On determinants (as functors) Fernando Muro Universitat de Barcelona Dept. Àlgebra i Geometria V Seminar on Categories and Applications Pontevedra, September 2008 Fernando Muro On determinants (as functors)
Categorification of determinants From Wikipedia : “In mathematics, categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues.” Crane–Yetter, Examples of categorification , Cahiers de Topologie et Géometrie Différentielle Catégoriques 39 (1998), no. 1, 3–25. Knudsen–Mumford, The projectivity of the moduli space of stable curves I . Math. Scand. 39 (1976), no. 1, 19–55. Deligne, Le déterminant de la cohomologie , Contemp. Math. 67 (1987), 93–177. Fernando Muro On determinants (as functors)
Categorification of determinants From Wikipedia : “In mathematics, categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues.” Crane–Yetter, Examples of categorification , Cahiers de Topologie et Géometrie Différentielle Catégoriques 39 (1998), no. 1, 3–25. Knudsen–Mumford, The projectivity of the moduli space of stable curves I . Math. Scand. 39 (1976), no. 1, 19–55. Deligne, Le déterminant de la cohomologie , Contemp. Math. 67 (1987), 93–177. Fernando Muro On determinants (as functors)
Categorification of determinants From Wikipedia : “In mathematics, categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues.” Crane–Yetter, Examples of categorification , Cahiers de Topologie et Géometrie Différentielle Catégoriques 39 (1998), no. 1, 3–25. Knudsen–Mumford, The projectivity of the moduli space of stable curves I . Math. Scand. 39 (1976), no. 1, 19–55. Deligne, Le déterminant de la cohomologie , Contemp. Math. 67 (1987), 93–177. Fernando Muro On determinants (as functors)
Categorification of determinants f : k n → k n homomorphism n × n matrix M � If k = R , | det ( M ) | is the scale factor for f . Let ω = e 1 ∧ · · · ∧ e n ∈ ∧ n k n be the volume form, ∧ n f : ∧ n k n ∧ n k n , − → det ( M ) ω. ω �→ Fernando Muro On determinants (as functors)
Categorification of determinants f : k n → k n homomorphism n × n matrix M � If k = R , | det ( M ) | is the scale factor for f . Let ω = e 1 ∧ · · · ∧ e n ∈ ∧ n k n be the volume form, ∧ n f : ∧ n k n ∧ n k n , − → det ( M ) ω. ω �→ Fernando Muro On determinants (as functors)
Categorification of determinants f : k n → k n homomorphism n × n matrix M � If k = R , | det ( M ) | is the scale factor for f . Let ω = e 1 ∧ · · · ∧ e n ∈ ∧ n k n be the volume form, ∧ n f : ∧ n k n ∧ n k n , − → det ( M ) ω. ω �→ Fernando Muro On determinants (as functors)
Categorification of determinants f : k n → k n homomorphism n × n matrix M � If k = R , | det ( M ) | is the scale factor for f . Let ω = e 1 ∧ · · · ∧ e n ∈ ∧ n k n be the volume form, ∧ n f : ∧ n k n ∧ n k n , − → det ( M ) ω. ω �→ Fernando Muro On determinants (as functors)
Categorification of determinants For any f. d. vector space A and any isomorphism f : A ∼ → B we set ( ∧ dim A A , dim A ) , det ( A ) = ∧ dim A f , det ( f ) = in the category lines Z of graded lines: Objects ( L , n ) are given by L a vector space of dim = 1 and n ∈ Z . Morphisms ( L , n ) → ( L ′ , n ′ ) are isomorphisms L → L ′ if n = n ′ and ∅ otherwise. The functor det : vect iso − → lines Z categorifies determinants. Fernando Muro On determinants (as functors)
Categorification of determinants For any f. d. vector space A and any isomorphism f : A ∼ → B we set ( ∧ dim A A , dim A ) , det ( A ) = ∧ dim A f , det ( f ) = in the category lines Z of graded lines: Objects ( L , n ) are given by L a vector space of dim = 1 and n ∈ Z . Morphisms ( L , n ) → ( L ′ , n ′ ) are isomorphisms L → L ′ if n = n ′ and ∅ otherwise. The functor det : vect iso − → lines Z categorifies determinants. Fernando Muro On determinants (as functors)
Categorification of determinants For any f. d. vector space A and any isomorphism f : A ∼ → B we set ( ∧ dim A A , dim A ) , det ( A ) = ∧ dim A f , det ( f ) = in the category lines Z of graded lines: Objects ( L , n ) are given by L a vector space of dim = 1 and n ∈ Z . Morphisms ( L , n ) → ( L ′ , n ′ ) are isomorphisms L → L ′ if n = n ′ and ∅ otherwise. The functor det : vect iso − → lines Z categorifies determinants. Fernando Muro On determinants (as functors)
Categorification of determinants For any f. d. vector space A and any isomorphism f : A ∼ → B we set ( ∧ dim A A , dim A ) , det ( A ) = ∧ dim A f , det ( f ) = in the category lines Z of graded lines: Objects ( L , n ) are given by L a vector space of dim = 1 and n ∈ Z . Morphisms ( L , n ) → ( L ′ , n ′ ) are isomorphisms L → L ′ if n = n ′ and ∅ otherwise. The functor det : vect iso − → lines Z categorifies determinants. Fernando Muro On determinants (as functors)
Categorification of determinants The functor det satisfies further properties. The category lines Z is a Picard groupoid, i.e. a symmetric categorical group, with tensor product ( L , n ) ⊗ ( L ′ , n ′ ) ( L ⊗ L ′ , n + n ′ ) , = and commutativity constraint twisted by a sign comm. ( L , n ) ⊗ ( L ′ , n ′ ) ( L ′ , n ′ ) ⊗ ( L , n ) , − → ( − 1 ) nn ′ w ⊗ v . v ⊗ w �→ Fernando Muro On determinants (as functors)
Categorification of determinants Given a s. e. s. p i ∆ = A B ։ B / A we have an additivity isomorphism det (∆): det ( B / A ) ⊗ det ( A ) − → det ( B ) defined as follows. Choose bases { v 1 , . . . , v p } of B / A and { w 1 , . . . , w q } of A , and set det (∆) v ′ 1 ∧ · · · ∧ v ′ ( v 1 ∧ · · · ∧ v p ) ⊗ ( w 1 ∧ · · · ∧ w q ) �→ p ∧ i ( w 1 ) ∧ · · · ∧ i ( w q ) , where p ( v ′ r ) = v r . Fernando Muro On determinants (as functors)
Categorification of determinants Given a s. e. s. p i ∆ = A B ։ B / A we have an additivity isomorphism det (∆): det ( B / A ) ⊗ det ( A ) − → det ( B ) defined as follows. Choose bases { v 1 , . . . , v p } of B / A and { w 1 , . . . , w q } of A , and set det (∆) v ′ 1 ∧ · · · ∧ v ′ ( v 1 ∧ · · · ∧ v p ) ⊗ ( w 1 ∧ · · · ∧ w q ) �→ p ∧ i ( w 1 ) ∧ · · · ∧ i ( w q ) , where p ( v ′ r ) = v r . Fernando Muro On determinants (as functors)
Categorification of determinants � � � � � � Additivity isomorphisms are natural with respect to s. e. s. isomorphisms, det (∆) � � B � � B / A det ( B / A ) ⊗ det ( A ) det ( B ) A � det ( h ) ⊗ det ( f ) det ( g ) ∼ f ∼ g ∼ h � � det ( B ′ ) � B ′ � � B ′ / A ′ det ( B ′ / A ′ ) ⊗ det ( A ′ ) det (∆ ′ ) A ′ Fernando Muro On determinants (as functors)
Categorification of determinants � � � � They are associative, i.e. for each 2-step filtration A B C the following diagram commutes det ( C ) � ���������������� � � � � det ( B C ։ C / B ) det ( A C ։ C / A ) � � � � � � � � � � � det ( C / B ) ⊗ det ( B ) det ( C / A ) ⊗ det ( A ) 1 ⊗ det ( A B ։ B / A ) det ( B / A C / A ։ C / B ) ⊗ 1 det ( C / B ) ⊗ ( det ( B / A ) ⊗ det ( A )) � ( det ( C / B ) ⊗ det ( B / A )) ⊗ det ( A ) assoc. of ⊗ Fernando Muro On determinants (as functors)
Categorification of determinants � � They are commutative, i.e. the following diagram commutes det ( A ⊕ B ) � ���������������� � � � � det ( B A ⊕ B ։ A ) det ( A A ⊕ B ։ B ) � � � � � � � � � � � � det ( B ) ⊗ det ( A ) det ( A ) ⊗ det ( B ) comm. of ⊗ Fernando Muro On determinants (as functors)
Determinant for exact categories What’s special about det above? lines Z is a Picard groupoid, vect has short exact sequences. Definition (Deligne’87) Let E be an abelian or exact category and P a Picard groupoid. A determinant is a functor det : E iso − → P together with an additivity isomorphism det (∆): det ( B / A ) ⊗ det ( A ) − → det ( B ) for each s. e. s. ∆ = A B ։ B / A in E satisfying naturality, associativity and commutativity. Fernando Muro On determinants (as functors)
Determinant for exact categories What’s special about det above? lines Z is a Picard groupoid, vect has short exact sequences. Definition (Deligne’87) Let E be an abelian or exact category and P a Picard groupoid. A determinant is a functor det : E iso − → P together with an additivity isomorphism det (∆): det ( B / A ) ⊗ det ( A ) − → det ( B ) for each s. e. s. ∆ = A B ։ B / A in E satisfying naturality, associativity and commutativity. Fernando Muro On determinants (as functors)
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