A universal property of the monoidal 2-category of cospans of finite linear orders and surjections

A univ ersal prop ert y of the monoidal 2-category of cospans of finite linear orders and surjections M. Menni ∗ N. Sabadini † R. F. C. W alters † Octob er 26, 2018 Abstract W e prov e that the mo noidal 2-category of cospans of finite linear orders and surjections is th e u niv ersal monoidal category with an ob ject X with a semigroup and a cosemigroup stru ct ures, where the tw o structures satisfy a certain 2-dimensional separable algebra condition. 1 In tro duction Univ ers al prop erties of cospan-like ca teg ories hav e b een studied in geometry and computer science. F or example, the category of 2-cob ordisms has been shown to b e the universal symmetric monoidal ca tegory with a symmetric F robenius algebra (see [3] for an expos ition and references). F urther, Lack show ed in [4] that the catego ry of co spans of finite sets is the universal symmetric monoidal category with a symmetric separa ble a lgebra. Rosebrug h, Sabadini and W a lters show ed in [6] a similar proper t y of the ca tegory of cospans of finite gr aphs. The aim of this pap er is to mak e a first step in extending these results to the 2 -dimensional str ucture of cospans. T o co ncen trate attention w e avoid symmetries and find that a very natural extensio n of Lack’s work characterizes the 2-categ o ry of cospa ns o f monoto ne surjections betw een tota lly o rdered sets, in the world of not-necessarily s ymmetric monoidal 2-categor ies. Part of the work in volv es describing univ ers al prop erties of bicatego ries of cospans. W ork alo ng these lines has b een a lready done in [2] and [1] a nd it is po ssible that some of the r esults in this paper can b e obtained as a b ypro duct of the work done in the pa pers just men tioned. On the o ther ha nd, our present concern allows us to make so me s implifying a ssumptions and we ha ve decided to prov e the univ ers a l pr operties w e need without app ealing to more gener al w or k. W e hop e that the mor e concrete proo fs presented here will make our work more accesible and, a t the same time, allow to see more clearly into the combin ato rics of the structures in volv ed. Another relev ant w ork is [7], which is how ever concerned with categor ies rather than 2-categ ories. ∗ F unded by Conicet, Universit´ a dell’Insubri a, ANPCyT and Lifia. † F unded by Universit´ a dell’Insubria and the Italian Go vernmen t PRIN pr o ject AR T ( Ana l- isi di sistemi di R iduzione me diante sistemi di T r ansizione ) 1 2 The univ ersal se migroup F or each n in N , let n b e the total orde r { 0 , 1 , . . . , n − 1 } . (So that 0 is the empty total order.) Denote b y Lin the catego ry whose ob jects ar e tota lly or dered sets n = { 0 < . . . < n − 1 } with n in N and whose morphisms ar e mono tone functions betw een these o rders. Ordinal a ddition is a functor + : Lin × Lin → Lin which together with the initial ob ject 0 induces a str ict monoidal category ( Lin , + , 0). This monoidal category is presented in detail in Section VII.5 o f [5] where, in particula r, it is pro ved that (1 , ∇ : 1 + 1 → 1 , ! : 0 → 1 ) is the un iversal monoid in the sense that for any str ic t monoidal category ( C , ⊗ , I ) to gether with a monoid ( C, m : C ⊗ C → C, u : I → C ) in C there exists a unique strict monoidal func- tor ( Lin , + , 0) → ( C , ⊗ , I ) which maps the monoid (1 , ∇ , !) to ( C, m, u ) . (See Prop osition 1 loc. c it.) Now let sLin b e the subcateg ory of Lin determined by the surjective maps. The monoidal structure o n Lin res tricts to sLin and exercise 3(b) of Sec - tion VII .5 of [5] states that ( sLin , + , 0 ) has the following universal prop ert y . A semigr oup in ( C , ⊗ , I ) is defined to be a pair ( C, m : C ⊗ C → C ) such tha t C is a n ob ject o f C and m is associa tiv e. Then (1 , ∇ ) is the univ ersal semigroup. The main results o f this pa per will also use the follo wing re sults concerning pushouts in sLin and their interaction with the tensor +. First let us say that a categor y has strict pushouts if e v ery diagram y ← x → z in the categor y can be completed to a unique pusho ut squar e. Lemma 2.1. sLin has strict pushouts. Pr o of. It is stra igh tforward to see that the pus ho ut (in the catego r y of finite or di- nals and all fun tions) of t wo surjections p ← m → n yields a pushout p → q ← n in the ca tegory of ordina ls and sur jectiv e functions. Among these pusho uts, there is a unique one making the function m → q or der preserving . Also, let us s a y that pushouts inter act with ⊗ in ( C , ⊗ , I ) if it holds that whenever the left and middle s quares b elo w are pushouts then so is the one on the righ t. x f   g / / z p 1   x f ′   g ′ / / z p ′ 1   x ⊗ x ′ f ⊗ f ′   g ⊗ g ′ / / z ⊗ z ′ p 1 ⊗ p ′ 1   y p 0 / / P y p ′ 0 / / P ′ y ⊗ y ′ p 0 ⊗ p ′ 0 / / P ⊗ P ′ Lemma 2.2. Pushouts inter act with + in ( s Lin , + , 0) . Pr o of. Ob vious. 3 Cospans In this s ection let C b e a categ ory with strict pusho uts. Then cospan ( C ) has the structure o f a 2-catego ry and ther e a r e obvious functors y : C → cospan ( C ) and z : C op → cospan ( C ) such that for every C in C , y C = z C . F or ev ery a r row 2 f : C → C ′ in C , y f is the cospan ( f : C → C ′ ← C ′ : id ) and z f is the cospan ( id : C ′ → C ′ ← C : f ). W e wr ite comp osition in ‘Pascal’ notation. So , for example, the comm utative square a bov e translates to the equation ( F 1 α ); ( F 0 β ) = ( F 0 p 0 ); ( F 1 p 1 ). Now let D be a 2-catego ry . Each 2-functor cospan ( C ) → D induces by comp osition functors C → D and C op → D which coincide a t ob jects. In this section w e descr ibe what else is nee de d go the other w ay around. Definition 3. 1. A pair of functors F 0 : C → D and F 1 : C op → D are called c omp atible if 1. they coincide at the level of ob jects (and so we write F 0 X = F X = F 1 X ) 2. for every pushout square as on the left below, X α   β / / B p 1   F X F 0 β / / F B A p 0 / / P F A F 1 α O O F 0 p 0 / / F P F 1 p 1 O O the square on the right ab ov e comm utes. Lemma 3.2. The functors y and z ar e c omp atible. Pr o of. Straigh tforward. Another s imple but importa n t fact is the following. Lemma 3.3. L et F 0 : C → D and F 1 : C op → D b e c omp atible functors and let G : D → E a functor. Then F 0 ; G and F 1 ; G ar e also c omp atible. Notice that in Definition 3.1 we are not requiring D to be a 2-categ ory . F or the next result let cosp an 0 ( C ) denote the underlying ordinary categor y of the 2-catego ry cos pan ( C ). Lemma 3. 4. L et D b e a c ate gory and let F 0 : C → D and F 1 : C op → D b e functors. Then ther e exists a un ique F : cospan 0 ( C ) → D such that y ; F = F 0 and z ; F = F 1 if and only if F 0 and F 1 ar e c omp atible. Pr o of. One direction is trivial by Lemma s 3 .2 and 3.3. O n the other hand, assume that F 0 and F 1 are compatible. The conditions y ; F = F 0 and z ; F = F 1 determine the definition o f F on ob jects. T o deal with 1-cells notice that every cospan p = ( p 0 : A → P ← B : p 1 ) is the res ult of the comp o sition ( y p 0 ); ( z p 1 ). So as F must pr eserve compo sition F p = ( F 0 p 0 ); ( F 1 p 1 ) : F A → F B . So the definition o f F is forced and it rema ins to chec k that so defined F is indeed a functor. Identities are preser ved beca use F 0 and F 1 preserve them. Concer ning comp osition, let f = ( f 0 : X → A ← Y : f 1 ) and g = ( g 0 : Y → B ← Z : g 1 ) b e a pair of compo sable co spans. If we let the following square be the pushout of f 1 and g 0 Y f 1   g 0 / / B p 1   A p 0 / / P 3 then f ; g is the co span ( f 0 ; p 0 ) : X → P ← Z : ( g 1 ; p 1 ). Now calculate using compatibility (and recall that F 1 is co n trav ariant): F ( f ; g ) = F 0 ( f 0 ; p 0 ); F 1 ( g 1 ; p 1 ) = ( F 0 f 0 ); ( F 0 p 0 ); ( F 1 p 1 ); ( F 1 g 1 ) = = ( F 0 f 0 ); ( F 1 f 1 ); ( F 0 g 0 ); ( F 1 g 1 ) = ( F f ); ( F g ) so the result is proved. A t gr eater generality , an a nalogous result is dealt with in Example 5.3 of [4]. W e prefer to b e somewhat more ex plic it as it will allow us to se e more clearly int o ho w to extend the results one dimension up. 4 The extension to 2-cells When considering 2 -categories, ι ( ) denotes the o peratio n pro viding ident ities for ho rizonal and vertical comp osition. That is, 2-c e lls of the form ι f act as units for vertical comp osition and those of the form ι id A act as units for horizontal comp osition. Also, vertical compo sition of 2-cells is denoted b y · and hor izon tal one by ∗ . In all cases w e write comp osition in ‘Pascal’ order . Definition 4. 1. Let D b e a 2-category and F 0 : C → D and F 1 : C op → D b e functors. A c omp atible sele ction of 2-c el ls is a function τ ( ) that as signs to ea c h map f : X → Y in C a tw o cell τ f : id F X ⇒ ( F 0 f ); ( F 1 f ) suc h that: 1. τ id X = ι ( id F X ) 2. τ α ; β = τ α · (( F 0 α ) ∗ τ β ∗ ( F 1 α )) 3. for every pushout in C as below X α   β / / B p 1   A p 0 / / P the follo wing identities ho ld: τ α ∗ ( F 0 β ) = ( F 0 β ) ∗ τ p 1 ( F 1 α ) ∗ τ β = τ p 0 ∗ ( F 1 α ) . The idea is, of course, that a co mpatible selection o f 2-cells is exactly w ha t is needed to extend Lemma 3 .4 to tw o dimensions. But b efore we prove the result le t us prov e a couple of tec hnical lemmas. First no tice that ea ch α : A → B in C induces a 2-cell α : id A ⇒ ( y α ); ( z α ) . Lemma 4.2. The assignment α 7→ α is c omp atible with y and z . Pr o of. Straigh tforward. Now w e need a result a na logous to Lemma 3.3. Lemma 4. 3. L et C b e a c ate gory and let D and E b e 2-c ate gories. Mor e over, let F 0 : C → D and F 1 : C op → D b e c omp atible functors and let G : D → E b e a 2-functor. If τ ( ) is a sele ction of 2-c el ls c omp atible with F 0 and F 1 then Gτ ( ) is c omp atible with F 0 ; G and F 0 ; G . 4 Pr o of. Easy . Lemma 4.4. L et p 0 and p 1 b e the p ushout of α and β as in D efinition 4.1 and let γ = α ; p 0 = β ; p 1 . Then τ α ∗ τ β = τ γ . Pr o of. Calculate: τ α ; p 0 = τ α · (( F 0 α ) ∗ τ p 0 ∗ ( F 1 α )) = τ α · (( F 0 α ) ∗ ( F 1 α ) ∗ τ β ) = = ( τ α ∗ ι X ) · ([( F 0 α ) ∗ ι A ∗ ( F 1 α )] ∗ τ β ) = ( τ α · [( F 0 α ) ∗ τ id A ∗ ( F 1 α )]) ∗ ( ι X · τ β ) = = τ ( α ; id A ) ∗ τ β = τ α ∗ τ β W e can now prove the following. Prop osition 4.5. L et D b e a 2-c ate gory, let F 0 : C → D and F 1 : C op → D b e functors and let τ ( ) b e a function assigning a 2-c el l to e ach map in C . Then the fol lowing ar e e qu iva lent: 1. t her e exists a unique 2-functor F : cosp an ( C ) → D such that the e quations y ; F = F 0 , z ; F = F 1 and τ ( ) = F ( ) hold; (her e and for the r est of the p ap er F ( ) is denoting the op er ation t hat t o e ach 1-c el l f in C assigns the 2-c el l F ( f ) ) 2. F 0 and F 1 ar e c omp atible and τ ( ) is a sele ction of 2-c el ls that is c omp atible with them. Pr o of. Assume that the first item holds . Lemma 3 .4 implies that y ; F = F 0 and z ; F = F 1 are co mpatible. Lemmas 4.2 and 4 .3 show that τ ( ) is a selection of 2-cells that is compatible with them. T o prov e the conv erse notice that we can apply Lemma 3 .4 aga in to co n- clude that ther e exists a unique or dinary functor F : cospan 0 ( C ) → D such that y ; F = F 0 and z ; F = F 1 . So w e ar e left to show that this F extends uniquely to a 2-functor in such a w ay that τ ( ) = F ( ) holds. First assume that the functor F do es extend to a 2-functor and consider an arbitrar y 2-cell α as b elow. A α   X f 0 > > ~ ~ ~ ~ ~ ~ ~ g 0 @ @ @ @ @ @ @ @ Y f 1 ` ` @ @ @ @ @ @ @ g 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ B If w e denote the cospans ( f 0 : X → A ← Y : f 1 ) and ( g 0 : X → A ← Y : g 1 ) b y f : X → Y and g resp ectively then it is easy to see that, for α consider ed as a 2-cell f ⇒ g , α = ( y f 0 ) ∗ α ∗ ( z f 1 ). So F ( α : f ⇒ g ) = F ( y f 0 ) ∗ F α ∗ F ( z f 1 ) and hence the definition of F at the lev el of 2-cells is completely de ter mined b y F ( α : f ⇒ g ) = ( F 0 f 0 ) ∗ τ α ∗ ( F 1 f 1 ). Finally w e are left to prove that if we define F ( α : f ⇒ g ) to b e the 2 -cell ( F 0 f 0 ) ∗ τ α ∗ ( F 1 f 1 ) as ab o ve then w e do obtain a 2- functor. 5 1. F ι f = ι F f F ι f = ( F 0 f 0 ) ∗ τ id A ∗ ( F 1 f 1 ) = ( F 0 f 0 ) ∗ ι id A ∗ ( F 1 f 1 ) = = ( F 0 f 0 ) ∗ ( F 1 f 1 ) = ι ( F 0 f 0 );( F 1 f 1 ) = ι ( F f ) 2. Consider maps α : A → B and β : B → C inducing 2-cells in unique p osi- ble way (starting from f ). Then calculate F ( α · β ) = ( F 0 f 0 ) ∗ τ ( α ; β ) ∗ ( F 1 f 1 ) = = ( F 0 f 0 ) ∗ ( τ α · (( F 0 α ) ∗ τ β ∗ ( F 1 α ))) ∗ ( F 1 f 1 ) = = ( ι ( F 0 f 0 ) · ι ( F 0 f 0 ) ) ∗ ( τ α · (( F 0 α ) ∗ τ β ∗ ( F 1 α ))) ∗ ( F 1 f 1 ) = = [( ι ( F 0 f 0 ) ∗ τ α ) · ( ι ( F 0 f 0 ) ∗ (( F 0 α ) ∗ τ β ∗ ( F 1 α )))] ∗ ( F 1 f 1 ) = = [( F 0 f 0 ) ∗ τ α ) · (( F 0 f 0 ) ∗ ( F 0 α ) ∗ τ β ∗ ( F 1 α ))] ∗ ( ι ( F 1 f 1 ) · ι ( F 1 f 1 ) ) = = [( F 0 f 0 ) ∗ τ α ∗ ( F 1 f 1 )] · [( F 0 f 0 ) ∗ ( F 0 α ) ∗ τ β ∗ ( F 1 α )) ∗ ( F 1 f 1 )] = ( F α ) · ( F β ) 3. Preserv a tion of horizontal comp osition. Supp ose we hav e 2-cells α : f ⇒ f ′ and β : g ⇒ g ′ as in the diagram b elow. A α   B β   X f 0 > > } } } } } } } } f ′ 0 A A A A A A A A Y f 1 ` ` A A A A A A A A f ′ 1 ~ ~ } } } } } } } Y g 0 > > } } } } } } } } g ′ 0 A A A A A A A Z g 1 ` ` A A A A A A A A g ′ 1 ~ ~ } } } } } } } A ′ B ′ In order to calcula te α ∗ β calculate the following pusho ut and resulting map (every small square is a push o ut). Y f 1   g 0 / / B p 1   β / / B ′ r   A α   p 0 / / P α ∗ β @ @ @ @ @ @ α ′   β ′ / / R p ′ 1   A ′ q / / Q p ′ 0 / / P ′ Now use Lemma 4.4 to calculate: F ( α ∗ β ) = F 0 ( f 0 ; p 0 ) ∗ τ α ∗ β ∗ F 1 ( g 1 ; p 1 ) = = ( F 0 f 0 ) ∗ ( F 0 p 0 ) ∗ τ α ′ ∗ τ β ′ ∗ ( F 1 p 1 ) ∗ ( F 1 g 1 ) = = ( F 0 f 0 ) ∗ τ α ∗ ( F 0 p 0 ) ∗ ( F 1 p 1 ) ∗ τ β ∗ ( F 1 g 1 ) = = ( F 0 f 0 ) ∗ τ α ∗ ( F 1 f 1 ) ∗ ( F 0 g 0 ) ∗ τ β ∗ ( F 1 g 1 ) = ( F α ) ∗ ( F β ) 6 5 Adding the monoidal stru cture In this section let ( C , ⊕ , 0) b e a strict monoidal ca tegory with strict pusho uts. W e ha ve already seen that cospan ( C ) is a 2 -category . W e w ant to ‘extend’ the tensor ⊕ on C to one on cospan ( C ). More precisely we will construct a 2-functor ⊕ : cospan ( C ) × cospan ( C ) → cospan ( C ). Lemma 5.1. Ther e ex ists a 2-iso cospan ( C ) × cospan ( C ) → cospan ( C × C ) such that the fol lowing diag r am c ommutes. C × C y ( ( R R R R R R R R R R R R R R y × y / / cospan ( C ) × cospan ( C ) ∼ =   C op × C op z × z o o ∼ =   cospan ( C × C ) ( C × C ) op z o o Pr o of. The obvious one. So we need o nly build a 2-functor ⊕ : cospan ( C × C ) → cospan ( C ) with the right prop erties. Lemma 5.2. The functors C × C ⊕ / / C y / / cospan ( C ) ( C × C ) op ⊕ op / / C op z / / cospan ( C ) ar e c omp atible if and o nly if pushouts inter act with ⊕ in ( C , ⊕ , 0) . Mor e over, in this c ase ( α, β ) 7→ α ⊕ β is a c omp atible sele ction of 2-c el ls. Pr o of. The functors coincide at the level of ob jects. Now, a pushout in C × C is a pair of pushouts α ; p 0 = β ; p 1 and α ′ ; p ′ 0 = β ′ ; p ′ 1 in C . The co mpa tibilit y condition reduces, in this case, to ( α ⊕ α ′ ); ( p 0 ⊕ p ′ 0 ) = ( β ⊕ β ′ ); ( p 1 ⊕ p ′ 1 ) b eing a pusho ut. So the first part of the result follows. F or the second part denote let σ ( α,β ) = α ⊕ β and recall that ( ) is a co mpat- ible selection of 2 -cells (Lemma 4.2). It is easy to show that σ ( id X ,id Y ) = ι id X ⊕ Y . In order to c heck the second conditon calculate: σ ( α,α ′ );( β ,β ′ ) = σ (( α ; β ) , ( α ′ ; β ′ )) = ( α ; β ) ⊕ ( α ′ ; β ′ ) = ( α ⊕ α ′ ); ( β ⊕ β ′ ) = = ( α ⊕ α ′ ) · ( y ( α ⊕ α ′ ) ∗ ( β ⊕ β ′ ) ∗ z ( α ⊕ α ′ )) = σ ( α,α ′ ) · ( y ( α ⊕ α ′ ) ∗ σ ( β ,β ′ ) ∗ z ( α ⊕ α ′ )) In or der to check the final condition as s ume that we have a pushout in C × C as on the left b elow ( α,α ′ )   ( β ,β ′ ) / / ( p 1 ,p ′ 1 )   α ⊕ α ′   β ⊕ β ′ / / p 1 ⊕ p ′ 1   ( p 0 ,p 1 ) / / p 0 ⊕ p 1 / / then the square on the right ab ov e is a pushout b ecause of in teraction. Then calculate: σ ( α,α ′ ) ∗ y ( β ⊕ β ′ ) = α ⊕ α ′ ∗ y ( β ⊕ β ′ ) = y ( β ⊕ β ′ ) ∗ p 1 ⊕ p ′ 1 = y ( β ⊕ β ′ ) ∗ σ ( p 1 ,p ′ 1 ) The equation z ( α ⊕ α ′ ) ∗ σ ( β ,β ′ ) = σ ( p 0 ,p ′ 0 ) z ( α ⊕ α ′ ) is dealt with simila rly . 7 Prop osition 5.3. L et ( C , ⊕ , 0 ) b e a st rict monoidal c ate gory with strict pushouts then the fol lowing ar e e quivalent: 1. t her e exists a unique 2-functor ⊕ : cospan ( C ) × cospan ( C ) → cospan ( C ) such that that the fol lowing dia gr ams c ommut e C × C y × y   ⊕ / / C y   cospan ( C ) × cospan ( C ) ⊕ / / cospan ( C ) C op × C op z × z   ⊕ op / / C op z   cospan ( C ) × cospan ( C ) ⊕ / / cospan ( C ) and such that α ⊕ β = α ⊕ β . 2. Pu sho ut s and ⊕ inter act in ( C , ⊕ , 0) . In t his c ase, the re sult ing structur e ( cospan ( C ) , ⊕ , 0 ) is a monoida l 2-c ate gory and the functors y and z extend to s t rict monoidal ( C , ⊕ , 0) → cospan ( C ) and ( C op , ⊕ , 0) → cospan ( C ) r esp e ctively. Pr o of. Lemma 5.2 to g ether with Pro position 4.5 show that the interaction of pushouts with ⊕ in the s tructure ( C , ⊕ , 0) is equiv ale nt to the existence of a 2-functor ⊕ : cospan ( C × C ) → cospan ( C ) satisfying a num b er of prop erties which, after precompo s ing with the isomorphism of Lemma 5.1, turn out to be exactly the ones in the statement of the present result. The re st of the statement is trivial by str ic tness. In par ticular: Corollary 5.4. ( cospan ( sLin ) , + , 0) is a strict monoida l 2-c ate gory. 5.1 Monoidal 2-functors from cospan ( C ) In this section let ( C , ⊕ , 0) be a s trict monoida l categor y suc h that C has strict pushouts tha t interact with ⊕ . B y Prop osition 5.3 w e ha ve the strict monoidal 2-catego ry ( cos pan ( C ) , ⊕ , 0). Lemma 5. 5. L et ( D , ⊗ , I ) b e a st rict monoida l 2-c ate gory. L et F 0 : C → D and F 1 : C op → D b e c omp atible functors. Final ly let τ ( ) b e a c omp atible sele ct ion of 2-c el ls. Then the induc e d 2-functor F : cosp an ( C ) → D is strict monoidal ( cospan ( C ) , ⊕ , 0) → ( D , ⊗ , I ) if and only if F 0 and F 1 ar e st rict m onoidal and τ α ⊕ β = τ α ⊗ τ β . 8 Pr o of. Assume that F is a s trict monoidal 2-functor. Then clearly F 0 = y ; F is a strict monoidal functor and s imilarly for F 1 . T o chec k the condition on the selection of 2-ce lls just calcula te: τ α ⊕ β = F ( α ⊕ β ) = F ( α ⊕ β ) = F α ⊗ F β = τ α ⊗ τ β Conv ersely , assume that the conditions stated for F 0 , F 1 and τ ( ) hold. Clearly F ( X ⊕ Y ) = F X ⊗ F Y b ecause F and F 0 coincide at the lev el of ob- jects. Now let f = ( f 0 : X → A ← Y : f 1 ) and f ′ = ( f ′ 0 : X ′ → A ′ ← Y ′ : f ′ 1 ) b e 1-cells. Then calculate F ( f ⊕ f ′ ) = F ( f 0 ⊕ f ′ 0 : X ⊕ X ′ → A ⊕ A ′ ← Y ⊕ Y ′ : f 1 ⊕ f ′ 1 ) = = F 0 ( f 0 ⊕ f ′ 0 ); F 1 ( f 1 ⊕ f ′ 1 ) = (( F 0 f 0 ) ⊗ ( F 0 f ′ 0 )); (( F 1 f 1 ) ⊗ ( F 1 f ′ 1 )) = = (( F 0 f 0 ); ( F 1 f 1 )) ⊗ (( F 0 f ′ 0 ); ( F 1 f ′ 1 )) = ( F f ) ⊗ ( F f ′ ) Finally , c onsider a 2-cells α from f and β fro m f ′ and calcula te using that ⊗ is a 2-functor: F ( α ⊕ β ) = F 0 ( f 0 ⊕ f ′ 0 ) ∗ τ α ⊕ β ∗ F 1 ( f 1 ⊕ f ′ 1 ) = = (( F 0 f 0 ) ⊗ ( F 0 f ′ 0 )) ∗ ( τ α ⊗ τ β ) ∗ (( F 1 f 1 ) ⊗ ( F 1 f ′ 1 )) = = (( F 0 f 0 ) ∗ τ α ∗ ( F 1 f 1 )) ⊗ (( F 0 f ′ 0 ) ∗ τ β ∗ ( F 1 f ′ 1 )) = ( F α ) ⊗ ( F β ) 6 Separable semi-algebras In this section we in tro duce the fundamen tal 1-dimensional str ucture to b e studied in the paper . Definition 6.1. Let ( D , ⊗ , I ) b e a strict monoida l categor y . A bi-semigr oup ( X, ∇ , ∆) is an ob ject X in D tog ether with morphisms ∇ : X ⊗ X → X and ∆ : X → X ⊗ X such that ( X , ∇ ) is a semigro up and ( X, ∆) is a ‘co-semigr oup’ in the sense that ∆ is co asso c iativ e. It is useful to ha ve a graphical no tation for ex pressions inv olving ∇ and ∆. A couple of ex amples will suffice to introduce it. C o nsider a bi-semigro up ( X, ∇ , ∆). The iden tity on X will b e denoted b y a straight line. On the other hand, id X ⊗ i d X will be denoted by tw o parallel horizontal lines. Mo r e imp or- tantly , ∇ will b e denoted as in the left diagram b elo w / /   ? ? ? ? ? ? ? / / / / / / ? ?          ? ? ? ? ? ? ? / / ? ?        / / 9 and ∆ will b e denoted as on the right ab ov e . So tha t, fo r e x ample, the diagr am / /   ? ? ? ? ? ? ? / / / /   ? ? ? ? ? ? ? / / ? ?          ? ? ? ? ? ? ? / / ? ?        / / / / ? ?        represents the expression ( id ⊗ ∇ ); ∇ ; ∆. Definition 6.2. Let ( D , ⊗ , I ) be a strict monoidal categ ory . A sep ar able semi- algebr a is a bi-semigro up ( D, ∇ , ∆) in D s uc h that: 1. (Separable) ∆; ∇ = id : D → D D ∆   id # # G G G G G G G G G D ⊗ D ∇ / / D 2. (F rob enius) (∆ ⊗ i d D ); ( id D ⊗ ∇ ) = ∇ ; ∆ = ( i d D ⊗ ∆); ( ∇ ⊗ id D ) D ⊗ D ∆ ⊗ id   ∇ / / D ∆   D ⊗ D id ⊗ ∆   ∇ / / D ∆   D ⊗ D ⊗ D id ⊗∇ / / D ⊗ D D ⊗ D ⊗ D ∇⊗ id / / D ⊗ D Graphically , separ abilit y can b e expressed as saying that the following tw o diagrams / /   ? ? ? ? ? ? ? / / ? ?          ? ? ? ? ? ? ? / / / / / / ? ?        are equa l. The authors hav e found it useful to think of separ abilit y as allo wing to p op the ‘bubble’ on the left. On the other hand, F rob enius says that the tw o diagrams b elow / / / /   ? ? ? ? ? ? ? / / ? ?          ? ? ? ? ? ? ? / / / /   ? ? ? ? ? ? ? / / ? ?        / / / / ? ?          ? ? ? ? ? ? ? / / ? ?        / / 10 which r epresent (∆ ⊗ id D ); ( id D ⊗ ∇ ) and ( id D ⊗ ∆); ( ∇ ⊗ id D ) resp ectively , are equal to / /   ? ? ? ? ? ? ? / / / / ? ?          ? ? ? ? ? ? ? / / ? ?        / / which represe nts ∇ ; ∆. Lemma 6.3. If we denote the c osp an ( id : 1 → 1 ← 1 + 1 : ∇ ) by ∆ : 1 → 1 + 1 then (1 + 1 , ∇ , ∆) is a sep ar able semialgeb r a in cospan 0 ( sLin ) . Pr o of. This is a simple exercis e left for the r eader. But it is imp ortan t to men- tion now that this separa ble semi-algebr a plays an impo rtan t role in everything that follo ws. 7 A univ ersal prop ert y of (1 + 1 , ∇ , ∆) The universal prop ert y we dis cuss in this section was indep enden tly observed by Lack on the one hand [4] and b y Rosebrug h, Saba dini and W a lter s o n the other [6]. It is impo rtan t to reca ll (see Lemma in Chapter VI I.5 o f [5]) the fact that every surjectio n in sLin can be factored in a unique way as a comp osition (satisfying certain conditions) of ma ps ( id + ∇ + i d ). (The conditions ensuring uniqueness will not be relev a n t for us here .) Let F 0 : C → D and F 1 : C op → D b e functor s agr e e ing on ob jects. W e say that F 0 and F 1 indulge a commutaiv e sq uare α   β / / p 1   p 0 / / if, just as in Definition 3.1, ( F 1 α ); ( F 0 β ) = ( F 0 p 0 ); ( F 1 p 1 ). (Notice that there is a handedness in this notio n. The fact that the functors indulge the squar e ab o ve does not seem to imply that it indulges the square obtained b y flipping the same sq ua re along its diag onal. Tha t is ( F 1 α ); ( F 0 β ) = ( F 0 p 0 ); ( F 1 p 1 ) does not seem to imply ( F 1 β ); ( F 0 α ) = ( F 0 p 1 ); ( F 1 p 0 ).) Lemma 7.1. If F 0 and F 1 indulge the two squar es b elow sep ar ately α   β / / α ′   β ′ / / p ′ 1   p 0 / / p ′ 0 / / then they indulge the r e ctangle. Pr o of. T rivial. 11 There is also a ‘vertical’ version which we shall use when necessar y . Lemma 7.2 (See [4]) . L et F 0 : sLi n → D and F 1 : sLin op → D b e monoidal functors agr e eing on obje cts. Then they ar e c omp atible if and only if they indulge the fol lowing pushout squar es 1 + 1 ∇   ∇ / / 1 id   1 + 1 + 1 ∇ +1   1+ ∇ / / 1 + 1 ∇   1 + 1 + 1 ∇ +1 / / 1+ ∇   1 + 1 ∇   1 id / / 1 1 + 1 ∇ / / 1 1 + 1 ∇ / / 1 Pr o of. One direction is tr ivial. Consider a pushout of the form below k f   g / / n p 1   m p 0 / / t If f or g ar e identities then the squar e is trivia lly indulged. So we can as- sume that f a nd g a re non-trivial compos itions. Say , f = ( l 0 + ∇ + l 1 ); f ′ and g = ( l ′ 0 + ∇ + l ′ 1 ); g ′ . The idea o f the pro of is to split the pushout into four smaller pushouts as below. k l 0 + ∇ + l 1   l ′ 0 + ∇ + l ′ 1 / / l ′ 0 + 1 + l ′ 1   g ′ / / n   l 0 + 1 + l 1 f ′   / /   / /   m / / / / t The inductive hypothesis can dea l with tw o b ottom squares and the top right one. If we can prov e that the top left one is indulged then Lemma 7.1 implies that the big pushout is indulged. So, co ncerning the top left pushout, the following things can happ en: 1. l 0 + 2 ≤ l ′ 0 , that is, f ’s first ∇ is strictly to the left of g ’s, 2. l 0 + 1 = l ′ 0 , that is, f ’s first ∇ “ touc hes” g ’s but f and g do not sta r t in the same wa y , 3. l 0 = l ′ 0 , that is, f and g start in the same way , 4. l 0 = l ′ 0 + 1, analogo us to the first item but to the r igh t, 5. l 0 ≥ l ′ 0 + 2, analogo us to the second item. Consider the first ca s e. Let k = k 0 + 2 + k 1 + 2 + k 2 , f = ( k 0 + ∇ + k ′ 1 ); f ′ and g = ( k ′ 0 + ∇ + k 2 ); g ′ where k ′ 0 = k 0 + 2 + k 1 and k ′ 1 = k 1 + 2 + k 2 . Then 12 the pushout is calculated as below: k 0 + 2 + k 1 + 2 + k 2 k 0 + ∇ + k ′ 1   k ′ 0 + ∇ + k 2 / / k 0 + 2 + k 1 + 1 + k 2 k 0 + ∇ + k 1 +1+ k 2   k 0 + 1 + k 1 + 2 + k 2 k 0 +1+ k 1 + ∇ + k 2 / / k 0 + 2 + k 1 + 1 + k 2 and it is indulged be c a use it is the sum of trivial pushouts (that are indulged) and mor e over F 0 and F 1 are monoida l so the tensor of indulged squares is indulged. (Should we sta te a Lemma analogous to Lemma 7.1 but for tensor ing squares?.) F or the second case let k = k 0 + 1 + 1 + 1 + k 1 , f = ( k 0 + ∇ + 1 + k 1 ); f ′ and g = ( k 0 + 1 + ∇ + k 1 ); g ′ . In this case the pusho ut in question is calculated as follo ws k 0 + 1 + 1 + 1 + k 1 k 0 + ∇ +1+ k 1   k 0 +1+ ∇ + k 1 / / k 0 + 1 + 1 + k 1 k 0 + ∇ + k 1   k 0 + 1 + 1 + k 1 k 0 + ∇ + k 1 / / k 0 + 1 + k 1 Again, the pushout is a sum o f tw o squar es that are tr iv ially indulged and one that is indulged b y assumption. T o deal with the third case let k = k 0 + 1 + 1 + k 1 , f = ( k 0 + ∇ + k 1 ); f ′ and g = ( k 0 + ∇ + k 1 ); g ′ . In this case the pus ho ut in question is calculated as follows k 0 + 1 + 1 + k 1 k 0 + ∇ + k 1   k 0 + ∇ + k 1 / / k 0 + 1 + k 1 id   k 0 + 1 + k 1 id / / k 0 + 1 + k 1 Again, the pushout is a sum o f tw o squar es that are tr iv ially indulged and one that is indulged b y assumption. The re maining tw o ca ses are analogous. Corollary 7. 3 (See [4] and [6]) . L et ( D , ⊗ , I ) b e a st rict monoidal c ate gory with a sep ar able semialgebr a ( D , ∇ , ∆) . Then ther e ex ists a unique strict monoidal functor ( cospan 0 ( sLin ) , + , 0) → ( D , ⊗ , I ) mapping (1 + 1 , ∇ , ∆) to ( D , ∇ , ∆) . Pr o of. The semigr oup ( D, ∆) is ess en tially the same thing as a strict mono idal functor F 0 : sLin → D (mapping ∇ to ∇ ) while the co -semigroup ( D , ∆) is essentially the same thing a s a stric t monoidal F 1 : sLin op → D (mapping ∇ to ∆). As F 0 and F 1 are strict mono idal and coincide on 1, they agree on o b jects. So we are left to prov e that F 0 and F 1 are compa tible. By Lemma 7.2 it is enough to c heck that F 0 and F 1 indulge thr ee pushout squar es. But notice that indulgence of these squares is equiv alent to Separability and F rob enius. 13 7.1 An alternative pro of of Corollary 7.3 Corollar y 7.3 can be in terpreted as saying that the free monoidal catego r y with a separ able se mi- algebra is cospan 0 ( sLin ). In this s hort s ection we sk etch a ‘graphical’ pro of which mak es a lot more evident the r e la tion betw een the result and the calculation of co limits. What is the free monoidal categ ory ge nerated by ∇ and ∆ sub ject to the equations in Definition 6.2? First, given only ∇ we can build dia grams of the form / / B B B B B B B B / / / / > > | | | | | | | | . . . . . . . . . / /   ? ? ? ? ? ? ? / / B B B B B B B B / / / / ? ?        / / > > | | | | | | | | The asso ciative law says that the order of applying ∇ s do es not matter so with only ∇ s we ca n can build exactly surjective monotone functions. Similarly , using only ∆ we can produce exa ctly the rev ers es of monoto ne sur jections. So using bo th we ca n pro duce cospans of monotone surjections. But per haps we can pro duce more? The ans w er is no . If in an expres sion o f ∇ s and ∆s a ∆ o ccurs to the left of a ∇ then only 4 cases ca n occur. The first one is when the 14 ∆ and the ∇ do not interact: / / / / > > | | | | | | | | B B B B B B B B / / . . . . . . . . . / /   ? ? ? ? ? ? ? / / / / ? ?        The se c ond case is given b y the bubble a s drawn after Definition 6.2. The third and four th cases are given b y the tw o diag rams representing the expressio ns in the F rob enius condition a nd dra wn b elow the bubble after Definition 6.2. In all four cases, the ∆s can b e mov ed to the rig h t of the ∇ s. In the fir s t case trivially , in the second by p oping the bubble (sepa r abilit y) and in the third and fourth ca ses by F ro benius. So the free mo noidal catego ry with a sepa rable semi-algebr a is cospan 0 ( sLin ). Remark 7.4. It is important to notice that the pro cess of moving ∆s to the right of ∇ s is rea lly calculating the pusho ut inv olved in the compos ition of cospans. In Section 8 .3 we add 2-dimensiona l data so tha t the free monoidal 2-categ ory on this data is cospan ( s Lin ). But first let us extract some mor e information from Le mma 7.2. 7.2 Monoidal 2-functors from cospan ( sLin ) Here we c har a cterize when t wo functors from sLin to a 2-category are compat- ible. Let us say that a selection of 2-cells τ ( ) indulges a square α ; p 0 = β ; p 1 if the t wo eq uations in Definit ion 4.1 relating the square and τ ( ) hold. Lemma 7.5. L et F 0 : sLin → D and F 1 : sLin op → D b e c omp atible monoidal functors. L et τ ( ) b e a sele ction of 2-c el ls satisfying the first two c onditions of Definition 4.1. Then τ ( ) is a c omp atible sele ction of 2-c el ls if and only if it indulges the squar es in the st atemen t of L emma 7.2. Pr o of. Analogous to that of Lemma 7 .2. 8 Adjoin t bi-semigroups In this section we intro duce what w e b eliev e are the rig h t lifting to 2-dimensions of the F r obenius and separ a bilit y conditions. 15 Definition 8.1. Let ( D , ⊗ , I ) be a s trict monoidal 2-c a tegory . An adjoint bi- semigr oup ( X , ∇ , ∆ , η , ǫ ) is a bi-semigro up ( X , ∇ , ∆) in D 0 together with 2-cells η : id X ⊗ X ⇒ ∇ ; ∆ and ǫ : ∆; ∇ ⇒ id X witnessing that ∇ ⊣ ∆. W e now lift the conditions of separa bilit y and F r obenius to the level of adjoint bi-semigroups . W e first deal with F rob enius. 8.1 F robenius adjoin t bi-semigroup s In order to justify the definition consider first the following res ult. Lemma 8.2. L et X = ( X, ∇ , ∆ , η , ǫ ) b e an adjoi nt bi-semigr oup such that t he (1-dimensional) structure ( X , ∇ , ∆) satisfies F r ob eniu s as a bi-semigr oup. Then the fol lowing two items ar e e quivalent: 1. t he mates of the asso ciative laws X ⊗ X ⊗ X ∇⊗ X   X ⊗∇ / / X ⊗ X ∇   X ⊗ X ⊗ X X ⊗∇   ∇⊗ X / / X ⊗ X ∇   X ⊗ X ∇ / / X X ⊗ X ∇ / / X ar e identity 2-c el ls 2. ( η ⊗ X ) ∗ ( X ⊗ ∇ ) = ( X ⊗ ∇ ) ∗ η and ( X ⊗ η ) ∗ ( ∇ ⊗ X ) = ( ∇ ⊗ X ) ∗ η . Pr o of. Consider the mate of one of the asso ciative laws X ⊗ X id ' ' N N N N N N N N N N N ∆ ⊗ X / / X ⊗ X ⊗ X ∇⊗ X   X ⊗∇ / / X ⊗ X ∇   id % % K K K K K K K K K K X ⊗ X ∇ / / X ∆ / / X ⊗ X where inside the triang les we hav e the 2- cells ǫ ⊗ X : (∆ ⊗ X ); ( ∇ ⊗ X ) ⇒ id and η : id ⇒ ∇ ; ∆. Notice that the outside of this diagr am is one of the F rob e- nius la ws. Now assume that ( η ⊗ D ) ∗ ( D ⊗ ∇ ) = ( D ⊗ ∇ ) ∗ η a nd calculate: [(∆ ⊗ X ) ∗ ( X ⊗ ∇ ) ∗ η ] · [( ǫ ⊗ X ) ∗ ∇ ∗ ∆] = = [(∆ ⊗ X ) ∗ ( η ⊗ X ) ∗ ( X ⊗ ∇ )] · [( ǫ ⊗ X ) ∗ (∆ ⊗ X ) ∗ ( X ⊗ ∇ )] = = [((∆ ∗ η ) ⊗ X ) ∗ ( X ⊗ ∇ )] · [(( ǫ ∗ ∆) ⊗ X ) ∗ ( X ⊗ ∇ )] = = [((∆ ∗ η ) ⊗ X ) · (( ǫ ∗ ∆) ⊗ X )] ∗ ( X ⊗ ∇ ) = = [((∆ ∗ η ) · ( ǫ ∗ ∆)) ⊗ X ] ∗ ( X ⊗ ∇ ) = (∆ ⊗ X ) ∗ ( X ⊗ ∇ ) which shows that the mate is the identit y 2 -cell. Similarly if one assumes that the other equation holds then the co rresp onding mate is the iden tit y . 16 Conv ersely , assume that the mates of a ssocia tivit y a re iden tity 2 -cells and contemplate the follo wing diagram: X 3 X ⊗∇   id ! ! C C C C C C C C X 2 id ! ! C C C C C C C C X ⊗ ∆ / / X 3 X ⊗∇   ∇⊗ X / / X 2 ∇   id ! ! C C C C C C C C X 2 ∇ / / X ∆ / / X 2 where the triangles ar e filled with the 2-cells X ⊗ η , X ⊗ ǫ and η . Pasting 2- cells one o btains that ( X ⊗ η ) ∗ ( ∇ ⊗ X ) = ( ∇ ⊗ X ) ∗ η . Indeed, one can calcula te: ( X ⊗ η ) ∗ ( ∇ ⊗ X ) = = [( X ⊗ η ) ∗ ( ∇ ⊗ X )] · [( X ⊗ ∇ ) ∗ ( X ⊗ ∆) ∗ ( ∇ ⊗ X ) ∗ η ] · [( X ⊗ ∇ ) ∗ ( X ⊗ ǫ ) ∗ ∇ ∗ ∆] = = [( ∇ ⊗ X ) ∗ η ] · [( X ⊗ η ) ∗ ( ∇ ⊗ X ) ∗ ∇ ∗ ∆] · [( X ⊗ ∇ ) ∗ ( X ⊗ ǫ ) ∗ ∇ ∗ ∆] = = [( ∇ ⊗ X ) ∗ η ] · [( X ⊗ η ) ∗ ( X ⊗ ∇ ) ∗ ∇ ∗ ∆] · [( X ⊗ ( ∇ ∗ ǫ )) ∗ ∇ ∗ ∆] = = [( ∇ ⊗ X ) ∗ η ] · [( X ⊗ ( η ∗ ∇ )) ∗ ∇ ∗ ∆] · [( X ⊗ ( ∇ ∗ ǫ )) ∗ ∇ ∗ ∆] = = [( ∇ ⊗ X ) ∗ η ] · [[( X ⊗ ( η ∗ ∇ )) · ( X ⊗ ( ∇ ∗ ǫ ))] ∗ ( ∇ ∗ ∆)] = = ( ∇ ⊗ X ) ∗ η The proo f of the other equa tion is analogous. Because o f this, w e find it natural to in tro duce the following definition. Definition 8.3. Let X = ( X , ∇ , ∆ , η , ǫ ) b e an adjoin t bi-semigroup such that the 1- dimensional str ucture ( X, ∇ , ∆) satisfies F rob enius as a bi-semigroup. W e say that X sa tisfies ∇ -F r ob enius if the equiv alent conditions of Lemma 8.2 hold. It is interesting a nd useful to notice that the equalities in the s econd item of Lemma 8.2 ca n b e thought o f as r ewrite rules. Indeed, notice that in the notation w e ha ve used fo r expressions with ∆s and ∇ s , the left 2- cell of the firs t equation of item 2 has domain the left hand diagr am below / /   ? ? ? ? ? ? ? / / / / ? ?          ? ? ? ? ? ? ? / /   ? ? ? ? ? ? ? / / ? ?        / /   ? ? ? ? ? ? ? / / / / / / ? ?        / / ? ?        and co domain the righ t hand diagram below. In other words, the 2-cell pinches the firs t tw o strings. The r eader is invited to draw the other 2 -cells and exercise in applying the pinc hing and p oping r ules. Back to the lifting of the F rob enius condition, it must be mentioned that one c a n prov e the following in a wa y ana logous to Lemma 8.2. 17 Lemma 8.4. L et X = ( X, ∇ , ∆ , η , ǫ ) b e an adjoi nt bi-semigr oup such that t he (1-dimensional) structure ( X , ∇ , ∆) satisfies F r ob eniu s as a bi-semigr oup. Then the fol lowing two items ar e e quivalent: 1. t he mates of X ∆   ∆ / / X ⊗ X ∆ ⊗ X   X ∆   ∆ / / X ⊗ X X ⊗ ∆   X ⊗ X X ⊗ ∆ / / X ⊗ X ⊗ X X ⊗ X ∆ ⊗ X / / X ⊗ X ⊗ X ar e identity 2-c el ls 2. (∆ ⊗ D ) ∗ ( D ⊗ η ) = η ∗ (∆ ⊗ D ) and ( D ⊗ ∆) ∗ ( η ⊗ D ) = η ∗ ( D ⊗ ∆) . So, just as in Definition 8.3 we say that X satisfies ∆ -F r ob enius if the equiv- alent conditions of Lemma 8.4 hold. Definition 8.5. Let X = ( X , ∇ , ∆ , η , ǫ ) b e an adjoin t bi-semigroup such that ( X, ∇ , ∆) satisfies F rob enius as a bi-semigr oup. W e say that X satisfies F r ob e- nius if it satisfies b oth ∇ -F rob enius and ∆-F rob enius. 8.2 Separable adjoin t bi-sem igroups In this se c tio n we in tro duce the notion o f separable adjoin t bi-se mig roup and show tha t for , these semi-groups, 1-dimensiona l F r obenius implies 2- dimens ional F rob enius. Definition 8.6. W e say that an adjoint bi- semigroup ( X , ∇ , ∆ , η , ǫ ) is sep ar able if ( X , ∇ , ∆) is separ able as a bi-semigroup and more o ver ǫ = ι id X . Notice that in a separa ble bi-semigroup, η ∗ ∇ = ι ∇ and ∆ ∗ η = ι ∆ . Lemma 8.7. L et X = ( X, ∇ , ∆ , η , ǫ ) b e an adjoi nt bi-semigr oup such that t he 1-dimensional structure ( X , ∇ , ∆) satisfies F r ob enius. If X is sep ar able t hen X satisfies F r ob enius. Pr o of. Under separability , the triangular identit ies w itnes sing that ∇ ⊣ ∆ b e- come η ∗ ∇ = ι ∇ and ∆ ∗ η = ι ∆ . T o prov e ∇ -F rob enius w e need to show that the mates of a s socia tivity are ident ity 2-cells. In particular, we need to show that [(∆ ⊗ X ) ∗ ( X ⊗ ∇ ) ∗ η ] · [( ǫ ⊗ X ) ∗ ∇ ∗ ∆] is ι ∇ ;∆ . Under se pa rabilit y , we need only pr o ve that [(∆ ⊗ X ) ∗ ( X ⊗ ∇ ) ∗ η ] · ι ∇ ;∆ = (∆ ⊗ X ) ∗ ( X ⊗ ∇ ) ∗ η is the iden tity 2-cell ι ∇ ;∆ . As ( X , ∇ , ∆) satisfies F rob enius a nd ∆ ∗ η = ι ∆ , w e can c a lculate: (∆ ⊗ X ) ∗ ( X ⊗ ∇ ) ∗ η = ∇ ∗ ∆ ∗ η = ∇ ∗ ∆ 18 so, indeed, the mate of as sociativity is the identit y . The other condition is dealt with in an analog ous w ay so X satis fie s ∇ -F rob enius. T o prov e ∆-F rob enius one uses the same idea. F o r ex ample, one of the conditions is proved as follows: [ η ∗ ( X ⊗ ∆) ∗ ( ∇ ⊗ X )] · [ ∇ ∗ ∆ ∗ ( ǫ ⊗ X )] = η ∗ ∇ ∗ ∆ = ∇ ∗ ∆ so, altogether, X satisfies F rob enius. Since the co unit is the iden tity , separa ble adjoint bi-semigro ups will usually be denoted b y ( X, ∇ , ∆ , η ). 8.3 A univ ersal prop erty of cospan ( sLin ) In this se c tio n w e pr o ve a universal pro perty of cospan ( sLin ) as a monoidal 2-catego ry . F or brevit y let us int ro duce the following definition. Definition 8.8. Let ( D , ⊗ , I ) b e a strict mo noidal 2-catego ry . A Como-algebr a is a separable adjoint bi- semigroup ( D , ∇ , ∆ , η ) such that ( D , ∇ , ∆) s a tisfies F rob enius. Alternatively , one can say that a Co mo-algebra is a structur e ( D , ∇ , ∆ , η ) such that ( D , ∇ , ∆) is a separable s emi-algebra in D 0 and η : id D ⊗ D ⇒ ∇ ; ∆ is a 2 -cell satisfying η ∗ ∇ = ι ∇ and ∆ ∗ η = ι ∆ (essentially saying ∇ ⊣ ∆). By Lemma 8.7 every Como-a lgebra satisfies F rob enius. Notice als o that 1 + 1 has an obvious Como- a lgebra s tructure: just take η = ∇ . Prop osition 8.9. L et ( D , ⊗ , I ) b e a strict monoidal 2-c ate gory with a Como- algebr a ( D , ∇ , ∆ , η ) . Then ther e exists a un ique strict monoida l 2-functor ( cospan ( sLin ) , + , 0) → ( D , ⊗ , I ) mapping (1 + 1 , ∇ , ∆ , η ) to ( D , ∇ , ∆ , η ) . Pr o of. By Co rollary 7.3 b e hav e a strict monoidal functor cospan 0 ( sLin ) → D mapping the universal separable semialgebra to the one in D . In order to ex tend this functor to a strict monoidal 2-functor w e need a compatible se lection of 2 - cells satisfying the co nditions o f Lemma 5.5 . That is, a compatible selection τ ( ) satisfying τ f + g = τ f ⊗ τ g . Now, Definition 4 .1 forces τ ( ) on identities and comp osition. As every map in sLin is built from ∇ and using tensor and comp osition, a selection of 2 -cells as the one we need is determined by its v alue τ ∇ : id D ⊗ D ⇒ ∇ ; ∆. Let us call this selection η . When do es the selection o f s uc h a 2-cell induces a co mpatible selection? The a nsw er is given b y Lemma 7.5. But indulgence of the three distinguished pushouts is equiv a le n t to the v a lidit y of the follo wing equations: 1. η ∗ ∇ = ι ∇ and ∆ ∗ η = ι ∆ (essentially saying ∇ ⊣ ∆) 2. ( η ⊗ D ) ∗ ( D ⊗ ∇ ) = ( D ⊗ ∇ ) ∗ η and (∆ ⊗ D ) ∗ ( D ⊗ η ) = η ∗ (∆ ⊗ D ) 3. ( D ⊗ η ) ∗ ( ∇ ⊗ D ) = ( ∇ ⊗ D ) ∗ η and ( D ⊗ ∆) ∗ ( η ⊗ D ) = η ∗ ( D ⊗ ∆) The first item is ex a ctly separability while the other t wo items are e xactly F rob enius (Definition 8.5). But a Como-a lgebra is separa ble by definition and it alw ays satisfies F rob enius by Lemma 8.7. So the res ult follows. 19 9 Como-units Let i Li n b e the full s ubcategor y of Lin de ter mined by injectiv e monotone func- tions. The monoidal structur e ( Lin , + , 0) r estricts to iLin a nd the inclusion iLin → Lin is strict monoidal. By results in [5 ], all maps in iLin are built out of ! : 0 → 1. Definition 9. 1. A unit in a monoidal ca tegory ( D , ⊗ , I ) is an ob ject X in D equipp e d with a map u : I → X . The ob ject 1 in i Lin together with ! : 0 → 1 is the universal o b ject with unit. Lemma 9.2. The c ate gory iLin has strict pul lb acks and they inter act with + . Lemma 9. 3. Every pul lb ack in iLi n is a c omp osition of trivial pul lb acks and pul lb acks of the form 0 id   id / / 0 !   0 ! / / 1 Pr o of. Similar to the pr oof of Lemma 7.2. Definition 9.4. Let ( D , ⊗ , I ) be a monoidal category . A split-unit is a str ucture ( X, s : I → X , r : X → I ) such that ( X , s : I → X ) is a unit and r : X → I is such that s ; r = id I . The ob ject 1 ha s a unique split-unit structure in the monoidal categor y ( cospan 0 ( iLin op ) , + , 0). Let us denote it b y (1 , ! : 0 → 1 , ? : 1 → 0). Corollary 9 .5. F or every strict m onoidal c ate gory ( D , ⊗ , I ) and every split-u nit ( X, s : I → X , r : X → I ) in it, ther e exists a u nique strict monoidal functor cospan 0 ( iLin op ) → D mapping (1 , ! : 0 → 1 , ? : 1 → 0 ) to ( X, s, r ) . Pr o of. The map r : X → I induces a functor strict monoidal F 0 : iLin op → D while the map s : I → X induces a strict monoidal F 1 : iLin → D . The functor s clearly agr ee on ob jects. By Lemma 9.3, the functors a r e compatible if and o nly if they indulge the pushout 1 ! op   ! op / / 0 id op   0 id op / / 0 in iLin op . This means e xactly that s ; r = i d . Definition 9.6. Let ( D , ⊗ , I ) b e a monoidal 2 -category . A Como-unit is a split-unit ( X , s : I → X , l : X → I ) to gether with a 2-cell η : ι X ⇒ l ; s suc h that l ⊣ s w ith unit η and counit ι id I . The split-unit (1 , ! : 0 → 1 , ? : 1 → 0) is a Como -unit when co ns idered as a n ob ject in ( cospan ( i Lin op ) , + , 0). W e denote the unit of the adjunction ? ⊣ ! b y η . In a wa y analo gous to Propo sition 8.9 we obta in the follo wing cor ollary . 20 Corollary 9.7. F or every strict monoidal 2-c ate gory ( D , ⊗ , I ) and Como-unit obje ct ( X, s, l , η ) in it, ther e exists a u nique strict monoida l 2-functor ( cospan ( iLin op ) , + , 0) → ( D , ⊗ , I ) mapping (1 , ! : 0 → 1 , ? : 1 → 0 , η ) to ( X , s, l , η ) . Pr o of. By Coro lla ry 9.5 we ha ve a unique strict mono idal functor ( cospan 0 ( iLin op ) , + , 0) → ( D 0 , ⊗ , I ) mapping the split-unit (1 , ! : 0 → 1 , ? : 1 → 0) to ( X , s, l ). In order to ex tend this functor to a strict monoidal 2-functor we need a selection τ ( ) of 2- c ells satisfying τ f + g = τ f ⊗ τ g . Such a selection o f 2 - cells is deter mined by its v alue τ (! op :1 → 0) : ι X ⇒ l ; s . Naturally , w e define τ (! op :1 → 0) = η . Is the resulting selec- tion compatible? W e need to check that τ ( ) indulges all pushout sq uares in iLin op . B y Lemma 9.3 we need o nly chec k that it indulges the squa re in the statement of that lemma. But this says ex actly that η ∗ l = l and s ∗ η = s . 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