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 n n n × n matrix M ! f : k ! k homomorphism If k = R, j det(M)j is the scale factor for f . n n Let ! = e1 ^ · · · ^ en 2 ^ k be the volume form, ^nf : ^n k n −! ^nk n; ! 7! det(M) !: Fernando Muro On determinants (as functors) Categorification of determinants n n n × n matrix M ! f : k ! k homomorphism If k = R, j det(M)j is the scale factor for f . n n Let ! = e1 ^ · · · ^ en 2 ^ k be the volume form, ^nf : ^n k n −! ^nk n; ! 7! det(M) !: Fernando Muro On determinants (as functors) Categorification of determinants n n n × n matrix M ! f : k ! k homomorphism If k = R, j det(M)j is the scale factor for f . n n Let ! = e1 ^ · · · ^ en 2 ^ k be the volume form, ^nf : ^n k n −! ^nk n; ! 7! det(M) !: Fernando Muro On determinants (as functors) Categorification of determinants n n n × n matrix M ! f : k ! k homomorphism If k = R, j det(M)j is the scale factor for f . n n Let ! = e1 ^ · · · ^ en 2 ^ k be the volume form, ^nf : ^n k n −! ^nk n; ! 7! det(M) !: Fernando Muro On determinants (as functors) Categorification of determinants ∼ For any f. d. vector space A and any isomorphism f : A ! B we set det(A)=( ^dim AA; dim A); det(f )= ^dim Af ; in the category linesZ of graded lines: Objects (L; n) are given by L a vector space of dim = 1 and n 2 Z. Morphisms (L; n) ! (L0; n0) are isomorphisms L ! L0 if n = n0 and ; otherwise. The functor det: vectiso −! linesZ 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 det(A)=( ^dim AA; dim A); det(f )= ^dim Af ; in the category linesZ of graded lines: Objects (L; n) are given by L a vector space of dim = 1 and n 2 Z. Morphisms (L; n) ! (L0; n0) are isomorphisms L ! L0 if n = n0 and ; otherwise. The functor det: vectiso −! linesZ 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 det(A)=( ^dim AA; dim A); det(f )= ^dim Af ; in the category linesZ of graded lines: Objects (L; n) are given by L a vector space of dim = 1 and n 2 Z. Morphisms (L; n) ! (L0; n0) are isomorphisms L ! L0 if n = n0 and ; otherwise. The functor det: vectiso −! linesZ 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 det(A)=( ^dim AA; dim A); det(f )= ^dim Af ; in the category linesZ of graded lines: Objects (L; n) are given by L a vector space of dim = 1 and n 2 Z. Morphisms (L; n) ! (L0; n0) are isomorphisms L ! L0 if n = n0 and ; otherwise. The functor det: vectiso −! linesZ categorifies determinants. Fernando Muro On determinants (as functors) Categorification of determinants The functor det satisfies further properties. The category linesZ is a Picard groupoid, i.e. a symmetric categorical group, with tensor product (L; n) ⊗ (L0; n0) = (L ⊗ L0; n + n0); and commutativity constraint twisted by a sign (L; n) ⊗ (L0; n0) comm.−! (L0; n0) ⊗ (L; n); 0 v ⊗ w 7! (−1)nn w ⊗ v: Fernando Muro On determinants (as functors) Categorification of determinants Given a s. e. s. i p ∆ = A B B=A we have an additivity isomorphism det(∆): det(B=A) ⊗ det(A) −! det(B) defined as follows. Choose bases fv1;:::; vpg of B=A and fw1;:::; wqg of A, and set det(∆) 0 0 (v1 ^ · · · ^ vp) ⊗ (w1 ^ · · · ^ wq) 7! v1 ^ · · · ^ vp ^ i(w1) ^ · · · ^ i(wq); 0 where p(vr ) = vr . Fernando Muro On determinants (as functors) Categorification of determinants Given a s. e. s. i p ∆ = A B B=A we have an additivity isomorphism det(∆): det(B=A) ⊗ det(A) −! det(B) defined as follows. Choose bases fv1;:::; vpg of B=A and fw1;:::; wqg of A, and set det(∆) 0 0 (v1 ^ · · · ^ vp) ⊗ (w1 ^ · · · ^ wq) 7! v1 ^ · · · ^ vp ^ i(w1) ^ · · · ^ i(wq); 0 where p(vr ) = vr . Fernando Muro On determinants (as functors) Categorification of determinants Additivity isomorphisms are natural with respect to s. e. s. isomorphisms, det(∆) A / / B / / B=A det(B=A) ⊗ det(A) / det(B) ∼ f ∼ g ∼ h det(h)⊗det(f ) det(g) A0 / / B0 / / B0=A0 det(B0=A0) ⊗ det(A0) / det(B0) det(∆0) Fernando Muro On determinants (as functors) Categorification of determinants They are associative, i.e. for each 2-step filtration ABC the following diagram commutes det(C) 9 e r LLL rrr LL det(BCC=B) rr LL det(ACC=A) rr LLL rrr LL rr LLL rrr L det(C=B) ⊗ det(B) det(C=A) ⊗ det(A) O O 1⊗det(ABB=A) det(B=AC=AC=B)⊗1 det(C=B) ⊗ (det(B=A) ⊗ det(A)) o (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) 9 e r LLL rrr LL det(BA⊕BA) rr LL det(AA⊕BB) rr LLL rrr LL rr LLL rrr L det(A) ⊗ det(B) / det(B) ⊗ det(A) comm. of ⊗ Fernando Muro On determinants (as functors) Determinant for exact categories What’s special about det above? linesZ 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: Eiso −! P together with an additivity isomorphism det(∆): det(B=A) ⊗ det(A) −! det(B) for each s. e. s. ∆ = ABB=A in E satisfying naturality, associativity and commutativity. Fernando Muro On determinants (as functors) Determinant for exact categories What’s special about det above? linesZ 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: Eiso −! P together with an additivity isomorphism det(∆): det(B=A) ⊗ det(A) −! det(B) for each s. e. s. ∆ = ABB=A in E satisfying naturality, associativity and commutativity. Fernando Muro On determinants (as functors) Determinant for exact categories What’s special about det above? linesZ 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: Eiso −! P together with an additivity isomorphism det(∆): det(B=A) ⊗ det(A) −! det(B) for each s. e. s. ∆ = ABB=A in E satisfying naturality, associativity and commutativity. Fernando Muro On determinants (as functors) Determinant for exact categories What’s special about det above? linesZ 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: Eiso −! P together with an additivity isomorphism det(∆): det(B=A) ⊗ det(A) −! det(B) for each s. e. s. ∆ = ABB=A in E satisfying naturality, associativity and commutativity. Fernando Muro On determinants (as functors) Determinants for exact categories Example E = vect(X) the exact category of vector bundles over X, P = Pic(X) the category of graded line bundles (L; n), with L a line bundle over X and n: X ! Z a locally constant map. One can define a determinant functor of vect(X) with values on Pic(X) as above, by using exterior powers. In the special case X = Spec(R), E = proj(R) and P = Pic(R) is the Picard groupoid of graded projective R-modules of constant rank 1. What if R is noncommutative? Do we have any canonical P in this case? Fernando Muro On determinants (as functors) Determinants for exact categories Example E = vect(X) the exact category of vector bundles over X, P = Pic(X) the category of graded line bundles (L; n), with L a line bundle over X and n: X ! Z a locally constant map.