AAN EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES
MINKYU KIMA bstract . Let A be a small abelian category. The purpose of this paper is to introduce andstudy a category A which implicitly appears in construction of some TQFT’s where A is de-termined by A . If A is the category of abelian groups, then the TQFT’s obtained by Dijkgraaf-Witten theory of abelian groups or Turaev-Viro theory of bicommutative Hopf algebras factorthrough A up to a scaling. In this paper, we go further by giving a su ffi cient condition for an A -valued Brown functor to extend to a homotopy-theoretic analogue of A -valued TQFT forarbitrary A . The results of this paper and our subsequent paper reproduces TQFT’s obtainedby DW theory and TV theory. C ontents
1. Introduction 1Acknowledgements 32. A general construction of (co)span categories 32.1. Definitions 32.2. Dagger structure 42.3. Symmetric monoidal category structure 43. A cospan category of pointed finite CW-spaces 54. A preorder of (co)span diagrams in an abelian category 74.1. Basic properties 74.2. Proof of Proposition 4.7 105. A (co)span category of an abelian category 115.1. Definitions 115.2. An isomorphism between cospan and span categories 126. Spanical and cospanical extensions 146.1. Brown functor 146.2. Proof of the second part of Theorem 1.1 156.3. Proof of the first part of Theorem 1.1 17References 191. I ntroduction
Let A be a small abelian category. The purpose of this paper is to introduce and study acategory A which implicitly appears in construction of some TQFT’s where A is determinedby A . If A “ Ab , i.e. the category of abelian groups, then the TQFT’s obtained by Dijkgraaf-Witten theory [2] [3] of abelian groups or Turaev-Viro theory [8] [1] of bicommutative Hopfalgebras factor through A up to a scaling. In this paper, we go further by giving a su ffi cientcondition for an A -valued Brown functor to extend to a homotopy-theoretic analogue of A -valued TQFT for arbitrary A . The results of this paper and our subsequent paper reproduceTQFT’s obtained by DW theory and TV theory. a r X i v : . [ m a t h . A T ] M a y MINKYU KIM
The category A is a dagger symmetric monoidal category. The category A is bijectivelyand faithfully embedded into A . We denote by ι A : A Ñ A the embedding functor. SeeRemark 5.15 for details.A d -dimensional A -valued Brown functor E is a functorial assignment of an object E p K q of A to a pointed finite CW-space K with dim K ď d . It assigns the direct sum in A to thewedge sum of spaces and satisfies the Mayer-Vietoris axiom (see Definition 6.4).For d P N Y t8u , a cospan diagram of pointed finite CW-spaces ´ K f Ñ L f Ð K ¯ is a d -dimensional if the dimension of the boundaries K , K and the bulk L is lower than or equalto p d ´ q and d respectively. Note that for n ď d , an n -dimensional cobordism induces such a d -dimensional cospan diagram. As a homotopy-theoretic analogue of cobordism categories,we introduce a d -dimensional cospan category of pointed finite CW-paces. We denote thecategory by Cosp »ď d p CW fin ˚ q (see Definition 3.10).Note that the homotopy category Ho p CW fin ˚ , ďp d ´ q q of pointed finite CW-spaces K withdim K ď p d ´ q is naturally a subcategory of Cosp »ď d p CW fin ˚ q . Then both of the sus-pension and the proper restriction of a d -dimensional A -valued Brown functor extend to Cosp »ď d p CW fin ˚ q : Theorem 1.1.
For d P N Y t8u , let E : Ho ` CW fin ˚ , ď d ˘ Ñ A be a d-dimensional A -valuedBrown functor. (1) There exists a unique dagger-preserving symmetric monoidal extension of ι A ˝ E ˝ Σ to Cosp »ď d p CW fin ˚ q . Here, Σ : Ho ´ CW fin ˚ , ďp d ´ q ¯ Ñ Ho ` CW fin ˚ , ď d ˘ denotes the suspensionfunctor. (2) There exists a unique dagger-preserving symmetric monoidal extension ι A ˝ E ˝ i to Cosp »ď d p CW fin ˚ q . Here, i : Ho ´ CW fin ˚ , ďp d ´ q ¯ Ñ Ho ` CW fin ˚ , ď d ˘ denotes the inclusionfunctor. The proof appears in section 6.We give some examples of -dimensional A -valued Brown functors in Example 6.6, 6.7.The categories Cosp »ď d p CW fin ˚ q and A are constructed from a general formulation on (co)spancategories (see section 2).In a forthcoming paper, we give a construction of a Vec ssd k -valued projective TQFT froma
Hopf bc , vol k -valued Brown functor based on the results in this paper. For a field k , we de-note by Vec ssd k the category of vector spaces over k equipped with a symmetric self-duality. Hopf bc , vol k denotes the category of bicommutative Hopf algebras with a finite volume [4]. Inparticular, we show that a Hopf bc k -valued homology theory induces a (possibly, empty) familyof projective TQFT’s. We also give some computations of the obstruction classes induced bythe scalars appearing from compositions. It gives a generalization of Dijkgraaf-Witten theoryof abelian groups and Turaev-Viro theory of bicommutative Hopf algebras.This paper is organized as follows. In section 2, we give a way to construct a (daggersymmetric monoidal) category whose morphisms consist of some equivalence classes of acospan diagram. We apply the results to obtain the category Cosp »ď d p CW fin ˚ q in section 3. Insection 4, we introduce a preorder of (co)span diagrams in an abelian category. In section5, we define Cosp « p A q , Sp « p A q via the general formulation in section 2. Furthermore, insubsection 5.2, we give an isomorphism between Cosp « p A q , Sp « p A q . In section 6, we proveTheorem 1.1. A finite CW-space is a topological space having, but not equipped with, a finite CW-complex structure. It preserves the composition up to a scaling
N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 3 A cknowledgements The author was supported by FMSP, a JSPS Program for Leading Graduate Schools inthe University of Tokyo, and JPSJ Grant-in-Aid for Scientific Research on Innovative AreasGrant Number JP17H06461.2. A general construction of ( co ) span categories Definitions.Definition 2.1.
Let C be a small category. A diagram Λ “ ´ x f Ñ y f Ð x ¯ is called a cospandiagram in C . Let Cosp p C q be the set consisting of cospan diagrams in the category C . Wedefine the source and target of Λ by s p Λ q def . “ x and t p x Ñ y Ð x q def . “ x . Definition 2.2.
Let C be a small category. Denote by ι : Mor p C q Ñ Cosp p C q the inducedcospan defined by ι p f q “ ´ x f Ñ y Id y Ð y ¯ . For an object x in C , we write ι p x q def . “ ι p Id x q where Id x : x Ñ x is the identity on x . Definition 2.3.
Consider an equivalence relation „ on the set Cosp p C q . The equivalencerelation „ is compatible with the source and target maps if Λ „ Λ for Λ , Λ P Cosp p C q implies s p Λ q “ s p Λ q and t p Λ q “ t p Λ q where s , t appear in Definition 2.1. Definition 2.4.
A subset U Ă Cosp p C q is admissible if the following conditions hold.(1) We have s p U q “ t p U q . Denote by B U def . “ s p U q “ t p U q .(2) For x , y P B U and a morphism f : x Ñ y , we have ι p f q P U . Definition 2.5.
Let „ be an equivalence relation on Cosp p C q compatible with the sourceand target maps. Let U be an admissible subset of Cosp p C q . Denote by U t ˆ s U “tp Λ , Λ q ; t p Λ q “ s p Λ qu . A map ˝ : U t ˆ s U Ñ U is a weak composition with respectto the pair p„ , U q if the following conditions hold.(1) For p Λ , Λ q , p Λ , Λ q P U t ˆ s U , if Λ „ Λ and Λ „ Λ , then we have Λ ˝ Λ „ Λ ˝ Λ .(2) For p Λ , Λ q , p Λ , Λ q P U t ˆ s U , we have p Λ ˝ Λ q ˝ Λ „ Λ ˝ p Λ ˝ Λ q .(3) For Λ P U , we have Λ ˝ ι p s p Λ qq „ Λ .(4) For Λ P U , we have ι p t p Λ qq ˝ Λ „ Λ .(5) For x , y , z P B U and morphisms f : x Ñ y , g : y Ñ z in C , we have ι p g q ˝ ι p f q „ ι p g ˝ f q . Definition 2.6.
Define a small category
Cosp „ U , ˝ p C q . Its object set is given by B U and mor-phism set is given by the quotient set U { „ . The source map s : p U { „q Ñ B U is inducedby the source map s : Cosp p C q Ñ Ob j p C q . The target map t : p U { „q Ñ B U is definedanalogously. The source map and target map are well-defined since U is admissible. Thecomposition of Cosp „ U , ˝ p C q is induced by the map ˝ : U t ˆ s U Ñ U as r Λ s ˝ r Λ s def . “ r Λ ˝ Λ s .The category Cosp „ U , ˝ p C q is well-defined since ˝ is a weak composition with respect to thepair p„ , U q . Definition 2.7.
Denote by C | B U the full subcategory of C whose objects are B U . We definea functor ι : C | B U Ñ Cosp „ U , ˝ p C q which is the identity on objects and assigns r ι p f qs to amorphism f of C | B U . Remark 2.8.
Note that Definition 2.7 is well-defined functor due to the second part of Defi-nition 2.4 and the latter three conditions in Definition 2.5.
MINKYU KIM
Dagger structure.
Recall that for a category D , a dagger operation on D is given byan involutive functor : : D op Ñ D which is an identity on objects. A category equipped witha dagger operation is a dagger category . Definition 2.9.
Let Λ “ ´ x f Ñ y f Ð x ¯ be a cospan in C . We define a dagger cospan Λ : by Λ : “ ´ x f Ñ y f Ð x ¯ : def . “ ´ x f Ñ y f Ð x ¯ . (1)The assignment of dagger cospan to cospans gives an involution on the set Cosp p C q . Thedagger operation on Cosp p C q is normal in the sense that ι p x q : “ ι p x q . Definition 2.10.
We say that p„ , U , ˝q is a triple if „ is an equivalence relation on Cosp p C q compatible with the source and target maps, U is an admissible subset of Cosp p C q and ˝ is aweak composition with respect to the pair p„ , U q . Definition 2.11.
A triple p„ , U , ˝q is compatible with the dagger operation on Cosp p C q if thefollowing conditions hold.(1) For Λ , Λ P Cosp p C q , an equivalence relation Λ „ Λ implies Λ : „ Λ : .(2) If Λ P U , then Λ : P U .(3) For Λ , Λ P U with t p Λ q “ s p Λ q , we have p Λ ˝ Λ q : „ Λ : ˝ Λ : .(4) For x , y P B U and an isomorphism f : x Ñ y in C , we have ι p f ´ q „ ι p f q : . Proposition 2.12.
If a triple p„ , U , ˝q is compatible with the dagger operation on the set Cosp p C q , then the dagger operation on Cosp p C q induces a dagger operation on the cate-gory Cosp „ U , ˝ p C q . The functor ι in Definition 2.7 assigns a unitary isomorphism to everyisomorphism in C | B U .Proof. For a morphism r Λ s of Cosp „ U , ˝ p C q , i.e. an equivalence class of Λ P U , let r Λ s : def . “r Λ : s . It induces an involutive functor : on the category Cosp „ U , ˝ p C q . It is immediate that thefunctor : is a dagger operation on the category Cosp „ U , ˝ p C q . (cid:3) Symmetric monoidal category structure.Definition 2.13.
Let Λ “ ´ x f Ñ y f Ð x ¯ , Λ “ ˆ x f Ñ y f Ð x ˙ be cospans in C . Wedefine a tensor product of cospans Λ b Λ by Λ b Λ def . “ ˆ x b x f b f Ñ y b y f b f Ð x b x ˙ (2) Definition 2.14.
A triple p„ , U , ˝q (see Definition 2.10) is compatible with the symmetricmonoidal category structure on C if the following conditions hold.(1) For Λ , Λ , Λ , Λ P Cosp p C q , equivalence relations Λ „ Λ and Λ „ Λ imply Λ b Λ „ Λ b Λ .(2) If Λ , Λ P U , then Λ b Λ P U .(3) For Λ , Λ , Λ , Λ P Cosp p C q with t p Λ q “ s p Λ q and t p Λ q “ s p Λ q , we have p Λ ˝ Λ q b p Λ ˝ Λ q „ p Λ b Λ q ˝ p Λ b Λ q . Proposition 2.15.
Let C be a symmetric monoidal category. If a triple p„ , U , ˝q is compatiblewith the symmetric monoidal structure, then the symmetric monoidal categoy structure of C induces a symmetric monoidal category structure on Cosp „ U , ˝ p C q . The functor ι : C | B U Ñ Cosp „ U , ˝ p C q in Definition 2.7 is enhanced to a symmetric monoidal functor. N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 5
Proof.
We define a tensor product on
Cosp „ U , ˝ p C q by r Λ s b r Λ s def . “ r Λ b Λ s . It gives awell-defined functor b : Cosp „ U , ˝ p C q ˆ Cosp „ U , ˝ p C q Ñ Cosp „ U , ˝ p C q due to Definition 2.14.We set the unit object of the SMC C as a unit object of Cosp „ U , ˝ . The associator, unitors andsymmetry on C induce them of Cosp „ U , ˝ p C q by the last part of Definition 2.5. For example,denote by a x , y , z : p x b y q b z Ñ x b p y b z q the associator of the SMC C . Then the inducedmorphism r ι p a x , y , z qs : p x b y q b z Ñ x b p y b z q for x , y , z P B U is an associator. In fact,the pentagon diagram with respect to r ι p a x , y , z qs is immeditate from that of the associator a x , y , z due to the last part of Definition 2.5. By the construction of symmetric monoidal categorystructure, the functor ι : C | B U Ñ Cosp „ U , ˝ p C q in Definition 2.7 is enhanced to a symmetricmonoidal functor in the obvious way. (cid:3) Proposition 2.16.
Let C be a symmetric monoidal category. If a triple p„ , U , ˝q is compatiblewith the dagger operation on Cosp p C q and the symmetric monoidal category structure of C ,then the symmetric monoidal category Cosp „ U , ˝ p C q with the dagger operation is a daggersymmetric monoidal category.Proof. It su ffi ces to prove that(1) pr Λ s b r Λ sq : “ r Λ s : b r Λ s : .(2) r ι p a x , y , z qs : “ r ι p a x , y , z qs ´ where a x , y , z is the associator of C .(3) r ι p l x qs : “ r ι p l x qs ´ where l x is the left unitor of C .(4) r ι p r x qs : “ r ι p r x qs ´ where r x is the right unitor of C .(5) r ι p s x , y qs : “ r ι p s x , y qs ´ where s x , y is the symmetry of C .These are immediate from their definitions. In particular, the claims from (2) to (5) followsfrom the fourth condition in Definition 2.11. (cid:3)
3. A cospan category of pointed finite
CW- spaces
In this section, we define a cospan category of pointed finite CW-spaces
Cosp »ď d p CW fin ˚ q by using the preliminaries in section 2. It is a homotopy theoretical analogue of cobordismcategories. In the set-theoretical sense, the category of all of topological spaces (or finiteCW-spaces) is not small. In this paper, we fix a small category of topological spaces (or finiteCW-spaces) which is categorically equivalent with the whole. Definition 3.1.
Denote the category of pointed topological spaces as
Top ˚ . Let Λ “ ´ K f Ñ L f Ð K ¯ , Λ “ ´ K f Ñ L f Ð K ¯ be cospans in Top ˚ with the same sources and targets. A map g : L Ñ L is a homotopy equivalence from Λ to Λ if(1) g : L Ñ L are pointed homotopy equivalences.(2) There exists a pointed homotopy g ˝ f » f .(3) There exists a pointed homotopy g ˝ f » f .If there exists a homotopy equivalence from Λ to Λ , then we write Λ » Λ .The homotopy equivalence relation » of cospans is an equivalent relation on Cosp p Top ˚ q .The equivalence relation » is compatible with the source and target maps. Definition 3.2.
For a pointed map f : K Ñ L , we define a mapping cylinder Cyl p f q by apointed space, Cyl p f q def . “ Cyl p K q ł f L . (3)In other words, it is the quotient pointed space by identifying r k , s P Cyl p K q with f p k q P L .By the canonical inclusion L Ñ Cyl p f q , we consider L as a subspace of Cyl p f q . We denote by i f : K Ñ Cyl p f q the inclusion where we identify K with K ^ t u ` Ă K ^ r , s ` “ Cyl p K q . MINKYU KIM
Definition 3.3.
Let Λ “ ´ K f Ñ L f Ð K ¯ , Λ “ ´ K g Ñ L g Ð K ¯ be cospans such that t p Λ q “ s p Λ q . We define a cospan Λ ˝ Λ in Top ˚ by ´ K k Ñ L k Ð K ¯ . Here, L isthe quotient space of Cyl p f q Ž Cyl p g q by identifying r k , s P Cyl p K q Ă Cyl p f q with r k , s P Cyl p K q Ă Cyl p g q . The quotient space L is equipped with the obvious base-point. The pointed maps k , k are given by compositions k “ ´ K f Ñ L i Ñ L ¯ and k “ ´ K g Ñ L j Ñ L ¯ where i , j are the canonical inclusions. The assignment p Λ , Λ q ÞÑ Λ ˝ Λ determines a map ˝ : U t ˆ s U Ñ U where U “ Cosp p Top ˚ q . Definition 3.4.
The symmetric monoidal category structure on the category
Top ˚ by thewedge sum induces a wedge sum of cospans in Top ˚ . In other words, for cospans Λ “ ´ K f Ñ L f Ð K ¯ , Λ “ ˆ K f Ñ L f Ð K ˙ , we define the wedge sum of cospans by Λ ł Λ def . “ ˆ K ł K f _ f Ñ L ł L f _ f Ð K ł K ˙ . (4) Proposition 3.5.
Let U “ Cosp p Top ˚ q . Recall the definition of triples in Definition 2.10. (1) The map ˝ : U t ˆ s U Ñ U in Definition 3.3 is a weak composition with respect to thepair p» , U q . (2) The triple p» , U , ˝q is compatible with the dagger operation on Cosp p Top ˚ q . (3) The triple p» , U , ˝q is compatible with the symmetric monoidal structure on Top ˚ .Proof. The proof is elementary so that we leave the proof to the readers. In particular, thefirst part is related with the homotopy invariance of homotopy colimits. (cid:3)
Definition 3.6.
Let X “ t X K u be a family of a pointed finite CW-complex structure X K foreach pointed finite CW-space K . Let d P N Y t8u . Denote by U d , X Ă Cosp p CW fin ˚ q a subsetconsisting of cospans Λ “ ´ K f Ñ L f Ð K ¯ in CW fin ˚ satisfying following conditions.(1) dim K ď p d ´ q , dim K ď p d ´ q and dim L ď d .(2) There exists a pointed finite CW-complex structure X L on L such that f and f arecellular with respect to the complex structures X K , X K , X L . Proposition 3.7.
The subset U d , X Ă Cosp p CW fin ˚ q is admissible in the sense of Definition 2.4.Moreover, B U d , X is the set of pointed finite CW-spaces with ďp d ´ q .Proof. It is immediate from definitions. (cid:3)
Proposition 3.8.
The map ˝ in Definition 3.3 induces a map ˝ : U t ˆ s U Ñ U whereU “ U d , X . The induced map gives a weak composition with respect to the pair p» , U d , X q .Proof. Consider p Λ , Λ q P U t ˆ s U . If Λ “ ´ K f Ñ L f Ð K ¯ and Λ “ ˆ K f Ñ L f Ð K ˙ ,then there exist pointed finite CW-complex structures X L , X L on L , L such that f , f , f , f are cellular maps with respect to the complex structure X K , X K , X K , X L , X L . Hence, themapping cylinders Cyl p f q , Cyl p f q have canonical pointed finite CW-complex structures, sothe glued space L “ Cyl p f q Ž K Cyl p f q does. Denote by X L the complex structure of L .Recall Definition 3.3. Then k : K Ñ L and k : K Ñ L are cellular maps with respect to X K , X K , X L , hence Λ ˝ Λ P U d , X “ U . It completes the proof. (cid:3) In the following proposition, recall the notations in Definition 2.6.
N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 7
Proposition 3.9.
For another family of pointed finite CW-complex structures Y “ t Y K u , wehave a canonical isomorphism of categories, Cosp » U d , X , ˝ p CW fin ˚ q – Cosp » U d , Y , ˝ p CW fin ˚ q . (5) Proof.
Note that B U d , X “ B U d , Y since B U d , X is the set of pointed finite CW-spaces with dim ďp d ´ q by Proposition 3.7. Hence, the object set of Cosp » U d , X , ˝ p CW fin ˚ q and Cosp » U d , Y , ˝ p CW fin ˚ q coincide with each other.On the other hand, the set U d , X and U d , Y essentially coincide with each other in the follow-ing sense : For any cospan Λ lying in U d , X , there exists a cospan Λ P U d , Y such that Λ » Λ .It follows from the definitions of U d , X and U d , Y .We construct a functor F X , Y : Cosp » U d , X , ˝ p CW fin ˚ q Ñ Cosp » U d , Y , ˝ p CW fin ˚ q as follows. Thefunctor F X , Y assigns K itself to an object K of Cosp » U d , X , ˝ p CW fin ˚ q . The functor F X , Y assigns amorphism r Λ s in Cosp » U d , Y , ˝ p CW fin ˚ q to a morphism r Λ s in Cosp » U d , X , ˝ p CW fin ˚ q where Λ » Λ .It is a well-defined functor due to the first part of Proposition 3.5.It follows from definitions that the functor F X , Y is an inverse functor of F Y , X . It completesthe proof. (cid:3) Definition 3.10.
Let X “ t X K u be a family of a pointed finite CW-complex structure X K foreach pointed finite CW-space K . We define a dagger symmetric monoidal category, Cosp »ď d p CW fin ˚ q def . “ Cosp » U d , X , ˝ p CW fin ˚ q . (6)By Proposition 3.9, the definition is independent of the choice of X up to a canonical isomor-phism. 4. A preorder of ( co ) span diagrams in an abelian category Basic properties.Definition 4.1.
For cospans Λ “ ´ A f Ñ B f Ð A ¯ and Λ “ ˆ A f Ñ B f Ð A ˙ , we denoteby Λ ĺ Λ if A “ A , A “ A and there exists a monomorphism g : B Ñ B in A such that g ˝ f “ f and g ˝ f “ f . For such a monomorphism g , we say that the monomorphism g gives Λ ĺ Λ . Proposition 4.2.
The relation ĺ gives a preorder of cospans in A , i.e. we have (1) Λ ĺ Λ . (2) Λ ĺ Λ and Λ ĺ Λ implies Λ ĺ Λ .Proof. The first part is proved by the fact that the identity morphism is a monomorphism.The second part is proved by the fact that the composition of monomorphisms is a monomor-phism. In fact, if a monomorphism g gives Λ ĺ Λ and a monomorphism g gives Λ ĺ Λ ,then the monomorphism g ˝ g gives Λ ĺ Λ . (cid:3) Lemma 4.3.
Let Λ , Λ be cospans in A . Then the following two conditions are equivalent. (1) There exists a lower bound of t Λ , Λ u . (2) There exists an upper bound of t Λ , Λ u .Proof. Let Λ “ ´ A f Ñ B f Ð A ¯ , Λ “ ˆ A f Ñ B f Ð A ˙ . With out loss of generality, weassume that A “ A and A “ A .We prove the second part starting from the first part. Suppose that a cospan Λ is a lowerbound of t Λ , Λ u where Λ “ ´ A g Ñ B g Ð A ¯ . Let m : B Ñ B be a monomorphismgiving Λ ĺ Λ and m : B Ñ B be a monomorphism giving Λ ĺ Λ . In other words, MINKYU KIM we have m ˝ g “ f , m ˝ g “ f , m ˝ g “ f , m ˝ g “ f . Define B to be thecokernel of u “ p m ‘ p´ m qq ˝ ∆ B : B Ñ B ‘ B . Put g : B Ñ B as the composition of B i Ñ B ‘ B cok p u q Ñ B , and g : B Ñ B as the composition of B i Ñ B ‘ B cok p u q Ñ B . Bydefinition, we have g ˝ m “ g ˝ m .(7) B B B ‘ B B C B ´ m u cok p u q p ee i g Note that g is a monomorphism since m is a monomorphism : If a morphism e : C Ñ B satisfies g ˝ e “
0, then there exists a unique morphism e : C Ñ B such that u ˝ e “ i ˝ e where i : B Ñ B ‘ B is the inclusion. Since we have 0 “ p ˝ i ˝ e “ p ˝ u ˝ e “ p´ m q ˝ e and m is a monomorphism, we obtain e “
0. Hence, g is a monomorphism.Similarly, the morphism g is a monomorphism since m is a monomorphismDefine h : A Ñ B and h : A Ñ B by h “ g ˝ f and h “ g ˝ f . Then we have g ˝ f “ h . In fact, g ˝ f “ g ˝ m ˝ g “ g ˝ m ˝ g “ g ˝ f “ h . Likewise g ˝ f “ h holds.Above all, the monomorphism g gives Λ ĺ Λ and the monomorphism g gives Λ ĺ Λ .The cospan Λ is an upper bound of t Λ , Λ u .We prove the first part starting from the second part. Suppose that a cospan Λ is an upperbound of t Λ , Λ u . Let Λ “ ´ A h Ñ B h Ð A ¯ . Let g : B Ñ B be a monomorphismgiving Λ ĺ Λ and g : B Ñ B be a monomorphism giving Λ ĺ Λ . In other words,we have g ˝ f “ h , g ˝ f “ h , g ˝ f “ h and g ˝ f “ h . Put B to be the kernel of v “ ∇ B ˝ p g ‘ p´ g qq : B ‘ B Ñ B . Put m : B Ñ B as the composition B ker p v q Ñ B ‘ B p Ñ B ,and m : B Ñ B as the composition B ker p v q Ñ B ‘ B p Ñ B . By definitions, we have g ˝ m “ g ˝ m .(8) B B B ‘ B B C B mker p v q vp l k i g The morphisms m is a monomorphism : Suppose that m ˝ l “ l : C Ñ B .Since m ˝ l “ p ˝ p ker p v q ˝ l q , there exists a unique morphism k : C Ñ B such that i ˝ k “ ker p v q ˝ l . Since g ˝ k “ v ˝ i ˝ k “ v ˝ ker p v q ˝ l “ g is a monomorphism,we obtain k “
0, hence ker p v q ˝ l “
0. Since ker p v q is a monomorphism, we obtain l “ m is a monomorphism. N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 9
Define g : A Ñ B by ker p v q˝ g “ p f ‘ f q˝ ∆ A . Then by definition we have m ˝ g “ f .In fact, we have m ˝ g “ p ˝ ker p v q ˝ g “ p ˝ p f ‘ f q ˝ ∆ A “ f . Likewise, we obtain m ˝ g “ f . Define g : A Ñ B by ker p v q ˝ g “ p f ‘ f q ˝ ∆ A . Then we also obtain m ˝ g “ f and m ˝ g “ f .Let Λ “ ´ A g Ñ B g Ð A ¯ . Then the monomorphism m gives Λ ĺ Λ and the monomor-phism m gives Λ ĺ Λ by the previous discussion. The cospan Λ is a lower bound of t Λ , Λ u . (cid:3) Proposition 4.4.
The preorder ĺ is compatible with the biproduct of cospans in A . In otherwords, for cospans Λ , Λ and cospans Λ , Λ , if Λ ĺ Λ and Λ ĺ Λ , then we have Λ ‘ Λ ĺ Λ ‘ Λ .Proof. Let g , h be a monomorphism which gives Λ ĺ Λ and Λ ĺ Λ respectively. Notethat the biproduct of monomorphisms is a monomorphism. Then the biproduct g ‘ h gives Λ ‘ Λ ĺ Λ ‘ Λ . (cid:3) Definition 4.5.
Let Λ , Λ be cospans in A with Λ “ ´ A f Ñ B f Ð A ¯ and Λ “ ˆ A f Ñ B f Ð A ˙ .We define a composition cospan Λ ˝ Λ “ ´ A g Ñ C g Ð A ¯ where C is the cokernel of thecomposition p f ‘ p´ f qq ˝ ∆ A : A Ñ B ‘ B and the morphisms g , g are given by thefollowing compositions, g “ A f Ñ B i Ñ B ‘ B , (9) g “ A f Ñ B i Ñ B ‘ B . (10) Proposition 4.6.
The preorder ĺ is compatible with the composition of cospans in A . Inother words, for cospans Λ , Λ from A to A and cospans Λ , Λ from A to A , if Λ ĺ Λ and Λ ĺ Λ , then we have Λ ˝ Λ ĺ Λ ˝ Λ .Proof. Let Λ “ ´ A f Ñ B f Ð A ¯ , Λ “ ´ A f Ñ B f Ð A ¯ , Λ “ ´ A f Ñ B f Ð A ¯ , Λ “ ´ A f Ñ B f Ð A ¯ . Suppose that monomorphisms g : B Ñ B and g : B Ñ B give Λ ĺ Λ and Λ ĺ Λ respectively.Denote by C , C the cokernels of u “ p f ‘ p´ f qq ˝ ∆ A : A Ñ B ‘ B and v “ p f ‘ p´ f qq ˝ ∆ A : A Ñ B ‘ B respectively. The biproduct g ‘ g : B ‘ B Ñ B ‘ B induces a morphism h : C Ñ C such that h ˝ cok p u q “ cok p v q ˝ p g ‘ g q . Themorphism h is a monomorphism. In fact, consider concentrated chain complexes D ‚ “ ´ ¨ ¨ ¨ Ñ Ñ A u Ñ B ‘ B Ñ Ñ ¨ ¨ ¨ ¯ , D “ ´ ¨ ¨ ¨ Ñ Ñ A v Ñ B ‘ B Ñ Ñ ¨ ¨ ¨ ¯ where the 0-th components are B ‘ B and B ‘ B respectively. The identity on A andthe biproduct g ‘ g gives a chain homomorphism j ‚ : D ‚ Ñ D which is a monomorphism.Then by the long exact sequence, we obtain an exact sequence H p D { D ‚ q B Ñ H p D ‚ q H p j ‚ q Ñ H p D q . We have H p D ‚ q “ Cok p u q “ C and H p D q “ Cok p v q “ C . Under the identifica-tions, we have H p j ‚ q “ h . Note that H p D { D ‚ q “
0. The morphism h is a monomorphism.The monomorphism h gives Λ ˝ Λ ĺ Λ ˝ Λ . Define m : B Ñ C by the composition B ã Ñ B ‘ B cok p u q Ñ C and m : B Ñ C by the composition B ã Ñ B ‘ B cok p v q Ñ C . Thenwe obtain h ˝ m “ m ˝ g . In particular, we obtain h ˝ p m ˝ f q “ p m ˝ f q . Similarly, wedefine m : B Ñ C by the composition B ã Ñ B ‘ B cok p u q Ñ C and m : B Ñ C by thecomposition B ã Ñ B ‘ B cok p v q Ñ C . Then we also have h ˝ p m ˝ f q “ m ˝ f . Above all,the monomorphism h gives Λ ˝ Λ ĺ Λ ˝ Λ . (cid:3) Proposition 4.7.
Consider a commutative diagram in A . (11) CB B A A A g hf f f f Let Λ “ ´ A f Ñ B f Ð A ¯ , Λ “ ˆ A f Ñ B f Ð A ˙ and Λ “ ˆ A g ˝ f Ñ C h ˝ f Ð A ˙ becospans in A . If the square diagram in (11) is exact, then we have Λ ˝ Λ ĺ Λ . Proof of Proposition 4.7.Definition 4.8.
Let A be an abelian category. A square diagram is a quadruple p g , f , g , f q ofmorphisms in A such that g , f and g , f are composable respectively. Consider a followingsquare diagram l in A .(12) B DA C gf f g The morphism f induces a morphism k l : Ker p f q Ñ Ker p g q . The morphism g induces amorphism c l : Cok p f q Ñ Cok p g q . The square diagram is exact if the morphism k l is anepimorphism and the morphism c l is a monomorphism. Definition 4.9.
Let l be a square diagram in A as (12). We define a chain complex C p l q by A u l Ñ B ‘ C v l Ñ D (13)where u l def . “ p f ‘ p´ f qq ˝ ∆ A and v l def . “ ∇ D ˝ p g ‘ g q . Proposition 4.10.
Consider a square diagram l in A . (14) B DA C gf f g (1) The square diagram l is exact. (2) The induced chain complex C p l q is exact.Proof. Denote by C ‚ the chain complex induced the morphism f : A Ñ B , i.e. C “ A , C “ B , C q “ ff erential is f . Similarly, we denote by D ‚ the chaincomplex induced by the morphism g . The morphisms g , f induce a chain homomorphism r E ‚ : C ‚ Ñ D ‚ . Then the mapping cone complex r E ‚ of the chain homomorphism r E ‚ consistsof (1) E “ A , E “ B ‘ C , and E “ D ,(2) the second di ff erential is u l : A Ñ B ‘ C ,(3) the first di ff erential is v l : B ‘ C Ñ D . N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 11
By the mapping cone exact sequence, we obtain an exact sequence :(15) H p C ‚ q H p D ‚ q H p r E ‚ q H p C ‚ q H p D ‚ q Ker p f q Ker p g q H p r E ‚ q Cok p f q Cok p g q H p r E ‚ q H p r E ‚ q k l c l By the exactness, the induced morphisms k l and c l are an epimorphism and a monomor-phism respectively if and only if H p r E ‚ q is a zero object. By the definition of the chaincomplex r E ‚ , H p r E ‚ q is a zero object if and only if the chain complex C p l q induced by thesquare diagram l is exact. (cid:3) Proof of Proposition 4.7.
Denote by l the square diagram in (11), i.e. l is a commu-tative subdiagram consisting of g , h , f , f . Suppose that the square diagram l is exact. ByProposition 4.10, the induced chain complex C p l q (see Definition 4.9) is an exact sequence.Equivalently, the induced morphism k l : Cok p u l q Ñ C is a monomorphism. Note that thecomposition of cospans Λ ˝ Λ is given by Λ ˝ Λ “ p A Ñ Cok p u l q Ð A q by definitions.Then the monomorphism k l gives Λ ˝ Λ ĺ Λ in the sense of Definition 4.1. It completesthe proof. 5. A ( co ) span category of an abelian category Definitions.Definition 5.1.
We define a relation « of cospan diagrams in A . We define Λ « Λ if thereexists an upper bound of t Λ , Λ u with respect to the preorder ĺ in Definition 4.1. By Lemma4.3, Λ « Λ is equivalent with the condition that there exists a lower bound of t Λ , Λ u . Proposition 5.2.
The relation defined in Definition 5.1 is an equivalence relation.Proof.
Since Λ ĺ Λ by Proposition 4.2, we have Λ « Λ .Suppose that Λ « Λ . Then there exists an upper bound Λ of a set t Λ , Λ u “ t Λ , Λ u .Hence, we have Λ « Λ .Suppose that Λ « Λ , Λ « Λ . Then we have upper bounds Λ , Λ of t Λ « Λ u , t Λ « Λ u respectively. Then the cospan Λ is a lower bound of t Λ , Λ u . By Lemma 4.3, the set t Λ , Λ u has an upper bound Λ . Since Λ ĺ Λ ĺ Λ and Λ ĺ Λ ĺ Λ , we have Λ ĺ Λ and Λ ĺ Λ . Hence, Λ is an upper bound of t Λ , Λ u . We obtain Λ « Λ . (cid:3) Proposition 5.3.
The equivalence relation « is compatible with the source and target mapsin the sense of Definition 2.3.Proof. It is immediate from definitions. (cid:3)
Proposition 5.4.
The composition in Definition 4.5 is a weak composition with respect to thepair p« , Cosp p A qq in the sense of Definition 2.5.Proof. We prove the first part of Definition 2.5. In fact, the equivalence relation « is pre-served under the composition of cospans by Proposition 4.6.The second and fifth parts of Definition 2.5 is obviously satisfied from Definition 4.5.We prove the third and fourth parts of Definition 2.5. Let Λ be a cospan from A to A . Wehave Λ ˝ ι p A q « Λ , (16) ι p A q ˝ Λ « Λ . (17) We prove the first claim and leave the second claim to the readers. Note that the cokernelof the composition A ∆ A Ñ A ‘ A Id A ‘p´ f q Ñ A ‘ B , which is the bulk part of the cospan Λ ˝ ι p A q , is naturally isomorphic to B . The natural isomorphism gives Λ ˝ ι p A q « Λ . (cid:3) Recall the definition of triples in Definition 2.10.
Proposition 5.5.
The triple p« , Cosp p A q , ˝q is compatible with the dagger operation on Cosp p A q in the sense of Definition 2.11.Proof. It is immediate from definitions. (cid:3)
Proposition 5.6.
The triple p« , Cosp p A q , ˝q is compatible with the symmetric monoidal cat-egory structure on A in the sense of Definition 2.14.Proof. The equivalence relation « is preserved under the biproduct of cospans as a corollaryof Proposition 4.4. (cid:3) In the following definition, recall the notations in Definition 2.6.
Definition 5.7.
Consider a triple p« , U , ˝q where the equivalence relation « is defined inDefinition 5.1, U “ Cosp p A q and the weak composition ˝ is defined in Definition 4.5. Wedefine a dagger symmetric monoidal category Cosp « p A q by Cosp « p A q def . “ Cosp « U , ˝ p A q (18)We denote by ι cosp : A Ñ Cosp « p A q the functor induced by ι in Definition 2.2. Definition 5.8.
By repeating a dual construction in this subsection, one can define a category Sp « p A q . In particular, we have Sp « p A q – Cosp « p A op q .The functor ι cosp : A op Ñ Cosp « p A op q is regarded as a functor from A to Cosp « p A op q op “ Sp « p A q op . The composition of the functor with the dagger : : Sp « p A q op Ñ Sp « p A q is de-noted by ι sp : A Ñ Sp « p A q . An isomorphism between cospan and span categories.
Recall Definition 5.7 and 5.8.In this subsection, we define the transposition of cospans to spans. It defines an isomorphismbetween the cospan category
Cosp « p A q and span category Sp « p A q . Definition 5.9.
Let Λ “ ´ A f Ñ B f Ð A ¯ be a cospan in A . Let C be the kernel of thecomposition v “ ∇ B ˝ p f ‘ f q : A ‘ A Ñ B and g , g be the components of the morphism v . We define a span T p Λ q by T p Λ q def . “ ´ A g Ð C g Ñ A ¯ . (19)We dually define a induced cospan T p V q for a span V in A . Lemma 5.10.
Consider a square diagram l in the sense of Definition 4.8 : (20) A BC A Let Λ “ p A Ñ B Ð A q and V “ p A Ð C Ñ A q be the cospan and span diagrams con-tained in the square diagram. If the square diagram l is exact, then we have T p Λ q ĺ V and T p V q ĺ Λ .Proof. We prove that T p Λ q ĺ V. By Proposition 4.10, the induced morphism u : C Ñ Ker p v q is an epimorphism where v “ ∇ B ˝ p f ‘ f q : A ‘ A Ñ B . It is easy to check thatthe epimorphism u gives T p Λ q ĺ V. The other claim T p V q ĺ Λ is proved dually. (cid:3) N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 13
Lemma 5.11.
Consider following three square diagrams l , l , l in A : (21) l “ B DA C gf f g , l “ D FC E hg g h , l “ B FA E h ˝ gg ˝ f f h Especially the morphism in the left side of l and that in the right side of l coincide witheach other, and the diagram l is induced by gluing l , l . If the square diagrams l , l are exact, then the square diagram l is exact.Proof. Suppose that the square diagrams l , l are exact. We first prove that the inducedmorphism k l is an epimorphism. Consider the following commutative diagram where thesequence in each row is the exact sequence induced by the composable morphisms g , h and f , g respectively :(22) Ker p g q Ker p h ˝ g q Ker p h q Cok p g q Ker p f q Ker p g ˝ f q Ker p g q Cok p f q k l k l k l c l Since k l , k l are epimorphisms and c l is a monomorphism, the morphism k l is an epi-morphism by the 4-lemma. We can dually prove that the morphism c l is an monomorphismby the 4-lemma. It completes the proof. (cid:3) Lemma 5.12.
Let Λ , Λ be composable cospans in A . We have T p Λ ˝ Λ q ĺ T p Λ q ˝ T p Λ q .In particular, we have T p Λ ˝ Λ q « T p Λ q ˝ T p Λ q .Proof. Consider cospans Λ “ p A Ñ B Ð A q and Λ “ p A Ñ B Ð A q . Let p A Ñ B Ð A q be the composition Λ ˝ Λ . Let T p Λ q “ p A Ð C Ñ A q and T p Λ q “ p A Ð C Ñ A q bethe induced spans. Let p A Ð C Ñ A q be the composition T p Λ q ˝ T p Λ q . Then we obtainthe following commutative diagram.(23) A B B C A B C C A The four square diagrams in the above commutative diagram are exact in the sense of Defi-nition 4.8 by definitions of B , B , B , C , C , C and Proposition 4.10. By Lemma 5.11 and itsvariants with respect to gluing sides, the induced square diagram below is exact.(24) A B C A By Lemma 5.10, we obtain T p Λ ˝ Λ q ĺ T p Λ q ˝ T p Λ q . (cid:3) Definition 5.13.
We define a functor T :
Cosp « p A q Ñ Sp « p A q as follows. It assigns theobject A itself to an object A of Cosp « p A q . It assigns a morphism r T p Λ qs in Sp « p A q to amorphism r Λ s in Cosp « p A q . It is a well-defined functor due to r T p ι cosp p A qqs “ r ι sp p A qs forany object A and Lemma 5.12. Theorem 5.14.
The functor
T :
Cosp « p A q Ñ Sp « p A q in Definition 5.13 gives an isomor-phism of dagger symmetric monoidal categories.Proof. For any cospan Λ , we have T p Λ q : “ T p Λ : q by definitions. Hence, the functor Tpreserves the dagger structure.The symmetric monoidal functor structure on the functor T is induced by the natural iso-morphism T p Λ ‘ Λ q – T p Λ q ‘ T p Λ q for cospans Λ , Λ in A .Note that the assignment r V s ÞÑ r T p V qs for a span V in A induces a dagger symmetricmonoidal functor T : Sp « p A q Ñ Cosp « p A q . This functor is an inverse functor of thefunctor in Definition 5.13 : For a cospan Λ in A , we have T p T p Λ qq ĺ Λ due to Lemma 5.10.In particular, we have T p T p Λ qq « Λ . Dually, we have T p T p V qq « V for a span V in A . (cid:3) Remark 5.15.
We give a remark about A in the introduction. The two categories Cosp « p A q and Sp « p A q are naturally isomorphic to each other by Theorem 5.14. The former one (latterone, resp.) consists of equivalence classes of cospan (span, resp.) diagrams in A . Then thecategory A is Cosp « p A q or equivalently Sp « p A q ; and ι A is given by an embedding functor ι sp : A Ñ Sp « p A q or equivalently ι cosp : A Ñ Cosp « p A q . See Definition 5.7, 5.8 for ι sp , ι cosp .
6. S panical and cospanical extensions
Brown functor.Definition 6.1.
Consider a diagram in CW fin ˚ , ď r which commutes up to a homotopy :(25) K LT K The diagram (25) is approximated by a triad of spaces if there exists a triad of pointed finiteCW-complexes p L , K , K q such that the following induced diagram (26) is homotopy equiv-alent with the diagram (25) ; there exist pointed homotopy equivalences K » K , K » K , T » K X K and L » K Y K which make the diagram (25) and (26) coincide up tohomotopies.(26) K K Y K K X K K Definition 6.2.
Let A be an abelian category. For r P N Y t8u , a functor E : Ho p CW fin ˚ , ď r q Ñ A satisfies the Mayer-Vietoris axiom if for an arbitrary diagram (26) in CW fin ˚ , ď r approximatedby a triad of spaces, the induced chain complex in A is exact. E p T q Ñ E p K q ‘ E p K q Ñ E p L q . (27) Remark 6.3.
The exactness in the previous definition could be equivalently rephrased asfollows : The induced square diagram (28) in A is exact in the sense of Definition 4.8. Theequivalence follows from Proposition 4.10. (28) E p K q E p L q E p T q E p K q N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 15
Definition 6.4. An A -valued Brown functor is a symmetric monoidal functor E : Ho p CW fin ˚ , ď r q Ñ A satisfying the Mayer-Vietoris axiom. Remark 6.5.
In Definition 6.1, we do not restrict the dimensions of L , K , K . It is su ffi cientto prove Lemma 6.14. Example 6.6.
Consider the category of abelian groups A “ Ab . A generalized homologytheory (for finite CW-spaces) induces a sequence of -dimensional Ab -valued Brown func-tors. In fact, if E ‚ is a generalized homology theory, then the q-th homology theory functorE q is an Ab -valued Brown functor. Example 6.7.
Let k be a field. Consider the category of bicommutative Hopf algebras overk, A “ Hopf bc k . In fact, it is known that the category Hopf bc k is an abelian category [6][7] . Analogously to Example 6.6, a Hopf bc k -valued homology theory induces a sequence of -dimensional Hopf bc k -valued Brown functors. There are various examples of Hopf bc k -valuedhomology theory [5] . Proof of the second part of Theorem 1.1.
For convenience, we introduce followingterminologies.
Definition 6.8.
Let F : Ho p CW fin ˚ , ď r q Ñ A be a symmetric monoidal functor.(1) A spanical extension of the symmetric monoidal functor F is a dagger-preservingsymmetric monoidal functor F : Cosp »ďp r ` q p CW fin ˚ q Ñ Sp « p A q with the followingcommutative diagram.(29) Ho p CW fin ˚ , ď r q A Cosp »ďp r ` q p CW fin ˚ q Sp « p A q F ι ι sp F (2) A cospanical extension of the symmetric monoidal functor F is a dagger-preservingsymmetric monoidal functor F : Cosp »ďp r ` q p CW fin ˚ q Ñ Cosp « p A q with the follow-ing commutative diagram.(30) Ho p CW fin ˚ , ď r q A Cosp »ďp r ` q p CW fin ˚ q Cosp « p A q F ι ι cosp F By using the above terminologies, Theorem 1.1 is reformulated as follows.
Theorem 1.1. (reformulation) For d P N Y t8u , let E : Ho ` CW fin ˚ , ď d ˘ Ñ A be a d -dimensional A -valued Brown functor.(1) There exists a unique spanical extension of E ˝ Σ .(2) There exists a unique cospanical extension E ˝ i .In the rest of this subsection, we prove the second part of theorem. The first part is provedin the next subsection. Lemma 6.9.
For a morphism r Λ s in Cosp »ď d p CW fin ˚ q , there exist cospans Λ , Λ , Λ of pointedfinite CW-complexes subject to following conditions. (1) The components of Λ j have dimension lower than or equal to p d ´ q . (2) We have r Λ s “ r Λ s ˝ r Λ s ˝ r Λ s in the category Cosp »ď d p CW fin ˚ q . Proof. (1) Let Λ P U d , X where U d , X is defined in Definition 3.6. In particular, Λ “ ´ K f Ñ L Ð pt ¯ be a cospan of pointed finite CW-complexes with dim K ď p d ´ q and dim L ď d . There exists a pointed finite CW-complex structure X L on L such that f is a cellular map with respect to X K , X L . If we denote by L p d ´ q the p d ´ q -skeletonof L with respect to X L , we have f p K q Ă L p d ´ q . Denote by f : K Ñ L p d ´ q theinduced map. Let ϕ j : D d Ñ L , r “ , , ¨ ¨ ¨ , k be characteristic maps of d -cellsof X L . Let ψ : Ž j p S d ´ q ` Ñ L p d ´ q be the pointed map induced by the wedge sum Ž j p ϕ j | S d ´ q . Let c be the pointed map induced by the collapsing maps S r Ñ pt.Then L “ L p d q is a homotopy pushout of L p d ´ q ψ Ð Ž j p S d ´ q ` c Ñ Ž j S , hence,we have r Λ s ˝ r Λ s “ r Λ s where Λ “ ´ K f Ñ L p d ´ q ψ Ð Ž j p S d ´ q ` ¯ and Λ “ ´Ž j p S d ´ q ` c Ñ Ž j S Ð pt ¯ .(2) We have r Λ s “ p ev K _ r ι p K qsq ˝ pr Λ s _ r ι p K qs _ r ι p K qsq ˝ pr ι p K qs _ coev K q . SeeFigure 1. We have p ev K _r ι p K qsq˝pr Λ s_r ι p K qs_r ι p K qsq “ p ev K ˝pr Λ s_r ι p K qsqq_r ι p K qs . By the previous discussion, there exist cospans Λ , Λ whose components aredim ď p d ´ q and p ev K ˝ pr Λ s _ r ι p K qsqq “ r Λ s ˝ r Λ s . Put Λ “ p ι p K q _ coev K q whose components also satisfy dim ď p d ´ q . Since r Λ s “ r Λ s ˝ r Λ s ˝ r Λ s , itcompletes the proof. F igure (cid:3) Lemma 6.10.
Let F : Ho p CW fin ˚ , ď r q Ñ A be a symmetric monoidal functor. If a cospanicalextension of the symmetric monoidal functor F exits, then it is unique.Proof. Let F be a cospanical extension of F . Let Λ “ ´ K f Ñ L f Ð K ¯ be a cospan ofpointed CW-spaces with dim ď r . Note that the homotopy equivalence classes of cospans r ι p f qs and r ι p f qs are morphisms of Cosp »ďp r ` q p CW fin ˚ q since we assume dim L ď r . Since wehave r Λ s “ r ι p f q : s ˝ r ι p f qs by definitions, we obtain F pr Λ sq “ F pr ι p f qs : q ˝ F pr ι p f qsq “ F pr ι p f qsq : ˝ F pr ι p f qsq “ ι p F pr f sqq : ˝ ι p F pr f sqq . Hence, F pr Λ sq is determined by thegiven symmetric monoidal functor F if Λ “ ´ K f Ñ L f Ð K ¯ is a cospan whose componentssatisfy dim ď r .By Lemma 6.9, all the morphisms in Cosp »ďp r ` q p CW fin ˚ q is decomposed into some mor-phisms of the above type. Thus, a cospanical extension F is determined uniquely by thesymmetric monoidal functor F . (cid:3) Lemma 6.11.
Let E : Ho p CW fin ˚ , ď d q Ñ A be an A -valued Brown functor. Let Λ , Λ P U d , X and Λ ˝ Λ be the composition. Then we haveE p Λ q ˝ E p Λ q ĺ E p Λ ˝ Λ q . (31) Proof.
Let Λ “ p K Ñ L Ð K q and Λ “ p K Ñ L Ð K q be cospans lying in U d , X . Thenthe composition Λ ˝ Λ “ p K Ñ L Ð K q is given by L “ Cyl p f q Ž K Cyl p f q . See N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 17
Definition 3.3 for details. It induces a commutative square diagram as follows.(32) E p L q E p L q E p K q E p L q Note that the square diagram (33) in CW fin ˚ , ď r is approximated by a triad of spaces with dim ď r in the sense of Definition 6.1. Hence, the square diagram is exact since E is a Brown functor.(33) L L K L Denote by u : E p K q Ñ E p L q ‘ E p L q the composition p E p f q ‘ p´ E p f qqq ˝ ∆ E p K q . The com-mutativity induces a morphism ¯ u : Cok p u q Ñ E p L q . The morphism ¯ u is a monomorphismby Proposition 4.10. Since E p Λ q ˝ E p Λ q “ p E p K q Ñ Cok p u q Ð E p K qq by definition, wehave E p Λ q ˝ E p Λ q ĺ E p Λ ˝ Λ q . (cid:3) Proof of the second part of Theorem 1.1.
We define a cospanical extension ˆ E : Cosp »ď d p CW fin ˚ q Ñ Cosp « p A q of E ˝ i as follows :(1) The functor ˆ E assigns E p K q to an object K of Cosp »ď d p CW fin ˚ q .(2) The functor ˆ E assigns the homotopy equivalence class r E p Λ qs of the induced cospan E p Λ q , which is a morphism in Cosp « p A q , to a morphism r Λ s of Cosp »ď d p CW fin ˚ q .We prove that the above assignment gives a well-defined functor. The assignment ˆ E assignsan identity in the target category to each identity in the source category. Let p Λ , Λ q P U t ˆ s U where U “ U d , X . We have E p Λ q ˝ E p Λ q ĺ E p Λ ˝ Λ q by Lemma 6.11. It implies ˆ E pr Λ sq ˝ ˆ E pr Λ sq “ ˆ E pr Λ s ˝ r Λ sq .The functor ˆ E is enhanced to a dagger-preserving symmetric monoidal functor : The func-tor preserves dagger structures by definitions. The symmetric monoidal functor structure of E naturally induces a symmetric monoidal functor structure on ˆ E .The dagger-preserving symmetric monoidal functor ˆ E is a cospanical extension by defini-tion. The uniqueness of a cospanical extension follows from Lemma 6.10. It completes theproof.6.3. Proof of the first part of Theorem 1.1.Definition 6.12.
Let τ K : Σ K Ñ Σ K ; r t , k s ÞÑ r ¯ t , k s be the conjugate for a pointed finiteCW-space K where we identify Σ K “ S ^ K . Let Λ “ ´ K f Ñ L f Ð K ¯ be a cospan ofpointed spaces. Denote by p , p the collapsing maps from the mapping cone C p f _ f q tothe suspensions Σ K and Σ K respectively. We define a span of pointed spaces T Σ p Λ q byT Σ p Λ q def . “ ´ Σ K τ K ˝ p ÐÝ C p f _ f q p ÝÑ Σ K ¯ . (34) Remark 6.13.
Recall that there is an assignment of spans T p Λ q to cospans Λ in an abeliancategory by Definition 5.9. It is not obvious that there is an analogous assignment in thecategory of (pointed finite CW-)spaces but it motivates the notation in Definition 6.12. Infact, if the symmetric monoidal functor E satisfies the Mayer-Vietoris axiom, then we have r E p T Σ p Λ qs “ r T p E p Σ p Λ qqqs , (35) for a cospan of pointed finite CW-spaces Λ . Here T in the right hand side is the transpositionin Definition 5.9 and Σ p Λ q denotes the suspension of cospan. The bracket r´s denotes theequivalence class of spans defined analogously to Definition 5.1. Lemma 6.14.
Let Λ , Λ P U d , X and Λ ˝ Λ be the composition cospan. Then we haveE p T Σ p Λ qq ˝ E p T Σ p Λ qq ĺ E p T Σ p Λ ˝ Λ qq . (36) Proof.
Let Λ “ ´ K f Ñ L f Ð K ¯ and Λ “ ˆ K f Ñ L f Ð K ˙ be cospans lying in U d , X .Then the composition Λ ˝ Λ “ ˆ K f Ñ L f Ð K ˙ is given by L “ Cyl p f q Ž K Cyl p f q .See Definition 3.3 for details. Denote by p , p the collapsing maps from the mapping cone C p f _ f q to the suspensions Σ K , Σ K , and by p , p the collapsing maps from the mappingcone C p f _ f q to the suspensions Σ K , Σ K . Denote by q , q the collapsing maps from themapping cone C p f _ f q to the mapping cones C p f _ f q and C p f _ f q respectively.It is easy to verify that the diagram (37) commutes up to a homotopy. In fact, we defineT p Λ q by using the conjugate τ K in Definition 6.12 for this diagram to commute. Note thatthe square diagram (37) in CW fin ˚ , ď r is approximated by a triad of spaces with dim ď p r ` q in the sense of Definition 6.1 (compare with the proof of Lemma 6.11).(37) C p f _ f q Σ K C p f _ f q C p f _ f q p q q τ K ˝ p It induces the following square diagram where the morphisms are induced by the canonicalcollapsing maps.(38) E p C p f _ f qq E p Σ K q E p C p f _ f qq E p C p f _ f qq E p p q E p q q E p q q E p τ K ˝ p q The square diagram (38) is exact since the symmetric monoidal functor E satisfies the Mayer-Vietoris axiom. The remaining proof is similar to that of Lemma 6.11. (cid:3) Proof of the first part of Theorem 1.1.
We define a spanical extension ˇ E : Cosp »ď d p CW fin ˚ q Ñ Sp « p A q of E ˝ Σ as follows.(1) The functor ˇ E assigns E p Σ K q to an object K of Cosp »ď d p CW fin ˚ q . It is well-definedsince the domain of E consists of complexes T with dim T ď d .(2) The functor ˇ E assigns the induced morphism r E p T Σ p Λ qqs in Sp « p A q to a morphism r Λ s of Cosp »ď d p CW fin ˚ q .The proof that the assignment ˇ E is a well-defined dagger-preserving symmetric monoidalfunctor is parallel with that of the second part of Theorem 1.1. Note that we apply Lemma6.14 instead of Lemma 6.11.We show that the dagger-preserving symmetric monoidal functor ˇ E is a spanical extensionof the symmetric monoidal functor E in the sense of Definition 6.8. It su ffi ces to prove thatˇ E p ι p f qq “ ι p E p f qq for a morphism f : K Ñ L in CW fin ˚ , ď d where ι p f q “ ´ K f Ñ L Id L Ð L ¯ .Denote by p , p the collapsing maps from the mapping cone C p f _ Id L q to Σ K and Σ L respectively. Note that the map p is a pointed homotopy equivalence, especially so the N EXTENSION OF BROWN FUNCTOR TO COSPAN DIAGRAMS OF SPACES 19 composition τ K ˝ p is. Since the following diagram commutes up to a homotopy, the spanT Σ p ι p f qq is homotopy equivalent with the span ´ Σ K Id Ð Σ K Σ f Ñ Σ L ¯ .(39) C p f _ Id L q Σ K Σ L Σ K τ K ˝ p p τ K ˝ p Σ fId Σ K As a result, we obtain ˇ E p ι p f qq “ r E p T Σ p ι p f qqqs “ ι p E p f qq . It completes the proof.R eferences [1] John Barrett and Bruce Westbury. Invariants of piecewise-linear 3-manifolds. Transactions of the AmericanMathematical Society , 348(10):3997–4022, 1996.[2] Robbert Dijkgraaf and Edward Witten. Topological gauge theories and group cohomology.
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