Yoneda Theory For Double Categories

Yoneda Theory for Double CategoriesRobert ParéDalhousie Universitypare@mathstat.dal.caJuly 23, 2011

IntroductionIIBasic tenet: The right framework for two-dimensionalcategory theory is that of double categoriesThe traditional approach was via:III(Weak) double categories are a lot like bicategoriesIIIIgood because much of the well-developed theory ofbicategories can be easily adapted to double categoriesinteresting when the theories differthe Yoneda lemma is an instance of thisThe Yoneda lemma is the cornerstone of category theoryIIIII2-categories (like Cat)bicategories (like Prof )categorical universal algebracategorical logicsheaf theoryrepresentability and adjointnessThe further development of double category theory dependson understanding the Yoneda lemma in this context

INot a priori clear what representables should beIIIThese questions lead to the double category SetIIIIWhere do they take their values?What kind of “functor” are they?sets, functions and spansthis is the double category version of the category of setsthe most basic double categoryRepresentables are lax functors into SetIIIIIIIhence the double category Lax(Aop , Set)a presheaf double categoryunderstand its propertieshorizontal and vertical arrowscomposition (not trivial)completenessetc.

(Weak) Double CategoriesIIIObjectsTwo kinds of arrows (horizontal and vertical)Cells that tie them togetherAv αCg IIIIf/B w /D/ w)(also denoted α : vHorizontal composition of arrows and cells give categorystructuresVertical composition gives “weak categories”, i.e. compositionis associative and unitary up to coherent special isomorphism(like for bicategories)Vertical composition of cells is as associative and unitary asthe structural isomorphisms allowInterchange holds

The Basic Example: SetIObjects are setsIHorizontal arrows are functionsIVertical arrows are spansICells are commutative diagramsAO/BOS/T /DCIVertical composition uses pullback – it is associative andunitary up to coherent special isomorphism

A Related Example: V-SetIV a -category with coproductsIObjects are setsIHorizontal arrows are functionsIVertical arrows AICells are matrices of V morphismsIVertical composition is matrix multiplication / B are A B matrices of objects of VWhen V Set, , we get SetWhen V 2, , we get sets with relations as vertical arrows

An Important Example: CatIObjects are small categoriesIHorizontal arrows are functorsIVertical arrows AICells / B are profunctors, i.e. Aop BAFP t BG/C Q /Dare natural transformationst : P( , )/ Q(F , G )/ Set

Three General Constructions: V, H, QIFor B a bicategory, VB is B made into a double categoryvertically, i.e. horizontal arrows are identitiesIFor C a 2-category, HC is C made into a double categoryhorizontally, i.e. vertical arrows are identitiesIFor C a 2-category, QC is Ehresmann’s double category of“quintets”. A general cell is/ C0C C̄{ / C̄ 0

/Lax Functors F : AAv Āfαf B/ A0 v0 / Ā0FA7 Fv FfFα F ĀF f / FA0 Fv 0 / F Ā0IPreserve horizontal compositions and identitiesIProvide comparison special cells for vertical composition andidentitiesφ(v̄ , v ) : F v̄ · Fv F (v̄ · v ),Iφ(A) : idFA F (idA )Satisfy naturality and coherence conditions, like for laxmorphisms of bicategoriesThere are also oplax, normal, strong, and strict double functors

ExamplesIF : VBIF : HCCat oIIIII1II/ VB 0 is a (lax) morphism of bicategories/ HC 0 is a 2-functor// Set/ Set is laxOb : Cat/ Cat is strongDisc : Set/ Set is oplax normalπ0 : Cat/ A is a vertical monad/ Set is a small category1/ V-Set is a small V-category1

/The Main Example: A( , A) : AopSet/ A}A(X , A) {f : X f / XA v α idAA(v , A) Y/g AThe span projections are domain and codomainI Horizontal functoriality is by compositionI Vertical comparisonsh(w , v ) : A(w , A) A(v , A) A(w · v , A)Xfv αgw βY Zh/AX idA /A idA /A7 f/Aw ·v β·α idA Zh /A

Natural Transformations of Lax Functorst:FIFor every A, tA : FAIFor every v : A /G/ GA (horizontal)/ Ā,FAFv F ĀIHorizontally naturalIVertically functorialtAtvt Ā/ GA Gv / G Ā

ExamplesIFor lax 1/ Set, we get functorsIFor lax 1/ V-Set, we get V-functorsI/ VB 0 , we get lax transformations which areFor VBidentities on objects/ HC 0 , we get 2-natural transformations/ A0 gives a natural transformationI Every horizontal f : AIFor HC/ A( , A0 )A( , f ) : A( , A)Xx/AXx/Av ξ Y idA /Ay7 7 xX/Af/ A0/ A0Xx/Afv ξ idf Y /Ayf idA0 / A0

The Yoneda LemmaTheorem/ Set and an object A of A, there is aFor a lax functor F : Aop/ F andbijection between natural transformations t : A( , A)elements x FA given by x t(A)(1A ).CorollaryEvery natural transformation t : A( , A)/ A0 .form A( , f ) for a unique f : A/ A( , A0 ) is of the

“Application”The theory of adjoints for double categories was set out in[Grandis-Paré, Adjoint for Double Categories, Cahiers (2004)]. Theleft adjoint is typically oplax and the right adjoint lax. It isexpressed in terms of conjoints in a strict double category Doub.Example:π0 a Disc a ObTheorem/ B oplax and U : B/ A lax, there is a bijectionFor F : Abetween adjunctions F a U and natural isomorphismsB(F , )of lax functors Aop B/ Set./ A( , U )

Vertical Structure of Lax(A, B)/ B, a module [Cockett, Koslowski,For F and G lax functors, ASeely, Wood – Modules, TAC 2003] m : F / G is given by thefollowing data.IFor every vertical arrow v : Amv : FA / G ĀIFor every cell α a cell mαAv Āfαf /C w / C̄/ Ā in A a vertical arrowFA7 mv G ĀFfmαG f / FC mw / G C̄

Modules (continued)IFor every pair of vertical arrows v : Aleft and right actionsFAFAmv / Ā and v̄ : Ā / Ã,FAFAFv G Ā m(v̄ ·v )λG v̄ G ÃG Ã F Āmv̄ ρ m(v̄ ·v ) G ÃG ÃsatisfyingIHorizontal functorialityINaturality of λ and ρILeft and right unit lawsILeft, right and middle associativity laws

Examples/ Set, modules are profunctors/ B lax, idF : F / F is given byI For F : AIFor 1FAidF (v ) Fv F ĀI(Main Example) For v : A / Ā in AA( , v ) : A( , A) X z A(z, v ) Y / A( , Ā)/A ξ v / Ā

ModulationsThe cells of Lax(A, B) are called modulations following [CKSW]Ftm µGs IFor every vertical v : A / F0 m0 / G0/ Ā we are givenFAmv G ĀtAµvs Ā/ F 0A m0 v / G 0 ĀsatisfyingI Horizontal naturalityI EquivarianceExample: A cell α of A produces a modulation A( , α)

The Yoneda Lemma IITheoremLet m : F / G be a module in Lax(Aop , Set) and v : A / Ā avertical arrow of A. Then there is a bijection between modulationsA( , A)A( ,v ) µA( , Ā)/F m /Gand elements r m(v ) given by r µ(v )(1r ).

Corollary/ Set lax, an element r F (v ) is uniquelyFor F : Aopdetermined by a modulation/FA( , A)A( ,v ) µ A( , Ā) idF /FCorollaryFor v : A / Ā and v 0 : A0 / Ā0 in A, every modulation/ A( , A0 )A( , A)A( ,v ) A( , Ā)µ A( ,v 0 ) / A( , Ā)0is of the form A( , α) for a unique cell α : v/ v 0.

Application: TabulatorsA tabulator for a vertical arrow v : Aan object T and a cell?AT / Ā in a double category isτ v Āwith universal properties:(T1) For every cell?AXξ v Āthere is a unique horizontal arrow x : X/ T such that τ x ξ

Tabulators: 2-Dimensional Property(T2) For every commutative tetrahedron of cells/A?Xx v / ĀX̄there is a unique cell ξ such thatXx ξ X̄?Av?T τ Āgives the tetrahedron in the “obvious” way.

Tabulators in Lax(Aop , Set)Let m : F / G be a module. If it has a tabulator, T , we can useYoneda to discover what it is. By Yoneda, elements of TA are in/ T which bybijection with natural transformations t : A( , A)T1 are in bijection with modulationsA( , A)IdA( ,A) µA( , A)/F m /Gand as IdA( ,A) A( , idA ), such µ are in bijection with elementsr m(idA ) by Yoneda II. So we defineT (A) m(idA )

By Yoneda II, elements of T (v ) are in bijection with modulationsA( , A)A( ,v ) µ A( , Ā)/T idT /Twhich correspond to commutative tetrahedraA( , A)A( ,v ) A( , Ā)/F? m /Gand this tells us that we must haveT (v ) (Gv m(idA )) m(v ) (m(idĀ ) Fv )It is now straightforward to check that with these definitions, T isindeed the tabulator.

Lax Double CategoriesAlso called fc-multicategories, virtual double categories,multicategories with several objectsLike double categories, except vertical arrows don’t compose.Instead multicells are givenB0f0w1 B1w2 vα .wk A0? A1f1Bkdenotedα : wk , . . . , w2 , w1/v

Lax Double Categories (continued)/ v and multicomposition: forThere are identities 1v : vcompatible/ wiβi : xi1 , . . . , xiliwe are givenα(βk , . . . , β1 ) : x11 , . . . , xklk/vwhich is associative and unitary in the appropriate sense.The composite wk , . . . , w1 exists (or is representable) if there is avertical arrow w and a special multicellι : wk , . . . , w1 wsuch that for every multicell α as above there exists a unique/ v such that ᾱι α.ᾱ : wThe composite wk , · . . . · w1 is strongly representable if ι has astronger universal property for α’s whose domain is a stringcontaining the w ’s as a substring.

MultimodulationsA multimodulationG0t0n1 G1n2 F0 mµ .? F1nk t1 GkIFor each path A0v1 / A1v2 / .µ(vk , . . . , v1 ) : nk vk · . . . · n1 v1vk / Ak , we are given/ m(vk · . . . · v1 )satisfyingI Horizontal naturalityI Left, right, inner equivariance (k 1 conditions)

The Multivariate Yoneda LemmaTheoremFor m : F / G in Lax(Aop , Set) and vk , . . . , v1 a path in A, wehave a bijection between multimodulationsA( , vk ), . . . , A( , v1 )/mand elementsr m(vk · . . . · v1 )CorollaryThe composite A( , vk ) · . . . · A( , v1 ) is represented byA( , vk · . . . · v1 ).TheoremAll composites (k-fold) are representable in Lax(Aop , Set).Remark: Don’t know if they are strongly representable. Don’tthink so, but we conjecture that they are if A satisfies a certainfactorization of cells condition.

The Yoneda EmbeddingY :A/ Lax(Aop , Set)Y (A) A( , A)Y (v ) A( , v )IY is a morphism of lax double categoriesIIt preserves identities and composition (up to iso)IIt is full on horizontal arrowsIIt is full on multicellsIIt is dense

DensityFor F : AopEl(F )/ Set construct the double category of elements of FIObjects are (A, x) with x FAIHorizontal arrows f : (A, x)that F (f )(x 0 ) xII/ A0 such/ (A0 , x 0 ) are f : AVertical arrows (v , r ) : (A, x) / (Ā, x̄) are v : Ar F (v ) such that r0 x and r1 x̄Cells are cells α of A such that F (α)(r 0 ) rThere is a strict double functor P : El(F )TheoremF lim YP /A / Ā and

Example: A Horizontally DiscreteLet A VA for a category A.I/ Set, El(F ) is also horizontallyFor a lax functor F : VAopdiscrete, i.e. VB. Thus F corresponds to an arbitrary/ A (Bénabou)category over A, BIThe representable A( , A) corresponds to A : 1IA natural transformation t : Fover A,B/A/ G corresponds to a functorT A/C

Example (continued)IA module m : FB P / G corresponds to a “profunctor over A”/C i.e. P AICa cell in CatThe representable A( , a) corresponds to1A1 1II/ABaA0/A IdA /AModulations are commutative prismsThus Lax(VAop , Set) ' Cat //A IdA /A

Introduction I Basic tenet: The right framework for two-dimensional category theory is that of double categories I The traditional approach was via: I 2-categories (like Cat) I bicategories (like Prof) I (Weak) double categories are a lot like bicategories I good because much of the well-developed theory o