Prime Spectra of 2-Categories Category theory Joint work with - PowerPoint PPT Presentation

Prime Spectra of 2-Categories Kent Vashaw Prime Spectra of 2-Categories Category theory Joint work with Milen Yakimov The prime spectra Applications to Richardson Kent Vashaw varieties Louisiana State University kvasha1@lsu.edu November 19, 2016

Overview Prime Spectra of 2-Categories Kent Vashaw Category 1 Category theory theory The prime spectra Applications to Richardson 2 The prime spectra varieties 3 Applications to Richardson varieties

2-Categories Prime Spectra of 2-Categories Kent Vashaw Definition Category A 2-category is a category enriched over the category of small theory categories. The prime spectra So a 2-category T has: Applications to Richardson varieties Objects, denoted by A 1 , A 2 etc; 1-morphisms between objects, denoted f , g , h , etc; set of 1-morphisms from A 1 to A 2 denoted T ( A 1 , A 2 ); 2-morphisms between 1-morphisms, denoted α, β, γ, etc; set of 2-morphisms from f to g denoted T ( f , g ) .

2-Categories Composition of 1-morphisms: Prime Spectra of 2-Categories g f A 1 − → A 2 − → A 3 . Kent Vashaw Vertical composition of 2-morphisms α ◦ β : Category theory f The prime spectra α A 1 A 2 Applications g to Richardson varieties β h Horizontal composition of 2-morphisms α 2 ∗ α 1 : f 1 f 2 A 1 α 1 A 2 α 2 A 3 g 1 g 2

2-Categories Prime Spectra of ( α 1 ◦ β 1 ) ∗ ( α 2 ◦ β 2 ) = ( α 1 ∗ α 2 ) ◦ ( β 1 ∗ β 2 ): 2-Categories Kent Vashaw f 1 f 2 Category theory α 1 α 2 A 1 A 2 ∗ A 2 A 3 The prime g 1 g 2 spectra β 1 β 2 Applications h 1 h 2 to Richardson varieties f 1 f 2 α 1 α 2 A 1 A 2 A 3 g 1 g 2 ◦ g 1 g 2 A 1 A 2 A 3 β 1 β 2 h 1 h 2

Exact categories Prime Spectra of 2-Categories Kent Vashaw Definition Category theory A 1-category is called exact if: The prime It is additive; spectra Applications It has a set of distinguished short exact sequences to Richardson varieties A 1 → A 2 → A 3 that obey some axioms. .

Exact categories Prime Spectra of 2-Categories Some exact 1-categories: Kent Vashaw An additive category with short exact sequences defined by Category theory A 1 → A 1 ⊕ A 3 → A 3 ; The prime spectra Abelian categories with traditional short exact sequences Applications (ker g ∼ to Richardson = im f ); varieties Full subcategories of abelian categories closed under extension. Definition A 2-category T is exact if each set T ( A , B ) is itself an exact 1-category.

Grothendieck group Prime Spectra of 2-Categories Kent Vashaw Definition Category theory Suppose C is an exact 1-category. Then the Grothendieck The prime group of C , denoted K 0 ( C ) , is defined by: spectra Take the free abelian group on objects of C ; Applications to Richardson For every exact sequence varieties 0 → A 1 → A 2 → A 3 → 0 , quotient by the relation [ A 1 ] + [ A 3 ] = [ A 2 ] .

Grothendieck group Prime Spectra of 2-Categories Kent Vashaw Definition Category theory Suppose T is an exact 2-category. Then the Grothendieck The prime group of T , denoted K 0 ( T ) is defined as the 1-category with: spectra Objects the same as T ; Applications to Richardson varieties Set of morphisms from X to Y given by K 0 ( T ( X , Y )) , the Grothendieck group of the 1-category T ( X , Y ) . Composition of morphisms induced from composition of morphisms in T .

Positive part of the Grothendieck group Prime Spectra of Definition 2-Categories Kent Vashaw The positive part of the Grothendieck group of an exact 1-category C , denoted K 0 ( C ) + , is defined as the subset of Category theory K 0 ( C ) forming a monoid under addition generated by the The prime indecomposable objects. spectra Applications In other words, while the Grothendieck group has all elements to Richardson varieties of the form � λ i [ b i ] , λ i ∈ Z , i the positive part of the Grothendieck group has elements of the form � λ i [ b i ] , λ i ∈ N . i

Positive part of the Grothendieck group Prime Spectra of 2-Categories Kent Vashaw Category theory Definition The prime spectra The positive part of the Grothendieck group of an exact Applications to Richardson 2-category T , denoted K 0 ( T ) + , has the same objects as T , varieties with hom spaces K 0 ( T ) + ( X , Y ) defined by K 0 ( T ( X , Y )) + .

Strong categorification Prime Spectra of 2-Categories Kent Vashaw Category theory Let A an algebra with orthogonal idempotents e i with The prime 1 = e 1 + e 2 + ... + e n . spectra A = � e i Ae j . Applications to Richardson varieties Consider A as a category: an object for each e i , set of morphisms from i to j given by e i Ae j . Composition of morphisms given by multiplication.

Strong categorification Prime Spectra of 2-Categories Kent Vashaw T Category theory The prime spectra Applications to Richardson varieties view as an algebra K 0 ( T ) A

Strong categorification Prime Spectra of 2-Categories Kent Vashaw Category Definition theory We call B + a Z + -ring if B + has a basis (as a monoid) { b i } The prime with relations b i b j = � m k spectra i , j b k where all coefficients are Applications positive. Elements are all positive linear combinations of basis to Richardson varieties elements, multiplication is extended from basis elements. So we can view Grothendieck groups of 2-categories as Z -algebras, and positive Grothendieck groups as Z + -rings.

Ideals Prime Spectra of 2-Categories Definition Kent Vashaw Let T be an exact 2-category where composition of Category 1-morphisms is an exact bifunctor. We call I a thick ideal of theory T if: The prime spectra I is a full subcategory of T such that if in T ( X , Y ) we Applications to Richardson have an exact sequence of 1-morphisms varieties 0 → f 1 → f 2 → f 3 → 0 , then f 2 is in I if and only if f 1 and f 2 are in I ; I is an ideal: if f ∈ ( X , Y ) is ∈ I and g ∈ T ( Y , Z ) then g ◦ f ∈ I ; and if h ∈ T ( W , X ) then f ◦ h ∈ I .

Ideals Prime Spectra of 2-Categories Kent Vashaw Definition Category Suppose M is any subset of 1-morphisms and 2-morphisms of theory a 2-category T . Then we define the thick ideal generated by The prime spectra M , denoted �M� , to be the smallest thick ideal that contains Applications M , which is the intersection of all thick ideals containing M . to Richardson varieties Definition Suppose B + is a Z + -ring. Then I ⊂ B + is a thick ideal if a + b is in I if and only if a and b are in I, and we also have that if i is in I, then ai and ia are in I for every a ∈ B + .

Prime and completely prime ideals Prime Spectra of 2-Categories Kent Vashaw Definition Category theory We call P a prime of T if P is a thick ideal of T such that if The prime I and J are thick ideals in T , then if I ◦ J ⊂ P , then either spectra I ⊂ P or J ⊂ P . We call I completely prime if it is a thick Applications to Richardson ideal such that f ◦ g ∈ I implies either f ∈ I or g ∈ I . varieties Definition The set of all primes P of a 2-category T is called the spectrum of T and is denoted Spec( T ) .

Prime and completely prime ideals Prime Spectra of 2-Categories Kent Vashaw Category theory The prime Definition spectra Suppose B + is a Z + -ring. Then we call P a prime if P is a Applications to Richardson thick ideal, and IJ ⊂ P implies I or J is in P for all thick ideals varieties I and J.

General results Prime Spectra of 2-Categories We obtain many results with respect to Spec( T ) that Kent Vashaw correspond to the prime spectra of noncommutative rings. Category theory Theorem The prime spectra A thick ideal P is prime if and only if: for all 1-morphisms m , n Applications of T with m ◦ T ◦ n ∈ P , either m ∈ P or n ∈ P . to Richardson varieties This corresponds to the result in the classical theory: Theorem An ideal P of a ring R is prime if and only if: for all x , y ∈ R, if xRy ⊂ P then x or y is in P.

General results Prime Spectra of 2-Categories Kent Vashaw Theorem Category A thick ideal P is prime if and only if: for all thick ideals I , J theory The prime properly containing P , we have that I ◦ J �⊂ P . spectra Applications to Richardson Theorem varieties Every maximal thick ideal is prime. Theorem The spectrum of an exact 2-category T is nonempty.

Relationship between the spectra Prime Spectra of 2-Categories Lemma Kent Vashaw There is a bijection between Spec( T ) and Spec( K 0 ( T ) + ) . Category theory Let T be a categorification of A . Consider the map The prime spectra φ : Spec( K 0 ( T ) + ) → Ideals( K 0 ( T )) = A defined by Applications φ ( P ) = { x − y : x , y ∈ P } . to Richardson varieties Lemma In general, φ is not a map Spec( K 0 ( T ) + ) → Spec( K 0 ( T )) . Example: let H be a Hopf algebra, T be the category of finitely generated H -modules. Then { 0 } is completely prime in K 0 ( T ) + but not in K 0 ( T ).

Relationship between the spectra Prime Spectra of 2-Categories Kent Vashaw Spec( T ) Category theory The prime spectra φ Applications Spec( K 0 ( T ) + ) Ideals( K 0 ( T )) to Richardson varieties Lemma Let T be a categorification of A. If φ ( P ) is a prime in K 0 ( T ) , and P is the prime in T corresponding to P, then A /φ ( P ) is categorified by the Serre quotient T / P .