Abelian categories and imaginaries Mike Prest Department of - - PowerPoint PPT Presentation

Abelian categories and imaginaries

Mike Prest Department of Mathematics Alan Turing Building University of Manchester Manchester M13 9PL UK mprest@manchester.ac.uk October 20, 2008

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R denotes a ring (associative, with 1)

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R denotes a ring (associative, with 1) Mod-R is the category of right R-modules

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R denotes a ring (associative, with 1) Mod-R is the category of right R-modules D ⊆ Mod-R is a definable subcategory if it is elementary and closed under finite (hence arbitrary) direct sums and under direct summands;

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R denotes a ring (associative, with 1) Mod-R is the category of right R-modules D ⊆ Mod-R is a definable subcategory if it is elementary and closed under finite (hence arbitrary) direct sums and under direct summands; equivalently it is a class closed under direct products, direct limits and pure submodules (and isomorphism)

() October 20, 2008 2 / 8

R denotes a ring (associative, with 1) Mod-R is the category of right R-modules D ⊆ Mod-R is a definable subcategory if it is elementary and closed under finite (hence arbitrary) direct sums and under direct summands; equivalently it is a class closed under direct products, direct limits and pure submodules (and isomorphism) A ≤ B is pure in B if for every pp formula φ, φ(A) = A ∩ φ(B)

() October 20, 2008 2 / 8

R denotes a ring (associative, with 1) Mod-R is the category of right R-modules D ⊆ Mod-R is a definable subcategory if it is elementary and closed under finite (hence arbitrary) direct sums and under direct summands; equivalently it is a class closed under direct products, direct limits and pure submodules (and isomorphism) A ≤ B is pure in B if for every pp formula φ, φ(A) = A ∩ φ(B) where a pp (positive primitive) formula is one of the form ∃y θ(x, y) where θ is a conjunction of atomic formulas (a system of R-linear equations in this case).

() October 20, 2008 2 / 8

R denotes a ring (associative, with 1) Mod-R is the category of right R-modules D ⊆ Mod-R is a definable subcategory if it is elementary and closed under finite (hence arbitrary) direct sums and under direct summands; equivalently it is a class closed under direct products, direct limits and pure submodules (and isomorphism) A ≤ B is pure in B if for every pp formula φ, φ(A) = A ∩ φ(B) where a pp (positive primitive) formula is one of the form ∃y θ(x, y) where θ is a conjunction of atomic formulas (a system of R-linear equations in this case). More generally, make the same definitions but now with R replaced by any skeletally small preadditive category R.

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Examples of definable (additive) categories: module categories Mod-R;

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Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R;

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Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field;

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Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets;

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Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets; categories of quasicoherent sheaves over nice enough schemes;

() October 20, 2008 3 / 8

Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets; categories of quasicoherent sheaves over nice enough schemes; locally finitely presented additive categories

() October 20, 2008 3 / 8

Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets; categories of quasicoherent sheaves over nice enough schemes; locally finitely presented additive categories, for instance the category of torsion abelian groups;

() October 20, 2008 3 / 8

Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets; categories of quasicoherent sheaves over nice enough schemes; locally finitely presented additive categories, for instance the category of torsion abelian groups; finitely accessible additive categories with products;

() October 20, 2008 3 / 8

Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets; categories of quasicoherent sheaves over nice enough schemes; locally finitely presented additive categories, for instance the category of torsion abelian groups; finitely accessible additive categories with products; any definable subcategory of a definable category.

() October 20, 2008 3 / 8

Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets; categories of quasicoherent sheaves over nice enough schemes; locally finitely presented additive categories, for instance the category of torsion abelian groups; finitely accessible additive categories with products; any definable subcategory of a definable category. A category C is finitely accessible if it has direct limits, if the subcategory Cfp of finitely presented objects is skeletally small and if every object of C is a direct limit

• f finitely presented objects.

() October 20, 2008 3 / 8

Examples of definable (additive) categories: module categories Mod-R; functor categories Mod-R; the category of C-comodules where C is a coalgebra over a field; the category of OX-modules where OX is a sheaf of rings over a space with a basis of compact open sets; categories of quasicoherent sheaves over nice enough schemes; locally finitely presented additive categories, for instance the category of torsion abelian groups; finitely accessible additive categories with products; any definable subcategory of a definable category. A category C is finitely accessible if it has direct limits, if the subcategory Cfp of finitely presented objects is skeletally small and if every object of C is a direct limit

• f finitely presented objects. Such a category is locally finitely presented if it is

also complete and cocomplete.

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To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries:

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To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ;

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To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

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To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

Let L(D)eq+ denote the corresponding language.

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To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

Let L(D)eq+ denote the corresponding language. Each D ∈ D has a canonical extension to an L(D)eq+-structure Deq+.

() October 20, 2008 4 / 8

To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

Let L(D)eq+ denote the corresponding language. Each D ∈ D has a canonical extension to an L(D)eq+-structure Deq+. In fact Deq+ is the (additive) functor evD, evaluation at D, from L(D)eq+ to Ab.

() October 20, 2008 4 / 8

To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

Let L(D)eq+ denote the corresponding language. Each D ∈ D has a canonical extension to an L(D)eq+-structure Deq+. In fact Deq+ is the (additive) functor evD, evaluation at D, from L(D)eq+ to Ab. More is true:

() October 20, 2008 4 / 8

To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

Let L(D)eq+ denote the corresponding language. Each D ∈ D has a canonical extension to an L(D)eq+-structure Deq+. In fact Deq+ is the (additive) functor evD, evaluation at D, from L(D)eq+ to Ab. More is true: L(D)eq+ is an abelian category;

() October 20, 2008 4 / 8

To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

Let L(D)eq+ denote the corresponding language. Each D ∈ D has a canonical extension to an L(D)eq+-structure Deq+. In fact Deq+ is the (additive) functor evD, evaluation at D, from L(D)eq+ to Ab. More is true: L(D)eq+ is an abelian category; evD is an exact functor;

() October 20, 2008 4 / 8

To D ⊆ Mod-R associate its category L(D)eq+ of pp-imaginaries: the objects are the pp-pairs φ/ψ; the morphisms from φ/ψ to φ′/ψ′ are the pp-definable maps - the equivalence classes of pp formulas ρ(x, y) such that in D, ∀x

• φ(x) → ∃y φ′(y) ∧ ρ(x, y)
• and ∀x y
• (ψ(x ∧ ρ(x, y)) → ψ′(y)
• .

Let L(D)eq+ denote the corresponding language. Each D ∈ D has a canonical extension to an L(D)eq+-structure Deq+. In fact Deq+ is the (additive) functor evD, evaluation at D, from L(D)eq+ to Ab. More is true: L(D)eq+ is an abelian category; evD is an exact functor; D ≃ Ex

• L(D)eq+, Ab
• , the category of exact functors from L(D)eq+ to Ab.

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The category L(Mod-R)eq+ has other realisations:

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The category L(Mod-R)eq+ has other realisations: as the category (mod-R, Ab)fp of finitely presented functors from the category mod-R = (Mod-R)fp of finitely presented modules to Ab;

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The category L(Mod-R)eq+ has other realisations: as the category (mod-R, Ab)fp of finitely presented functors from the category mod-R = (Mod-R)fp of finitely presented modules to Ab; as the free abelian category Ab(Rop) on R;

() October 20, 2008 5 / 8

The category L(Mod-R)eq+ has other realisations: as the category (mod-R, Ab)fp of finitely presented functors from the category mod-R = (Mod-R)fp of finitely presented modules to Ab; as the free abelian category Ab(Rop) on R; and if D is a definable subcategory of Mod-R then L(D)eq+ is the quotient category/localisation of (mod-R, Ab)fp by the Serre subcategory of those functors which are 0 on D.

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L(D)eq+ can be any small abelian category:

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L(D)eq+ can be any small abelian category: given a skeletally small abelian category A, set D = Ex(A, Ab)

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L(D)eq+ can be any small abelian category: given a skeletally small abelian category A, set D = Ex(A, Ab). Then D is a definable subcategory of (A, Ab) = A-Mod = Mod-Aop and A ≃ L(D)eq+.

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Let C, D be definable additive categories;

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Let C, D be definable additive categories; an interpretation of D in C is given by specifying:

() October 20, 2008 7 / 8

Let C, D be definable additive categories; an interpretation of D in C is given by specifying:

• an axiomatisable subcategory C ′ of C
• a “structure-preserving” I : C ′ → D which is an interpretation in the usual sense

() October 20, 2008 7 / 8

Let C, D be definable additive categories; an interpretation of D in C is given by specifying:

• an axiomatisable subcategory C ′ of C
• a “structure-preserving” I : C ′ → D which is an interpretation in the usual sense
• except that we insist on the additive structure being preserved, hence C ′ should

be a definable subcategory and I should be an additive functor, and this forces everything to be given by pp formulas. In particular

() October 20, 2008 7 / 8

Let C, D be definable additive categories; an interpretation of D in C is given by specifying:

• an axiomatisable subcategory C ′ of C
• a “structure-preserving” I : C ′ → D which is an interpretation in the usual sense
• except that we insist on the additive structure being preserved, hence C ′ should

be a definable subcategory and I should be an additive functor, and this forces everything to be given by pp formulas. In particular to each sort of L(D) there will correspond a pp pair in L(C)eq+

() October 20, 2008 7 / 8

Let C, D be definable additive categories; an interpretation of D in C is given by specifying:

• an axiomatisable subcategory C ′ of C
• a “structure-preserving” I : C ′ → D which is an interpretation in the usual sense
• except that we insist on the additive structure being preserved, hence C ′ should

be a definable subcategory and I should be an additive functor, and this forces everything to be given by pp formulas. In particular to each sort of L(D) there will correspond a pp pair in L(C)eq+ and to each basic function or relation symbol of L(D) there will correspond some pp formula of L(C)eq+ which, when applied to members of C ′, will define it

() October 20, 2008 7 / 8

Let C, D be definable additive categories; an interpretation of D in C is given by specifying:

• an axiomatisable subcategory C ′ of C
• a “structure-preserving” I : C ′ → D which is an interpretation in the usual sense
• except that we insist on the additive structure being preserved, hence C ′ should

be a definable subcategory and I should be an additive functor, and this forces everything to be given by pp formulas. In particular to each sort of L(D) there will correspond a pp pair in L(C)eq+ and to each basic function or relation symbol of L(D) there will correspond some pp formula of L(C)eq+ which, when applied to members of C ′, will define it

• and such that every object of D is thus obtained.

() October 20, 2008 7 / 8

Let C, D be definable additive categories; an interpretation of D in C is given by specifying:

• an axiomatisable subcategory C ′ of C
• a “structure-preserving” I : C ′ → D which is an interpretation in the usual sense
• except that we insist on the additive structure being preserved, hence C ′ should

be a definable subcategory and I should be an additive functor, and this forces everything to be given by pp formulas. In particular to each sort of L(D) there will correspond a pp pair in L(C)eq+ and to each basic function or relation symbol of L(D) there will correspond some pp formula of L(C)eq+ which, when applied to members of C ′, will define it

• and such that every object of D is thus obtained.

C C ′

• D

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Theorem

Let I : C ′ → D be an additive functor between definable additive categories;

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Theorem

Let I : C ′ → D be an additive functor between definable additive categories; then I is an interpretation functor iff I commutes with direct products and direct limits.

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Theorem

Let I : C ′ → D be an additive functor between definable additive categories; then I is an interpretation functor iff I commutes with direct products and direct limits. There is a natural bijection between interpretation functors from C ′ to D and exact functors from L(D)eq+ to L(C ′)eq+.

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Theorem

Let I : C ′ → D be an additive functor between definable additive categories; then I is an interpretation functor iff I commutes with direct products and direct limits. There is a natural bijection between interpretation functors from C ′ to D and exact functors from L(D)eq+ to L(C ′)eq+. Let ABEX denote the 2-category whose objects are the skeletally small abelian categories, whose (1-)arrows are the exact functors and whose 2-arrows are the natural transformations; let DEF denote the 2-category whose objects are the definable additive categories, whose (1-)arrows are the functors which commute with direct products and direct limits (i.e. the interpretation functors) and whose 2-arrows are the natural transformations.

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Theorem

Let I : C ′ → D be an additive functor between definable additive categories; then I is an interpretation functor iff I commutes with direct products and direct limits. There is a natural bijection between interpretation functors from C ′ to D and exact functors from L(D)eq+ to L(C ′)eq+. Let ABEX denote the 2-category whose objects are the skeletally small abelian categories, whose (1-)arrows are the exact functors and whose 2-arrows are the natural transformations; let DEF denote the 2-category whose objects are the definable additive categories, whose (1-)arrows are the functors which commute with direct products and direct limits (i.e. the interpretation functors) and whose 2-arrows are the natural transformations.

Theorem

The above gives an equivalence of 2-categories.

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