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Graded Eilenberg-Watts theorem with coherent shifts

Statement

Let $k$ be a field and $A, B$ graded $k$ -algebras. Fix a uniformly definable family $J$ as in clause 4 of Coherently shift-compatible functors and transformations form k-linear hom categories, containing the canonical tensor generator labelled by $(A, B, M)$ for every graded $(B, A)$ -bimodule $M$ . In this statement $C o h F u n (A, B)$ denotes its actual word-coded category ${C o h F u n}_{J} (A, B)$ ; a word is interpreted as its composite coherent functor. Write $G r B i m o d (B, A)$ for the category of graded $(B, A)$ -bimodules and degree-zero bimodule maps and $C o h F u n (A, B)$ for the locally small $k$ -linear category of words representing $k$ -linear right exact coproduct-preserving coherently shift-compatible functors ${GrMod}_{0} (A) \to {GrMod}_{0} (B)$ with coherent natural transformations (Coherently shift-compatible functors and transformations form k-linear hom categories) — equivalently, by Colimits of a graded additive functor equal right exactness plus coproduct preservation, the $k$ -linear cocontinuous functors with coherent comparisons. Then $Φ : G r B i m o d (B, A) ⟶ C o h F u n (A, B), Φ (M) = (T_{M}, θ^{M}), Φ (f) = f \otimes 1,$ is an equivalence of $k$ -linear categories. Explicitly:

$Φ$ is well defined on objects and morphisms by Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent;
(essential surjectivity) for every $F$ in $C o h F u n (A, B)$ , $M : = F (A)$ carries the graded $(B, A)$ -bimodule structure of Homogeneous right multiplication reconstructs the graded kernel action and the comparison $τ : T_{M} \Rightarrow F$ of Homogeneous free presentations prove the graded comparison is an isomorphism is a coherent natural isomorphism;
(quasi-inverse) the unit isomorphism $M \otimes_{A} A \to M$ , $m \otimes a \mapsto m a$ , is a degree-zero $(B, A)$ -bimodule isomorphism $T_{M} (A) ≅ M$ , so the object map $F \mapsto F (A)$ with the action of 2 is inverse to $Φ$ up to coherent natural isomorphism, and the unit and comparison isomorphisms satisfy the triangle identities;
(full and faithful) for all graded $(B, A)$ -bimodules $M, M^{'}$ the map $η \mapsto η_{A}$ , read through the unit isomorphisms, is a $k$ -linear bijection ${Nat}^{c o h} (T_{M}, T_{M^{'}}) \to {Hom}_{B - A} (M, M^{'})$ with inverse $f \mapsto f \otimes 1$ .

Any separately supplied definable coherent functor may be included in $J$ using finitely many fixed formulas, so its tensor representation is still covered; no category of all proper-class functor graphs is formed. The statement assumes no commutativity beyond the field $k$ and uses no choice; specialising $F$ to an equivalence recovers the tensor-representation statement of Hazrat's Theorem 2.3.7 for shift-commuting equivalences, while the statement here classifies all right exact coproduct-preserving shift-coherent functors and their transformations.

Facts & Assumptions

Given: A field $k$ , graded $k$ -algebras $A, B$ , graded $(B, A)$ -bimodules $M, M^{'}$ , a degree-zero $(B, A)$ -bimodule map $f : M \to M^{'}$ , and $F \in C o h F u n (A, B)$ .

[L1]

For the specified uniformly definable family $J$ , the word-coded ${C o h F u n}_{J} (A, B) = C o h F u n (A, B)$ is a locally small $k$ -linear category whose morphisms are the coherent transformations and whose composition is vertical composition, the map $η \mapsto η_{A}$ is injective for $k$ -linear right exact coproduct-preserving $F, G$ , and composite and identity functors carry coherent data (Coherently shift-compatible functors and transformations form k-linear hom categories).

[L2]

$M : = F (A)$ carries a graded $(B, A)$ -bimodule structure with $m \cdot a = F (r_{a}) θ_{A, d}^{- 1} (m)$ , $m \cdot 1 = m$ and $(m \cdot a) \cdot b = m \cdot (a b)$ (Homogeneous right multiplication reconstructs the graded kernel action).

[L3]

The comparison $τ : T_{F (A)} \Rightarrow F$ is a coherent natural isomorphism for $k$ -linear right exact coproduct-preserving coherent $F$ (Homogeneous free presentations prove the graded comparison is an isomorphism).

[L4]

$T_{M}$ is $k$ -linear, right exact and coproduct preserving with the coherent comparisons $θ^{M}$ , the components $f \otimes 1_{X}$ are degree-zero $B$ -linear and define the coherent transformation $f \otimes 1$ , and $f \mapsto f \otimes 1$ preserves identities and composition (Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent).

[L6]

Coherently shift-compatible functors carry natural degree-zero isomorphisms satisfying the unit and cocycle, coherent transformations satisfy the equivariance square, and $C o h F u n$ is the $k$ -linear right exact coproduct-preserving sub-class (Coherently shift-compatible functors and natural transformations).

[L7]

A graded $(B, A)$ -bimodule is a $(B, A)$ -bimodule whose graded pieces are homogeneous under both actions, with degree-zero maps as morphisms (Associative graded algebras, bimodules, and internal shifts).

[L10]

The tensor-unit maps $λ_{M} : M \otimes_{A} A \to M$ , $m \otimes a \mapsto m a$ , are group isomorphisms with inverse $m \mapsto m \otimes 1$ , natural in $M$ , and respect every displayed outer module structure (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[L11]

A natural transformation has components satisfying the naturality equation and isomorphisms of functors are its natural isomorphisms (Natural transformation and its components, Natural isomorphism).

[L12]

An equivalence of categories consists of quasi-inverse functors with natural isomorphisms $η : 1 \Rightarrow G F$ and $ε : F G \Rightarrow 1$ ; an adjoint equivalence additionally satisfies the triangle identities $G ε \circ η G = 1_{G}$ and $ε F \circ F η = 1_{F}$ (Equivalence, quasi-inverse, and adjoint equivalence of categories).

[L13]

A functor between $k$ -linear categories is $k$ -linear when each induced map of hom-spaces is $k$ -linear (k-linear categories and k-linear functors).

[L14]

A $k$ -vector space has a pointwise abelian group structure and scalar action (Vector space over a field).

[L15]

A field has a commutative multiplication (Field); every field is a commutative ring (Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring).

[L16]

A functor assigns objects and morphisms compatibly with identities and composites (Covariant functor, identity functor, composite functor, and contravariant functor).

[L17]

A category consists of objects and morphisms with associative unital composition (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).

Proof

technique · direct

1.1L4L6L16L17

Represent $Φ (M)$ by its canonical one-letter tensor word, and read $f \otimes 1$ as the transformation code with those source and target words. The generator assignments are uniformly definable in the set parameter $M$ , so [L1] supplies the actual target category. Clause 1: by [L4] the assignment $M \mapsto (T_{M}, θ^{M})$ takes each graded $(B, A)$ -bimodule to a $k$ -linear right exact coproduct-preserving coherently shift-compatible functor, and $f \mapsto f \otimes 1$ takes degree-zero bimodule maps to coherent transformations and preserves identities and composition; hence $Φ$ is a well-defined functor $G r B i m o d (B, A) \to C o h F u n (A, B)$ [L16, L17].

1.2L1L2L4L6L7L10L11L13L14L16

For $F, G \in C o h F u n (A, B)$ and a coherent $η : F \Rightarrow G$ , naturality at $r_{a} : A {d} \to A$ and coherence at $A, d$ give $η_{A} F (r_{a}) θ_{A, d}^{F, - 1} = G (r_{a}) θ_{A, d}^{G, - 1} (η_{A} {d})$ . Hence $η_{A} (m \cdot a) = η_{A} (m) \cdot a$ for homogeneous $a$ , and therefore for all $a$ by additivity. Since $η_{A}$ is degree-zero $B$ -linear, evaluation defines a $k$ -linear functor $Ψ (F) = F (A)$ , $Ψ (η) = η_{A}$ into graded bimodules, preserving identities and composition componentwise. For $F = T_{M}$ its reconstructed action on $T_{M} (A) = M \otimes_{A} A$ sends $m \otimes b$ to $m \otimes b a$ ; the unit $λ_{M}$ sends this to $m b a = λ_{M} (m \otimes b) a$ , so $λ_{M}$ identifies the reconstructed action with that on $M$ . Evaluation on tensor functors thus lands in ${Hom}_{B - A} (M, M')$ . It is injective by [L1] and surjective with inverse $f \mapsto f \otimes 1$ by [L4] and [L10]. Both maps are $k$ -linear componentwise.

2.1step 1.1L2L3L7

Clause 2: for $F \in C o h F u n (A, B)$ the module $M = F (A)$ with the reconstructed action is a graded $(B, A)$ -bimodule by [L2, L7], and the comparison $τ : T_{M} \Rightarrow F$ of [L3] is a coherent natural isomorphism; hence every object of $C o h F u n (A, B)$ is isomorphic to $Φ (F (A))$ , which is essential surjectivity.

2.2step 1.2L2L3L4L6L10L11L12

The degree-zero bimodule isomorphisms $λ_{M} : Ψ Φ (M) \to M$ are natural in $M$ , since on $m \otimes a$ both paths for a bimodule map $f$ give $f (m) a = f (m a)$ . Put $η_{M} = λ_{M}^{- 1} : m \mapsto m \otimes 1_{A}$ . The coherent comparisons $ε_{F} = τ^{F} : Φ Ψ (F) \Rightarrow F$ of [L3] are natural in $F$ : for coherent $ζ : F \Rightarrow G$ and homogeneous $x \in X_{d}$ , naturality at $ℓ_{x}$ and coherence at $A, d$ give $ζ_{X} τ_{X}^{F} (m \otimes x) = G (ℓ_{x}) θ_{A, d}^{G, - 1} (ζ_{A} (m)) = τ_{X}^{G} (ζ_{A} (m) \otimes x)$ . Thus $η : 1 \Rightarrow Ψ Φ$ and $ε : Φ Ψ \Rightarrow 1$ are natural isomorphisms. The first triangle sends $m \otimes x$ to $(m \otimes 1_{A}) \otimes x$ and then to $m \otimes x$ , because $τ_{X}^{T_{M}} ((m \otimes 1_{A}) \otimes x) = m \otimes x$ . The second sends $m \in F (A)$ to $m \otimes 1_{A}$ and then to $F (ℓ_{1_{A}}) (m) = m$ , since $ℓ_{1_{A}} = {i d}_{A}$ and $θ_{A, 0} = 1$ . Hence both triangle identities hold and $Φ, Ψ$ form an adjoint equivalence.

3.1step 1.1step 2.1step 1.2step 2.2L11L12L13L15∎

Collecting steps 1.1, 1.2, 2.1 and 2.2: $Φ$ is well defined, essentially surjective, full and faithful, and admits the quasi-inverse $Ψ$ with unit $λ^{- 1}$ and counit $τ$ ; hence $Φ$ is an equivalence of categories [L12], and it is an equivalence of $k$ -linear categories because the bijection of clause 4 is $k$ -linear [L13]. Every construction used the canonical tensor product, the canonical reconstructed action and the canonical free covers, so no choice is used, and no commutativity of $A$ or $B$ beyond the central field $k$ , which is a commutative ring [L15], entered.

Depends on

Used by

Graded bimodule maps classify shift-compatible transformations Corollary
Graded Eilenberg-Watts respects bicategorical coherence Corollary
The degree-zero projection is exact and cocontinuous but not a graded tensor functor Counterexample
The internal shift as a graded Eilenberg-Watts kernel Example

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