts to be abstracted and uniﬁed. What makes category theory more than just an organizational system, however, is its abil- ity to generate information about these abstract objects by studying their relations to each other. This ability comes from what Emily Riehl calls \arguably the most important result in category theory"[4], the Yoneda Lemma. The Yoneda Lemma allows us to formally deﬁne an object by its relations to other objects, which is central to the relation-oriented perspective taken by category theory.
[Show full text]
• Yoneda's Lemma for Internal Higher Categories
YONEDA'S LEMMA FOR INTERNAL HIGHER CATEGORIES LOUIS MARTINI Abstract. We develop some basic concepts in the theory of higher categories internal to an arbitrary 1- topos. We deﬁne internal left and right ﬁbrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories. Contents 1. Introduction 2 Motivation 2 Main results 3 Related work 4 Acknowledgment 4 2. Preliminaries 4 2.1. General conventions and notation4 2.2. Set theoretical foundations5 2.3. 1-topoi 5 2.4. Universe enlargement 5 2.5. Factorisation systems 8 3. Categories in an 1-topos 10 3.1. Simplicial objects in an 1-topos 10 3.2. Categories in an 1-topos 12 3.3. Functoriality and base change 16 3.4. The (1; 2)-categorical structure of Cat(B) 18 3.5. Cat(S)-valued sheaves on an 1-topos 19 3.6. Objects and morphisms 21 3.7. The universe for groupoids 23 3.8. Fully faithful and essentially surjective functors 26 arXiv:2103.17141v2 [math.CT] 2 May 2021 3.9. Subcategories 31 4. Groupoidal ﬁbrations and Yoneda's lemma 36 4.1. Left ﬁbrations 36 4.2. Slice categories 38 4.3. Initial functors 42 4.4. Covariant equivalences 49 4.5. The Grothendieck construction 54 4.6. Yoneda's lemma 61 References 71 Date: May 4, 2021. 1 2 LOUIS MARTINI 1. Introduction Motivation. In various areas of geometry, one of the principal strategies is to study geometric objects by means of algebraic invariants such as cohomology, K-theory and (stable or unstable) homotopy groups.
[Show full text]
• Notes on Categorical Logic
Notes on Categorical Logic Anand Pillay & Friends Spring 2017 These notes are based on a course given by Anand Pillay in the Spring of 2017 at the University of Notre Dame. The notes were transcribed by Greg Cousins, Tim Campion, L´eoJimenez, Jinhe Ye (Vincent), Kyle Gannon, Rachael Alvir, Rose Weisshaar, Paul McEldowney, Mike Haskel, ADD YOUR NAMES HERE. 1 Contents Introduction . .3 I A Brief Survey of Contemporary Model Theory 4 I.1 Some History . .4 I.2 Model Theory Basics . .4 I.3 Morleyization and the T eq Construction . .8 II Introduction to Category Theory and Toposes 9 II.1 Categories, functors, and natural transformations . .9 II.2 Yoneda's Lemma . 14 II.3 Equivalence of categories . 17 II.4 Product, Pullbacks, Equalizers . 20 IIIMore Advanced Category Theoy and Toposes 29 III.1 Subobject classiﬁers . 29 III.2 Elementary topos and Heyting algebra . 31 III.3 More on limits . 33 III.4 Elementary Topos . 36 III.5 Grothendieck Topologies and Sheaves . 40 IV Categorical Logic 46 IV.1 Categorical Semantics . 46 IV.2 Geometric Theories . 48 2 Introduction The purpose of this course was to explore connections between contemporary model theory and category theory. By model theory we will mostly mean ﬁrst order, ﬁnitary model theory. Categorical model theory (or, more generally, categorical logic) is a general category-theoretic approach to logic that includes inﬁnitary, intuitionistic, and even multi-valued logics. Say More Later. 3 Chapter I A Brief Survey of Contemporary Model Theory I.1 Some History Up until to the seventies and early eighties, model theory was a very broad subject, including topics such as inﬁnitary logics, generalized quantiﬁers, and probability logics (which are actually back in fashion today in the form of con- tinuous model theory), and had a very set-theoretic ﬂavour.
[Show full text]
• GROUPOID SCHEMES 022L Contents 1. Introduction 1 2
GROUPOID SCHEMES 022L Contents 1. Introduction 1 2. Notation 1 3. Equivalence relations 2 4. Group schemes 4 5. Examples of group schemes 5 6. Properties of group schemes 7 7. Properties of group schemes over a ﬁeld 8 8. Properties of algebraic group schemes 14 9. Abelian varieties 18 10. Actions of group schemes 21 11. Principal homogeneous spaces 22 12. Equivariant quasi-coherent sheaves 23 13. Groupoids 25 14. Quasi-coherent sheaves on groupoids 27 15. Colimits of quasi-coherent modules 29 16. Groupoids and group schemes 34 17. The stabilizer group scheme 34 18. Restricting groupoids 35 19. Invariant subschemes 36 20. Quotient sheaves 38 21. Descent in terms of groupoids 41 22. Separation conditions 42 23. Finite ﬂat groupoids, aﬃne case 43 24. Finite ﬂat groupoids 50 25. Descending quasi-projective schemes 51 26. Other chapters 52 References 54 1. Introduction 022M This chapter is devoted to generalities concerning groupoid schemes. See for exam- ple the beautiful paper [KM97] by Keel and Mori. 2. Notation 022N Let S be a scheme. If U, T are schemes over S we denote U(T ) for the set of T -valued points of U over S. In a formula: U(T ) = MorS(T,U). We try to reserve This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 GROUPOID SCHEMES 2 the letter T to denote a “test scheme” over S, as in the discussion that follows. Suppose we are given schemes X, Y over S and a morphism of schemes f : X → Y over S.
[Show full text]
• 4 Fibered Categories (Aaron Mazel-Gee) Contents
4 Fibered categories (Aaron Mazel-Gee) Contents 4 Fibered categories (Aaron Mazel-Gee) 1 4.0 Introduction . .1 4.1 Deﬁnitions and basic facts . .1 4.2 The 2-Yoneda lemma . .2 4.3 Categories ﬁbered in groupoids . .3 4.3.1 ... coming from co/groupoid objects . .3 4.3.2 ... and 2-categorical ﬁber product thereof . .4 4.0 Introduction In the same way that a sheaf is a special sort of functor, a stack will be a special sort of sheaf of groupoids (or a special special sort of groupoid-valued functor). It ends up being advantageous to think of the groupoid associated to an object X as living \above" X, in large part because this perspective makes it much easier to study the relationships between the groupoids associated to diﬀerent objects. For this reason, we use the language of ﬁbered categories. We note here that throughout this exposition we will often say equal (as opposed to isomorphic), and we really will mean it. 4.1 Deﬁnitions and basic facts φ Deﬁnition 1. Let C be a category. A category over C is a category F with a functor p : F!C. A morphism ξ ! η p(φ) in F is called cartesian if for any other ζ 2 F with a morphism ζ ! η and a factorization p(ζ) !h p(ξ) ! p(η) of φ p( ) in C, there is a unique morphism ζ !λ η giving a factorization ζ !λ η ! η of such that p(λ) = h. Pictorially, ζ - η w - w w 9 w w ! w w λ φ w w w w - w w w w ξ w w w w w w p(ζ) w p(η) w w - w h w ) w (φ w p w - p(ξ): In this case, we call ξ a pullback of η along p(φ).
[Show full text]
• Notes on Moduli Theory, Stacks and 2-Yoneda's Lemma
Notes on Moduli theory, Stacks and 2-Yoneda's Lemma Kadri lker Berktav Abstract This note is a survey on the basic aspects of moduli theory along with some examples. In that respect, one of the purpose of this current document is to understand how the introduction of stacks circumvents the non-representability problem of the corresponding moduli functor F by using the 2-category of stacks. To this end, we shall briey revisit the basics of 2-category theory and present a 2-categorical version of Yoneda's lemma for the "rened" moduli functor F. Most of the material below are standard and can be found elsewhere in the literature. For an accessible introduction to moduli theory and stacks, we refer to [1,3]. For an extensive treatment to the case of moduli of curves, see [5,6,8]. Basics of 2-category theory and further discussions can be found in [7]. Contents 1 Functor of points, representable functors, and Yoneda's Lemma1 2 Moduli theory in functorial perspective3 2.1 Moduli of vector bundles of xed rank.........................6 2.2 Moduli of elliptic curves.................................7 3 2-categories and Stacks 11 3.1 A digression on 2-categories............................... 11 3.2 2-category of Stacks and 2-Yoneda's Lemma...................... 14 1 Functor of points, representable functors, and Yoneda's Lemma Main aspects of a moduli problem of interest can be encoded by a certain functor, namely a moduli functor of the form F : Cop −! Sets (1.1) where Cop is the opposite category of the category C, and Sets denotes the category of sets.
[Show full text]
• THE YONEDA LEMMA MATH 250B 1. the Yoneda Lemma the Yoneda
THE YONEDA LEMMA MATH 250B ADAM TOPAZ 1. The Yoneda Lemma The Yoneda Lemma is a result in abstract category theory. Essentially, it states that objects in a category C can be viewed (functorially) as presheaves on the category C. Before we state the main theorem, we introduce a bit of notation to make our lives easier. C For a category C and an object A 2 C, we denote by hA the functor Hom (•;A) (recall that this is a contravariant functor). Similarly, we denote the functor HomC(A; •) by hA. The statement of the Yoneda Lemma (in contravariant form) is the following. Theorem: Let C be a category. If F is an arbitrary contravariant functor Cop ! Set, then one has Fun ∼ Hom (hA;F ) = F (A): This isomorphism is functorial in A in the sense that one has an isomorphism of functors: Fun ∼ Hom (h•;F ) = F (•): op Before we prove the statement, let us show that h• is actually a functor C! Fun(C ; Set). Namely, suppose that A ! B is a morphism in C, we must show how to construct a natural transformation hf : hA ! hB. Namely, for an object C 2 C, we need to associate a hf (C): hA(C) ! hB(C): C C This map is the only natural choice: one has hA(C) = Hom (C; A) and hB(C) = Hom (C; B). C C C The map hf (C) : Hom (C; A) ! Hom (C; B) is deﬁned to be the map h (f) (post- composition with f). The fact that this makes h• into a functor is left as an (easy) exercise.
[Show full text]
• Ends and Coends
THIS IS THE (CO)END, MY ONLY (CO)FRIEND FOSCO LOREGIAN† Abstract. The present note is a recollection of the most striking and use- ful applications of co/end calculus. We put a considerable eﬀort in making arguments and constructions rather explicit: after having given a series of preliminary deﬁnitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse ﬁelds like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu, can regard basically every statement as a guided exercise. Contents Introduction. 1 1. Dinaturality, extranaturality, co/wedges. 3 2. Yoneda reduction, Kan extensions. 13 3. The nerve and realization paradigm. 16 4. Weighted limits 21 5. Profunctors. 27 6. Operads. 33 Appendix A. Promonoidal categories 39 Appendix B. Fourier transforms via coends. 40 References 41 Introduction. The purpose of this survey is to familiarize the reader with the so-called co/end calculus, gathering a series of examples of its application; the author would like to stress clearly, from the very beginning, that the material presented here makes arXiv:1501.02503v2 [math.CT] 9 Feb 2015 no claim of originality: indeed, we put a special care in acknowledging carefully, where possible, each of the many authors whose work was an indispensable source in compiling this note.
[Show full text]
• Naturality and the Yoneda Lemma
NATURALITY AND THE YONEDA LEMMA J. WARNER 1. Naturality Let C and D be two categories, and let F; G : C!D be covariant functors. We would like to deﬁne the notion of a "map of functors" that is consistent with the usual requirements of a map between two algebraic structures. That is, a map between F and G should in some sense preserve the structure of a functor. What is included in this structure? First, the functors F and G assign to each object X 2 C objects in D, denoted F (X) and G(X) respectively. Thus, if we desire a map of functors η : F ! G to respect the structure of the functors' values on objects, η should include the data of a morphism ηX 2 HomD(F (X);G(X)) for all X 2 C. Second, the functors F and G assign to each pair of objects X; Y 2 D and each morphism f 2 HomC(X; Y ) morphisms in HomD(F (X);F (Y )) and HomD(G(X);G(Y )) denoted F (f) and G(f) respectively. So far, our idea of a map of functors leads us to consider two morphisms in HomD(F (X);G(Y )): G(f) ◦ ηX and ηY ◦ F (f). Now, if we desire our map η to respect the structure of the functors' values on morphisms, these two morphisms should be equal in HomD(F (X);G(Y )). We have thus arrived at the appropriate deﬁnition of a map of functors, which we now deﬁne formally. Deﬁnition 1.1.
[Show full text]
• The Yoneda Isomorphism Commutes with Homology
THE YONEDA ISOMORPHISM COMMUTES WITH HOMOLOGY GEORGE PESCHKE AND TIM VAN DER LINDEN Abstract. We show that, for a right exact functor from an abelian category to abelian groups, Yoneda’s isomorphism commutes with homology and, hence, with functor derivation. Then we extend this result to semiabelian domains. An interpretation in terms of satellites and higher central extensions follows. As an application, we develop semiabelian (higher) torsion theories and the associated theory of (higher) universal (central) extensions. Introduction Yoneda’s lemma establishes for any Set-valued functor F : X Ñ Set and any object U of X the bijection NatpHomX pU; ´q;F q ÝÑ F U; τ ÞÑ τp1U q: We refer to this bijection as the Yoneda isomorphism [39, pp. 59ﬀ.], and note that it follows without any further assumptions from the structural axioms of categories. It therefore belongs to the very foundations of the theory of categories. If X is enriched in the category of abelian groups Ab, and F is additive and Ab-valued, then the Yoneda isomorphism is a morphism of abelian groups. This may be veriﬁed directly, or seen as a special case of results in [36]. Extending work of Mac Lane [37], as a critical reﬁnement of Ab-enriched cat- egories, Buchsbaum [10] and Grothendieck [24] introduced abelian categories to provide a purely axiomatic foundation for homological algebra. The key result of the present paper establishes a homological Yoneda isomorphism, Lemma 1.1: For a chain complex C in an abelian category A , and a right exact functor T from A to the category of abelian groups, n – NatpH HomA pC; ´q;T q ÝÝÝÝÝÑ HnT C: This result belongs to the very foundations of abelian categories as it follows without arXiv:1505.05406v2 [math.KT] 6 Sep 2015 any further assumptions from their structural axioms, and this is visible in its proof: Yoneda’s lemma turns the claim into a tautology upon contemplating the self-dual nature of the concept of ‘chain complex’ and what this means to ‘homology’.
[Show full text]
• Contents 1. the Yoneda Lemma 1 2
A NOTE ON THE YONEDA LEMMA ALEX MASSARENTI Abstract. We state, without proofs, the Yoneda lemma, and the 2-Yoneda lemma. Then we prove that a functor is representable if and only if it admits a universal object using the Yoneda lemma, and that a bered category is representable if and only if it is bered in groupoids using the 2-Yoneda lemma. Contents 1. The Yoneda Lemma 1 2. The 2-Yoneda Lemma 3 References 4 1. The Yoneda Lemma Let C be a category. The functors from Cop to (Set) can be thought as a category Hom(Cop; (Set)), in which the arrows are the natural transformations. To any op object X 2 C we can associate a functor hX : C ! (Set), which maps an object Y 2 C in HomC(Y; X), and a morphism f : Y ! Z in the map hX f : HomC(Z; X) ! HomC(Y; X); α 7! α ◦ f: Now let now ρ : X ! Y be an arrow in C. For any object Z 2 C we get a function hf Z : hX Z ! hY Z; α 7! ρ ◦ α. This denes a morphism hf : hX ! hY , i.e. for any arrow β : Z ! W in C the following diagram commute hf Z hX Z hY Z hX β hY β hf W hX W hY W In this way we have dened a functor op op H : C ! Hom(C ; (Set));X ! hX ; f ! hf : Lemma 1.1. (Weak Yoneda Lemma) The functor H is fully faithful, that is the function Hom(X; Y ) ! Hom(hX ; hY ); f 7! hf is bijective.
[Show full text]