Monoidal Structures on Higher Categories

Transcription

1 Monoidal Structures on Higer Categories Paul Ziegler Monoidal Structures on Simplicial Categories Let C be a simplicial category, tat is a category enriced over simplicial sets. Suc categories are a model for, -categories. For tem, one can define monoidal structures by enricing te definition of a monoidal structure on a category: Definition.. A symmetric monoidal structure on C consists of a functor of enriced categories : C C C, an object C togeter wit natural isomorpisms X Y Z = X Y Z X = X X = X X Y = Y X for all X, Y, Z C wic satisfy te same coerence axioms as te analogous isomorpisms in a usual symmetric monoidal category. As an example we consider te category C C of cain complexes over a field k. First we define a simplicial enricment: For n let R n be te k-algebra k[x,..., x n ]/ n i= x i. Tis sould be tougt of as te algebraic n-simplex over k. For varying n tese algebras form a simplicial object by associating to a map ρ: [n] [m] te omomorpism R m R n given by x i j : ρj=i x j. For n let Ω n be te algebraic de Ram complex of R n over k: R n Ω R n/k Ω2 R... n/k Ωn R n/k By functoriality of te algebraic de Ram complex for varying n tese form a simplicial object. Now for any X, Y C C we let Hom C C X, Y n := Hom C C X, Y Ω n. By functoriality tese sets form a simplicial set Hom X, Y. We define a simplicial category C C wic as te same objects as C C and tese simplicial sets as om objects. Tis definition is motived by te following fact: Claim.2. For any X, Y C C, elements of Hom C C X, Y correspond to omotopies between two cain maps X Y. Proof. We identify R wit k[x] via x x. Ten Ω can be written as k[x] k[x]dx. Consequently, using te notation X[x] := X k[x] wit k[x] concentrated in degree, a cain map : X Y Ω is given by a map f : X Y [x] of degree and a map X Y [x]dx wic we write as gxdx wit g : X Y [x] of degree. Using te fact tat te differential

2 on Y Ω is given by d Y Ω := d Y id +id d dr, were d dr is te de Ram differential, te condition d Y Ω = d x is equivalent to f dx + d Y f d Y gdx = fd X + gd X dx, were f denotes te formal differential of f wit respect to x. By splitting tis up into graded pieces we get te following two conditions:.3. d Y f = fd X f = gd X + d Y g Te first condition means tat f is a family of cain maps, and we sow now tat te second condition gives us a omotopy between f and f: Tere is a formal k-linear integration map : k[x] k wic sends xn to x n+ /n + = /n +. By tensoring wit te identity on Y we get a map : Y [x] Y. Applying tis operator to. yields f f = gd X + d Y Tus any as above yields a omotopy between f and f. On te oter and, assume tat we ave cain maps f, f : X Y as well as a omotopy g : X Y of degree suc tat f f = gd X +d Y g. Ten if we define f := f x+f x and := f + gdx we see tat conditions.3 and. are satisfied, so tat we obtain Hom X, Y. Tis sows te claim. Similarly one can see tat elements of Hom C C X, Y n for n 2 correspond to omotopies between omotopies between... between cain maps. Now we can define a monoidal structure on C C as follows: On objects we take te usual tensor product C C C C C C. On morpism, we act by te composition Hom C C X, Y Ω n Hom C C Z, W Ω n g. Hom C C X Z, Y W Ω n Ω n Hom C C X Z, Y W Ω n were te first map is given by te usual tensor product in C C and te second map is induced by te wedge product map Ω n Ω n Ω n. 2 Monoidal Structures on n-fold Segal Spaces Giving a monoidal category is te same as a 2-category wit a single object: If one as te latter, te endomorpism of te unique object form a category on wic one as te additional structure given by te composition of endomorpisms. Tis structure amounts to a monoidal structure on te category. Tis motivates te following definition: Definition 2.. A monoidal n-fold Segal space is a n + -fold Segal space X,..., suc tat te space X,..., is connected. Recall: Definition 2.2. Let X,..., be an n-fold Segal space. For objects x, y X,...,, te n - fold Segal space of morpisms from x to y is given by Hom X x, y,..., := {x},..., X,,...,,..., {y}. 2

3 Ten te underlying n-fold Segal space of a monoidal n-fold Segal space X is te space Hom X, for any X,...,. To define symmetric monoidal Segal spaces, we pus te above idea furter: Instead of considering iger categories wit just a single object, we consider iger categories wit a single object, a single morpism, a single 2-morpism and so on up to some level k. Te following example gives an indication tat tis is te rigt idea: Example 2.3. Recall tat a monoid is te same as a category wit a single object. Going one level iger, a commutative monoid is te same as a 2-category wit a single object and a single morpism: Suc a category is determined by G := Homid, id, were is te unique object. On G we ave two different monoid structures G G G given by orizontal and vertical composition. Tey are compatible in te sense tat if one considers G as a monoid troug one of tese two maps, te map G G G giving te oter multiplication is a omomorpism. Tus te Eckmann-Hilton argument implies tat tese two monoid structures coincide and are commutative. In general, it is not known weter going up to some level k as above suffices to give a symmetric monoidal structure on a, n-category. Tus we go all te way up to infinity: Definition 2.. For k, a k-tuply monoidal n-fold Segal space is a n+k-fold Segal space X suc tat for all l k te space X,...,,,... wit exactly l ones is connected. A symmetric monoidal n-fold Segal space consists of a sequence X k k wit X k a k- tuply monoidal n-fold Segal space togeter wit weak equivalences X k Hom X k, for objects X k,...,. Now we sow ow to make te bordism categories into symmetric monoidal spaces in tis sense. Recall: Definition 2.5. Let V be a finite-dimensional vector space. Let n and l n. For every n-tuple k,..., k n, we let PBord l,v n k,...,k n l be te collection of tuples M V R n l, t i... ti k i i=,...,n l satisfying. M is a closed n-dimensional submanifold of V R n l, 2. te composition π : M V R n l R n l is a proper map, 3. for every S {,..., n l} and for every collection {j i } i S, were j i k i, te composition p S : M π R n l R S does not ave t ji i S as a critical value.. for every x M suc tat p {i} x {t i,..., ti n}, te map p {i+,...,n l} is submersive at x. PBord l n = lim PBord l,v V n Remark 2.6. Tis defines an, n l-category and is well-defined even for negative l. Note tat l corresponds to te level we re extending down to, so PBord n = PBord n. We will omit te vector space V in our notation from now on and just remember tat in te end we need to take te limit over vector spaces. Let := V,,..., PBord n,...,. By critical value we mean tat dp S is surjective at te preimages of t ji i S. 3

4 Claim 2.7. Hom PBord n, PBordn. Sketc of proof. Te map sends an element in Hom PBord, k n,...,k n represented by M V R R n, s s, t i t i k i i=,...,n PBord n,k,...,k n to M V }{{ R } R n, t i t i k i i=,...,n PBord n k,...,k n. =Ṽ Conversely, we can send to M V R n, t i t i k i i=,...,n M V R R n, s < s, t i t i k i i=,...,n by te inclusion i : R n R n+, x,..., x n s, x,..., x n, were s < s < s. Tis depends on te coice of s, s R + 2 and s, but te set of coices is contractible. One can ceck easily tat te maps are well-defined up to a contractible coice in te second case. Tis endows PBord n wit te structure of a monoidal n-fold Segal space. Tis procedure can be iterated: Similarly te te above one as PBord l n Hom PBord l+, for l. Tis n endows PBord n wit te structure of a symmetric monoidal n-fold Segal space. Claim 2.8. Te symmetric monoidal structure on PBord endows Cob, wic is equivalent to its omotopy category, wit te symmetric monoidal structure given by disjoint union. Proof. Consider two objects or morpisms M and N in Cob. Tey come from elements M := M V R, t t k N := N V R, t t k in Bord k for k = or. Fixing real numbers s < s < s, tey are sent to elements im := M V R R, s s, t t k in := N V R R, s s, t t k in Hom Bord, k. Recall te composition law in a 2-fold Segal space X: {x} X, {y} {y} X, {z} {x} X, X, {x} X 2, {z} {x} X, {z}. {z}

5 Here te second arrow goes te wrong way but it is a weak equivalence. We coose a differentiable pat in,k from te second element to an element N V R R, s s 2, t t k, by moving N as illustrated in te following drawings i.e. we re just canging te embedding of N into V R R: R s R s s 2 s 2 s s s s s s t t t 2 t 3 t t t t 2 t 3 t Ten im, in is an element in,k wic lies in te image of R t t t t 2 t 3 t t 2,k and terefore can be composed to an element of,k t t 2 t 3 t R t,k by forgetting s. By going back under te bijection of Claim 2.7 and going to te omotopy category of PBord we get te tensor product of M and N. Tis amounts to forgetting s and s and taking te fiber over t resp. [t, t ]. Tus we see tat tis tensor product is te disjoint union of M and N. 5