Incidence bialgebras of monoidal categories Lucia Rotheray TU Dresden Oslo, 30.07.2019 Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 1 / 13
Idea For a (nice) monoidal category ( C , ⊗ , 1) and a field k we will construct a bialgebra on the k -vector space span k ( Mor C ). Monoidal product � multiplication Unit object � unit element Composition � coproduct Identity morphisms � counit. Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 2 / 13
Motivation There are several combinatorial Hopf algebras with coproducts of the form � ∆(Γ) = Γ /γ ⊗ γ. ���� γ ⊆ Γ some sort of contraction For example: ⊗ 1 + 1 ⊗ ⊗ ∆( ) = + ∆( ) = ⊗ 1 + 1 ⊗ + ⊗ Slight change of perspective: � γ 1 ⊗ γ 2 . ∆(Γ) = γ 1 ◦ γ 2 =Γ � �� � some sort of composition Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 3 / 13
The algebra structure Lemma Let ( C , · , 1) be a (small, strict) monoidal category, k a field and k C the k-vector space spanned by Mor C . Then ( k C , · , i 1 ) defines an associative unital k-algebra. Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 4 / 13
The coalgebra structure Definition (Decompositions and length) Given f ∈ Mor C and n ∈ N , we define: N n ( f ) = { ( f 1 , . . . , f n ) ∈ C × n | f 1 ◦ . . . ◦ f n = f } N n ( f ) = { ( f 1 , . . . , f n ) ∈ ( C \ Id C ) × n | f 1 ◦ . . . ◦ f n = f } ˆ ℓ ( f ) = sup { n ∈ N | ˆ N n ( f ) � = ∅} . Definition (Locally finite category) C is called locally finite if | N 2 ( f ) | is finite for every f ∈ Mor C . Definition (M¨ obius category) C is called M¨ obius if it is locally finite and every morphism has finite length. Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 5 / 13
The coalgebra structure Theorem (Joni and Rota ’79) Let C be a category, k a field and k C the k-vector space spanned by Mor C . The following defines a coassociative counital k-coalgebra structure on k C iff C is locally finite: � 1 f ∈ Id C � ∆( f ) = f 1 ⊗ f 2 , ε ( f ) = ∈ Id C 0 , f / ( f 1 , f 2 ) ∈ N 2 ( f ) Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 6 / 13
Examples Example Let ( P , � ) be a poset and C P the category defined by � ( x , y ) , x � y Ob C P = P , C P ( x , y ) = . ∅ , else C P is M¨ obius and locally finite iff every interval [ x , y ] := { z ∈ P | x � z � y } is finite. Example The path category of a quiver is locally finite and M¨ obius. Example The PROP whose morphisms are (operadic) rooted forests is locally finite. Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 7 / 13
The bialgebra structure Definition (ULF property) A functor F : C → D has the unique lifting of factorisations property if the map N 2 ( f ) → N 2 ( Ff ) ( f 1 , f 2 ) �→ ( Ff 1 , Ff 2 ) is bijective for all f ∈ C . Lemma ULF functors reflect identity morphisms, i.e. f ∈ Id C ⇔ Ff ∈ Id D . We are interested in the case F = · , i.e. we want a bijection N 2 ( f ) × N 2 ( g ) → N 2 ( f · g ) (( f 1 , f 2 ) , ( g 1 , g 2 )) �→ ( f 1 · g 1 , f 2 · g 2 ) for all f , g ∈ C . Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 8 / 13
The bialgebra structure Theorem Let ( C , · , 1) be a monoidal category, k a field and k C the k-vector space spanned by Mor C and ( k C , ∆ , ε ) denote the incidence coalgebra structure and ( k C , · , i 1 ) the k-algebra structure defined above. If C is M¨ obius and · has the ULF property, the following hold: 1 ( k C , · , i 1 , ∆ , ε ) is a k-bialgebra. 2 This bialgebra is a Hopf algebra if and only if ( Ob C , · ) forms a group. Proof. ε ( i 1 ) = 1 as i 1 is an identity. ULF functors reflect identities ⇒ ǫ ( f · g ) = ǫ ( f ) ǫ ( g ). C M¨ obius ⇒ ∆( i x ) = i x ⊗ i x ∀ x ∈ Ob C , in particular ∆(1) = 1 ⊗ 1. · ULF ⇒ ∆( f · g ) = ∆( f )∆( g ) and ε ( f · g ) = ε ( f ) ε ( g ). If C is M¨ obius, ( k C , ∆ , ε ) is pointed.Then we can apply Theorem: A pointed bialgebra is a Hopf algebra if and only if every group-like element is invertible. Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 9 / 13
Example Let ( M , · , 1) be a monoid with a partial relation � satisfying x � y ∧ z � t ⇒ x · z � y · t and C M the category defined by � ( x , y ) , x � y Ob C M = M , C M ( x , y ) = . 0 , else The ULF condition for · becomes “[ x , y ] × [ z , t ] → [ x · z , y · t ] is a bijection”. Example The path category of a quiver does not admit a ULF monoidal product. Example The disjoint union of planar rooted forests is ULF. Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 10 / 13
A more in-depth example Let S be the monoid of (0 , 1) strings of finite length where the product is tacking one string onto the end of another, e.g. · = We impose the following partial order on S : If p , q have the same number of 1s and 0s, we define i i � � q � p ⇔ p j ≥ q j ∀ i = 1 , . . . , h + w j =1 j =1 This gives us a category C S as previously described. Viewing the strings as paths lets us view the morphisms as skew-shapes (with extra group like ”legs”). Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 11 / 13
Skew shape category/bialgebra Composition: ◦ = monoidal product: · = Coproduct: ∆( ) = ⊗ + ⊗ + ⊗ ⊗ ⊗ + + Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 12 / 13
Further reading (Weak) incidence bialgebras of monoidal categories (Ulrich Kraehmer, L.R.), arXiv:1803.07897v4. S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics , Stud. Appl. Math., 61 (1979), pp. 93–139. P. Leroux Les Categories de M¨ obius Cahiers de Topologie et G` eometrie Diff` erentielle Cat` egoriques, 16(3):280–282, 1975. F. W. Lawvere and M. Menni, The Hopf Algebra of M¨ obius Intervals Theory and Applications of Categories, Vol. 24, No. 10, 2010, pp. 221–265. Imma G` alvez-Carillo, Joachim Kock and Andrew Tonks, Decomposition Spaces, incidence algebras and M¨ obius inversion III: the decomposition space of M¨ obius intervals. 2015 arXiv:1512.07580 Ralph M. Kaufmann, Benjamin C. Ward Feyman Categories arXiv:1312.1269v3 Claude Cibils and Marc Rosso, Hopf quivers , 2000 arXiv:math/0009106. Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 13 / 13
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