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Module reconstruction from a small projective generator with supplied copowers and cokernels

Statement

Let $C$ be a locally small cocomplete abelian category and let $P$ be a small projective generator of $C$ . Assume in addition that definable assignments of copowers of $P$ (including their injections) for every set, and of cokernels (including their quotient maps) for every morphism of $C$ , are supplied. Cocompleteness alone asserts their existence individually, not such simultaneous choices. Put $E = {End}_{C} (P)$ and $A = E^{o p}$ , and let $H = C (P, -) : C \to A -Mod$ and $L : A -Mod \to C$ be the functors of The copower presentation construction is left adjoint to the generator Hom functor, so that $L ⊣ H$ . Then the unit $η : 1_{A - M o d} \Rightarrow H L$ and the counit $ε : L H \Rightarrow 1_{C}$ of this adjunction are natural isomorphisms. Consequently $H$ is an equivalence of categories with quasi-inverse $L$ , and $C$ is equivalent to the module category $A -Mod$ (Equivalence, quasi-inverse, and adjoint equivalence of categories). No commutativity of rings is assumed and no choice is used.

Facts & Assumptions

Given: A locally small cocomplete abelian category $C$ , a small projective generator $P$ of $C$ , the supplied definable copower and cokernel assignments of the Statement, $E = {End}_{C} (P)$ , $A = E^{o p}$ , $H = C (P, -)$ , and the functor $L : A -Mod \to C$ of The copower presentation construction is left adjoint to the generator Hom functor with its natural bijections and the adjunction $L ⊣ H$ with unit $η$ and counit $ε$ .

[F1]

$H$ is additive, exact, preserves every set-indexed coproduct, is faithful and satisfies $H (Z) = 0 \Rightarrow Z = 0$ (The Hom functor of a small projective generator is exact, coproduct-preserving, and faithful, Endomorphisms of an object of a preadditive category form a ring, Small projective generators and progenerators).

[F2]

$L (V)$ is constructed as the cokernel of the transposed canonical presentation of $V$ , so that $L$ is left adjoint to $H$ with unit $η : 1_{A - M o d} \Rightarrow H L$ and counit $ε : L H \Rightarrow 1_{C}$ satisfying the triangle identities (The copower presentation construction is left adjoint to the generator Hom functor, Adjunction by unit, counit, and the triangle identities).

[F3]

$H (P) ≅_{A} A$ and $H (P^{(I)}) ≅ A^{(I)}$ for every set $I$ , and the adjunction bijection $C (L (V), Y) ≅ {Hom}_{A} (V, H (Y))$ identifies the unit at $V$ with the transpose of $1_{L (V)}$ (The copower presentation construction is left adjoint to the generator Hom functor).

[F4]

A left $A$ -module $V$ has a canonical presentation $A^{(J)} \overset{d}{\to} A^{(I)} \overset{q}{\to} V \to 0$ with $V ≅ coker d$ ; a functor preserving cokernels carries it to $coker H (τ)$ for the transposed map $τ$ , and the transposition is the identity on the matrix entries under $H (P^{(I)}) ≅ A^{(I)}$ (Canonical free presentations force the comparison to be an isomorphism, The copower presentation construction is left adjoint to the generator Hom functor).

[F5]

An additive functor is exact exactly when it preserves kernels and cokernels (An additive functor is exact exactly when it preserves kernels and cokernels, Abelian category, Module homomorphism and isomorphism, kernel, image and cokernel).

[F6]

An adjoint equivalence is an adjunction whose unit and counit are natural isomorphisms (Equivalence, quasi-inverse, and adjoint equivalence of categories, Adjunction by unit, counit, and the triangle identities).

[F7]

In an abelian category a monic and epic morphism is an isomorphism (An abelian category is balanced).

Proof

technique · direct

1.1F2F3givenalgebra

(The unit at free modules.) For every set $I$ , the copower universal property gives $C (P^{(I)}, Y) ≅ {Hom}_{A} (A^{(I)}, H (Y))$ . Explicitly a map $t : P^{(I)} \to Y$ corresponds to the map sending the basis vector $e_{i}$ to $t ȷ_{i}$ . This is the representation used to define $L$ in [F2], so uniqueness of representing objects identifies $L (A^{(I)})$ with $P^{(I)}$ compatibly with that bijection. The transpose of its identity is therefore $e_{i} \mapsto ȷ_{i}$ in $H (P^{(I)})$ , the canonical isomorphism $A^{(I)} \to H (P^{(I)})$ of [F3]. Hence $η_{A^{(I)}}$ is an isomorphism, including $I = \emptyset$ .

2.1F1F2F3F4F5step 1.1givenalgebra

(The unit is an isomorphism at every module.) Let $V$ be a left $A$ -module with canonical presentation $A^{(J)} \overset{d}{\to} A^{(I)} \overset{q}{\to} V \to 0$ of [F4], so $V ≅ coker d$ and, by construction of $L$ , the object $L (V)$ is the cokernel of the transposed map $τ = τ (d)$ . Since $H$ preserves cokernels by [F1] and [F5], $H (L (V)) ≅ coker H (τ)$ , and by [F4] the map $H (τ)$ corresponds to $d$ under the free identifications $H (P^{(I)}) ≅ A^{(I)}$ , $H (P^{(J)}) ≅ A^{(J)}$ ; hence $H (L (V)) ≅ coker d ≅ V$ . The unit $η_{V}$ is natural and its component at $A^{(I)}$ is an isomorphism by step 1.1; the cokernel descriptions identify $η_{V}$ with the identity of $coker d$ up to these isomorphisms, so $η_{V}$ is an isomorphism for every $V$ .

3.1F1F2F5F7step 2.1given

(The counit is an isomorphism.) Let $X \in C$ . The triangle identity gives $H (ε_{X}) \circ η_{H (X)} = 1_{H (X)}$ by [F2]; since $η_{H (X)}$ is an isomorphism by step 2.1, $H (ε_{X})$ is an isomorphism. Exactness of $H$ [F1] gives $H (\ker ε_{X}) ≅ \ker H (ε_{X}) = 0$ and $H (coker ε_{X}) ≅ coker H (ε_{X}) = 0$ ; by the zero-detection property of [F1], $\ker ε_{X} = 0$ and $coker ε_{X} = 0$ , so $ε_{X}$ is monic and epic and therefore an isomorphism by [F7].

4.1F2F6step 2.1step 3.1∎

(The equivalence.) Steps 2.1 and 3.1 show that the unit and counit of the adjunction $L ⊣ H$ of [F2] are natural isomorphisms, so $L$ and $H$ form an adjoint equivalence and in particular an equivalence of categories by [F6]; hence $C$ is equivalent to the module category $A -Mod$ through $H = C (P, -)$ with quasi-inverse $L$ . No commutativity of rings is assumed, all objects used are the given $P$ , its copowers and the supplied cokernels, and no choice principle is used.

Depends on

Used by

Morita equivalence is invertibility of a bimodule Theorem

Dependency tree · two levels

65 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

W. Crawley-Boevey, Noncommutative Algebra, §3.12, Theorem and proof (A is cocomplete with a finitely generated projective generator iff A is equivalent to R-Mod) (standard reference, not scraped)
P. Etingen, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, printed p.10 (finite-dimensional special case of the reconstruction) (standard reference, not scraped)