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Tensoring defines a schematic pseudofunctor with interchange

Statement

Write $R n g M o d$ for the schematic strict 2-category with object labels the unital rings $A$ , interpreted as the module categories $A -Mod$ , whose 1-cells are the additive cocontinuous functors $A -Mod \to B -Mod$ (Additive cocontinuous module functors and their schematic category), and whose 2-cells are all natural transformations, with vertical and horizontal composition as in Whiskering and horizontal composition of natural transformations, so that $R n g M o d (A -Mod, B -Mod)$ is the functor category ${F u n}_{a d d}^{c o c} (A -Mod, B -Mod)$ of Natural transformations of additive cocontinuous module functors are determined at the regular module. Here the category and strict 2-category laws are interpreted componentwise for fixed definable functor schemas, as in Additive cocontinuous module functors and their schematic category: no category whose objects are proper-class functors is formed. The pseudofunctor terminology asserts the equations of Bicategories, pseudofunctors, and biequivalences in this schematic sense. Then the assignment $Φ (A) = A -Mod, Φ (M) = T_{M} = M \otimes_{A} -, Φ (f) = f \otimes 1,$ for a unital ring $A$ , a $(B, A)$ -bimodule $M$ , and a bimodule map $f$ , is a pseudofunctor $Φ : B i m o d \to R n g M o d$ (Bicategories, pseudofunctors, and biequivalences): its composition comparison is the associativity isomorphism $T_{N} \circ T_{M} ≅ T_{N \otimes_{B} M}$ with components $N \otimes_{B} (M \otimes_{A} X) ≅ (N \otimes_{B} M) \otimes_{A} X$ , its identity comparison is $X \to A \otimes_{A} X$ , $x \mapsto 1_{A} \otimes x$ , and the pseudofunctor coherence equations hold. Moreover horizontal composition of 2-cells corresponds to tensoring bimodule maps, $ϕ_{N', M'} (Φ (g) * Φ (f)) = Φ (g \otimes f) ϕ_{N, M}$ , so vertical and horizontal composition satisfy the interchange law (Horizontal and vertical composition of natural transformations satisfy the interchange law). No commutativity and no choice are used.

Facts & Assumptions

Given: The Morita bicategory $B i m o d$ of The Morita bicategory of rings and bimodules, the schematic strict 2-category $R n g M o d$ of the Statement, with object labels the rings $A$ , interpreted as $A -Mod$ , whose 1-cells $A -Mod \to B -Mod$ are the additive cocontinuous functors, and whose 2-cells are all natural transformations, and the assignment $Φ (A) = A -Mod$ , $Φ (M) = T_{M} = M \otimes_{A} -$ , $Φ (f) = f \otimes 1$ .

[F1]

The hom-category $R n g M o d (A -Mod, B -Mod)$ is the schematic category with set-coded fixed Hom-collections ${F u n}_{a d d}^{c o c} (A -Mod, B -Mod)$ of additive cocontinuous functors and all natural transformations, with componentwise vertical composition and composition of functors, and horizontal composition of 2-cells is whiskering (Additive cocontinuous module functors and their schematic category, Natural transformations of additive cocontinuous module functors are determined at the regular module, Natural transformation and its components, Whiskering and horizontal composition of natural transformations).

[F2]

For every $(B, A)$ -bimodule $M$ the functor $T_{M} = M \otimes_{A} -$ is additive, right exact and coproduct-preserving, hence additive cocontinuous; a bimodule map $f : M \to M^{'}$ induces the natural transformation $f \otimes 1 : T_{M} \Rightarrow T_{M^{'}}$ , and these assignments preserve identities and composition (Eilenberg-Watts theorem for arbitrary unital rings, Module homomorphisms induce tensor-product homomorphisms functorially).

[F3]

There is a canonical natural isomorphism $α_{N, M, X} : (N \otimes_{B} M) \otimes_{A} X \to N \otimes_{B} (M \otimes_{A} X)$ with $α ((n \otimes m) \otimes x) = n \otimes (m \otimes x)$ for a $(C, B)$ -bimodule $N$ , a $(B, A)$ -bimodule $M$ and a left $A$ -module $X$ , and it respects the outer actions (Associativity of tensor products for compatible bimodules).

[F4]

There is a canonical natural isomorphism $λ_{X} : A \otimes_{A} X \to X$ , $a \otimes x \mapsto a x$ , respecting the outer actions (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[F5]

A pseudofunctor carries invertible composition and identity comparisons satisfying the associativity and unit coherence equations of Bicategories, pseudofunctors, and biequivalences.

[F6]

The associator and unitors of the Morita data satisfy the pentagon and triangle identities, checked on elementary tensors and extended by additivity (The Morita data satisfy the bicategory coherence axioms, The Morita bicategory of rings and bimodules).

[F7]

Horizontal and vertical composition of natural transformations satisfy the interchange law (Horizontal and vertical composition of natural transformations satisfy the interchange law).

Proof

technique · direct

1.1F1F2given

( $Φ$ is well defined on 1-cells and 2-cells.) The object map sends a unital ring $A$ to the module category $A -Mod$ . A 1-cell $A \to B$ , that is, a $(B, A)$ -bimodule $M$ , is sent to the additive cocontinuous functor $T_{M} = M \otimes_{A} -$ by [F2], an object of $R n g M o d (A -Mod, B -Mod)$ by [F1]; and a 2-cell $f : M \to M^{'}$ is sent to $f \otimes 1$ , a natural transformation $T_{M} \Rightarrow T_{M^{'}}$ by [F2] and hence a morphism of the same hom-category. Identity 2-cells $1_{M}$ are sent to the identity transformation $1_{M} \otimes 1_{X}$ by [F2], and composition of bimodule maps is preserved, so $Φ$ is a well-defined assignment on both sorts of cells.

1.2F1F3given

(Composition comparison.) For a $(C, B)$ -bimodule $N$ and a $(B, A)$ -bimodule $M$ , the comparison $ϕ_{N, M} : T_{N} \circ T_{M} \Rightarrow T_{N \otimes_{B} M}$ has component at a left $A$ -module $X$ the inverse of the canonical associativity isomorphism of [F3], $ϕ_{N, M, X} : N \otimes_{B} (M \otimes_{A} X) \to (N \otimes_{B} M) \otimes_{A} X$ , natural in $X$ and compatible with the outer actions; it is an isomorphism of functors with inverse given componentwise by $α_{N, M, X}$ .

1.3F1F4given

(Identity comparison.) For a unital ring $A$ , the comparison $ϕ_{A} : 1_{A - M o d} \Rightarrow T_{A} = A \otimes_{A} -$ has component at $X$ the inverse of the unitor $λ_{X}$ of [F4], $x \mapsto 1_{A} \otimes x$ , a natural isomorphism of functors.

2.1F1F2F3step 1.2givenalgebra

(Naturality of the composition comparison.) For bimodule maps $f : M \to M'$ and $g : N \to N'$ , the horizontal composite $Φ (g) * Φ (f)$ has component $n \otimes (m \otimes x) \mapsto g (n) \otimes (f (m) \otimes x)$ . Consequently $ϕ_{N', M'} \circ (Φ (g) * Φ (f)) = Φ (g \otimes f) \circ ϕ_{N, M}$ : both sides send $n \otimes (m \otimes x)$ to $(g (n) \otimes f (m)) \otimes x$ , and these tensors generate. This is the required naturality, with domains and codomains related by the comparison rather than literally equal.

2.2F3F4F5F6step 1.2step 1.3givenalgebra

(Coherence equations.) Evaluate the pseudofunctor associativity equation of [F5] at a variable left $A$ -module $X$ . The two sides become composites of the comparisons of steps 1.2 and 1.3 with whiskered identities; after rewriting the comparison components as inverses of the associator of [F3] and applying the naturality of the associator, both sides reduce to the two paths of the pentagon of [F6] with the extra variable $X$ appended, which agree on elementary tensors and hence, by additivity of the functors, on all elements. The two unit equations similarly reduce to the triangle of [F6] with $X$ appended: on an elementary tensor involving a unit factor both sides multiply the unit with the neighbouring factor. Hence the coherence equations hold.

3.1F7step 2.1givenalgebra

(Interchange.) For composable bimodule maps the comparison identity of step 2.1 identifies horizontal composition of 2-cells with tensoring maps, while vertical composition is componentwise composition of natural transformations by [F1]; the interchange law for these operations is [F7], and the component computation of step 2.1 shows the two whiskerings of the induced transformations agree on $n \otimes m \otimes x$ and hence, by generation of tensor products by elementary tensors, on all elements.

4.1step 1.1step 1.2step 1.3step 2.1step 2.2step 3.1∎

Steps 1.1-1.3 give the object, 1-cell and 2-cell maps together with the invertible composition and identity comparisons, and steps 2.2 and 3.1 verify the pseudofunctor coherence equations and the interchange behaviour; hence $Φ : B i m o d \to R n g M o d$ is a pseudofunctor and horizontal composition of 2-cells corresponds to $g \otimes f$ under the composition comparisons. All comparisons are the canonical ones, no commutativity of rings is assumed, and no choice is used.

Depends on

Used by

Dependency tree · two levels

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Sources

N. Johnson and D. Yau, 2-Dimensional Categories, Example 2.1.26 (Bimod) and Definition 4.1.2/Explanation 4.1.5 (pseudofunctor) (standard reference, not scraped)
Fuchs-Schaumann-Schweigert, Eilenberg-Watts calculus for finite categories, introduction (Morita-invariant Eilenberg-Watts equivalences) (standard reference, not scraped)