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Eilenberg-Watts schematic biequivalence between the Morita bicategory and module categories

Statement

The pseudofunctor $Φ : B i m o d \to R n g M o d$ of Tensoring defines a schematic pseudofunctor with interchange, sending a ring $A$ to $A -Mod$ , a $(B, A)$ -bimodule $M$ to $T_{M} = M \otimes_{A} -$ , and a bimodule map to the corresponding natural transformation, is a schematic biequivalence. This means the local equivalence data and coherence equations of Bicategories, pseudofunctors, and biequivalences for fixed functor schemas, with the set-coded natural transformations of Additive cocontinuous module functors and their schematic category; it does not form a category with proper-class functors as objects. Explicitly:

for all unital rings $A, B$ the local assignment $B i m o d (A, B) \to R n g M o d (A -Mod, B -Mod)$ , $M \mapsto T_{M}$ , is a schematic equivalence, so full, faithful, and essentially surjective (Eilenberg-Watts is a schematic equivalence of Hom categories);
every object label of $R n g M o d$ is the image of its ring, hence equivalent to an image object. Consequently, under the composition comparison, a natural transformation between two composite tensor functors is exactly a bimodule map between their composite kernels, including all source and target actions. No commutativity and no choice are used.

Facts & Assumptions

Given: The pseudofunctor $Φ : B i m o d \to R n g M o d$ of Tensoring defines a schematic pseudofunctor with interchange, sending a unital ring $A$ to $A -Mod$ , a $(B, A)$ -bimodule $M$ to $T_{M} = M \otimes_{A} -$ and a bimodule map to the corresponding natural transformation.

[F1]

$Φ$ is a pseudofunctor whose object map is $A \mapsto A -Mod$ , whose map on 1-cells is $M \mapsto T_{M}$ , and whose composition comparison identifies $T_{N} \circ T_{M}$ with $T_{N \otimes_{B} M}$ through the canonical associativity isomorphism (Tensoring defines a schematic pseudofunctor with interchange, Bicategories, pseudofunctors, and biequivalences).

[F2]

For all unital rings $A, B$ the assignment $M \mapsto T_{M}$ is a schematic equivalence from the $(B, A)$ -bimodules with bimodule maps to the additive cocontinuous functors $A -Mod \to B -Mod$ with natural transformations; it is full and faithful with $Nat (T_{M}, T_{M^{'}}) ≅ {Hom}_{B - A} (M, M^{'})$ , and it is essentially surjective (Eilenberg-Watts is a schematic equivalence of Hom categories).

[F3]

The objects of $R n g M o d$ are exactly the module categories $A -Mod$ for unital rings $A$ , the 1-cells are fixed additive cocontinuous functor schemas and the 2-cells are their set-coded natural transformations, with horizontal composition given by whiskering (Strict 2-category, Additive cocontinuous module functors and their schematic category, Natural transformations of additive cocontinuous module functors are determined at the regular module, Functor category $[C, D]$ , Natural transformation and its components).

[F4]

A pseudofunctor is a biequivalence when every local functor is an equivalence of categories and every target object is equivalent to an image object; the local equivalences used here come with the supplied quasi-inverse $F \mapsto F (A)$ of [F2] (Bicategories, pseudofunctors, and biequivalences, Equivalence, quasi-inverse, and adjoint equivalence of categories).

Proof

technique · direct

1.1F1F2F4given

(The local functors are equivalences.) Fix unital rings $A, B$ . The local map $B i m o d (A, B) \to R n g M o d (A -Mod, B -Mod)$ is the functor $M \mapsto T_{M}$ on objects and $f \mapsto f \otimes 1$ on morphisms, which is exactly the assignment of the pseudofunctor [F1] restricted to this pair of objects. By [F2] this functor is a schematic equivalence with the specified evaluation quasi-inverse, and in particular is full, faithful and essentially surjective.

1.2F1F3given

(Every target object is an image.) Let $A$ be an object of $R n g M o d$ . By [F3] there is a unital ring $A$ with $A = A -Mod$ , and $Φ (A) = A -Mod = A$ by [F1], so $A$ is (trivially) equivalent to the image of an object of $B i m o d$ .

1.3F1F2given

(Composite kernels.) Let $M, M^{'}$ be $(B, A)$ -bimodules and $N, N^{'}$ $(C, B)$ -bimodules. By the composition comparisons of [F1] the composite functors $T_{N} \circ T_{M}$ and $T_{N^{'}} \circ T_{M^{'}}$ are naturally isomorphic to $T_{N \otimes_{B} M}$ and $T_{N^{'} \otimes_{B} M^{'}}$ . Composing with these natural isomorphisms gives a bijection $Nat (T_{N} \circ T_{M}, T_{N^{'}} \circ T_{M^{'}}) ≅ Nat (T_{N \otimes_{B} M}, T_{N^{'} \otimes_{B} M^{'}})$ , and by the full faithfulness of [F2] the right-hand side is ${Hom}_{C - A} (N \otimes_{B} M, N^{'} \otimes_{B} M^{'})$ ; under the bijection a natural transformation of composites corresponds to a bimodule map of the composite kernels preserving both the left $C$ -action and the right $A$ -action.

2.1F4step 1.1step 1.2step 1.3∎

Steps 1.1 and 1.2 verify the two clauses of the definition of biequivalence for $Φ$ , so by [F4] the pseudofunctor $Φ$ is a schematic biequivalence between the Morita bicategory and the schematic strict 2-category of module categories and additive cocontinuous functors; step 1.3 identifies the natural transformations between composite tensor functors with bimodule maps of composite kernels. No commutativity of rings is assumed and no choice is used.

Depends on

Used by

Morita equivalence is invertibility of a bimodule Theorem

Dependency tree · two levels

40 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

nLab, Morita equivalence, Idea and Classical Morita theorem (the 2-category of rings, bimodules and intertwiners; equivalence with linear equivalences of module categories) (standard reference, not scraped)
Fuchs-Schaumann-Schweigert, Eilenberg-Watts calculus for finite categories, introduction (Morita-invariant form of Eilenberg-Watts) (standard reference, not scraped)