Background Koszul theory of directed graded k -linear categories Type A ∞ categories Kosulity of directed categories in representation stability theory Wee Liang Gan and Liping Li University of California, Riverside November 23, 2014 Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI ◮ Objects: finite sets. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI ◮ Objects: finite sets. ◮ Morphisms: injections. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI ◮ Objects: finite sets. ◮ Morphisms: injections. ◮ Equivalently, objects are [ n ], n ∈ N ∪ { 0 } Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI ◮ Objects: finite sets. ◮ Morphisms: injections. ◮ Equivalently, objects are [ n ], n ∈ N ∪ { 0 } ◮ End C ([ n ]) is precisely S n . Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI q ◮ Objects: finite dimensional spaces over a finite field F q . Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI q ◮ Objects: finite dimensional spaces over a finite field F q . ◮ Morphisms: linear injections. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI q ◮ Objects: finite dimensional spaces over a finite field F q . ◮ Morphisms: linear injections. ◮ Equivalently, objects are F n , n ∈ N ∪ { 0 } Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories The category FI q ◮ Objects: finite dimensional spaces over a finite field F q . ◮ Morphisms: linear injections. ◮ Equivalently, objects are F n , n ∈ N ∪ { 0 } ◮ End C ([ n ]) is precisely the general linear group. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Applications ◮ These two categories, as well as a lot of variations, are introduced and studied by CEFN, Putman, Sam, Snowden, Wilson, etc. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Applications ◮ These two categories, as well as a lot of variations, are introduced and studied by CEFN, Putman, Sam, Snowden, Wilson, etc. ◮ They are used to study representations of a family of groups simultaneously , in particular the representation stability when n increases. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Applications ◮ These two categories, as well as a lot of variations, are introduced and studied by CEFN, Putman, Sam, Snowden, Wilson, etc. ◮ They are used to study representations of a family of groups simultaneously , in particular the representation stability when n increases. ◮ They have many applications in representation theory, algebraic topology, geometry, combinatorics, etc. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Properties ◮ Theorem (CEFN): FI is locally Noetherian over any left Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Properties ◮ Theorem (CEFN): FI is locally Noetherian over any left Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated. ◮ Theorem (GL, PS): FI q is locally Noetherian over any left Noetherian ring. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Properties ◮ Theorem (CEFN): FI is locally Noetherian over any left Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated. ◮ Theorem (GL, PS): FI q is locally Noetherian over any left Noetherian ring. ◮ Theorem (SS): Every finitely generated projective FI -module is also injective over the complex field. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Properties ◮ Theorem (CEFN): FI is locally Noetherian over any left Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated. ◮ Theorem (GL, PS): FI q is locally Noetherian over any left Noetherian ring. ◮ Theorem (SS): Every finitely generated projective FI -module is also injective over the complex field. ◮ Many proofs use representations of these particular groups. Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories EI categories ◮ All above categories are examples of locally finite EI categories of type A ∞ , which are small categories such that every endomorphism is invertible and satisfy: Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories EI categories ◮ All above categories are examples of locally finite EI categories of type A ∞ , which are small categories such that every endomorphism is invertible and satisfy: ◮ for every pair x , y ∈ Ob C , |C ( x , y ) | is finite; Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories EI categories ◮ All above categories are examples of locally finite EI categories of type A ∞ , which are small categories such that every endomorphism is invertible and satisfy: ◮ for every pair x , y ∈ Ob C , |C ( x , y ) | is finite; ◮ objects are indexed by N ∪ { 0 } , and C ( j , s ) ◦ C ( i , j ) = C ( i , s ). Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories EI categories ◮ All above categories are examples of locally finite EI categories of type A ∞ , which are small categories such that every endomorphism is invertible and satisfy: ◮ for every pair x , y ∈ Ob C , |C ( x , y ) | is finite; ◮ objects are indexed by N ∪ { 0 } , and C ( j , s ) ◦ C ( i , j ) = C ( i , s ). ◮ Therefore, it is natural to consider them from the viewpoint of representation theory of categories, and characterize these properties using certain conditions independent of particular groups . Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Graded k -linear categories ◮ Let C be a small skeletal k -linear category. We assume: Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Graded k -linear categories ◮ Let C be a small skeletal k -linear category. We assume: ◮ C ( x , y ) = � i � 0 C ( x , y ) i and C ( y , z ) j · C ( x , y ) i ⊆ C ( x , z ) i + j ; Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Background Koszul theory of directed graded k -linear categories Type A ∞ categories Graded k -linear categories ◮ Let C be a small skeletal k -linear category. We assume: ◮ C ( x , y ) = � i � 0 C ( x , y ) i and C ( y , z ) j · C ( x , y ) i ⊆ C ( x , z ) i + j ; ◮ For any objects x and y , C ( x , y ) is finite dimensional; Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory
Recommend
More recommend
