Theory and Applications of Categories, Vol. 1, No. 4, 1995, pp. 72–77.

KAN EXTENSIONS ALONG PROMONOIDAL FUNCTORS

BRIAN DAY AND ROSS STREET

Transmitted by R. J. Wood

ABSTRACT. Strong promonoidal functors are defined. Left Kan extension (also called “existential quantification”) along a strong promonoidal functor is shown to be a strong monoidal functor. A construction for the free monoidal category on a promonoidal category is provided. A Fourier-like transform of presheaves is defined and shown to take convolution product to cartesian product.

Let V be a complete, cocomplete, symmetric, closed, monoidal category. We intend that all categorical concepts throughout this paper should be V-enriched unless explicitly declared to be “ordinary”. A reference for enriched category theory is [10], however, the reader unfamiliar with that theory can read this paper as written with V the category of sets and ⊗ for V as cartesian product; another special case is obtained by taking all categories and functors to be additive and V to be the category of abelian groups. The reader will need to be familiar with the notion of promonoidal category (used in [2], [6], [3], and [1]): such a category A is equipped with functors P : A^{op}⊗A^{op}⊗A⟶V, J : A⟶V, together with appropriate associativity and unit constraints subject to some axioms. Let C be a cocomplete monoidal category whose tensor product preserves colimits in each variable. If A is a small promonoidal category then the functor category [A, C] has the convolution monoidal structure given by

F ∗G = ∫ ^{A,A′} P (A, A′, −)⊗(F A⊗GA′)

(see [7], Example 2.4).

Suppose A and B are promonoidal categories. A promonoidal functor is a functor Φ : A⟶B together with natural transformations

ϕ_{AA′A′′} : P (A, A′, A′′)⟶P (ΦA, ΦA′, ΦA′′), ϕ_{A} : JA⟶JΦA

satisfying two axioms; see [2], [5] for details. When A, B are small it means that the functor

[Φ, 1] : [B, V]⟶[A, V]

is canonically (via the natural transformations ϕ) a monoidal functor in the sense of [8]. In particular, if A, B are monoidal categories, promonoidal functors Φ : A⟶B are precisely monoidal functors.

Received by the editors 19 May 1995.

Published on 15 June 1995

1991 Mathematics Subject Classification : 18D10.

Key words and phrases: monoidal functor, promonoidal category, Kan extension, Fourier transform, convolution tensor product.

\begin{matrix}c \\ ◯ \end{matrix} Brian Day and Ross Street 1995. Permission to copy for private use granted.

Our purpose here is to define and discuss “existential quantification” along promonoidal functors. For any promonoidal functor Φ : A⟶B, the natural transformations

P (A, A′, A′′)⊗B(ΦA′′, B) ⟶^{ϕ⊗1} P (ΦA, ΦA′, ΦA′′)⊗B(ΦA′′, B) ⟶^{μ} P (ΦA, ΦA′, B)

JA⊗B(ΦA, B) ⟶^{ϕ⊗1} JΦA⊗B(ΦA, B) ⟶^{μ} JB

(where the arrows μ are part of the functoriality of P, J) induce natural transformations

∫ ^{A′′} P (A, A′, A′′)⊗B(ΦA′′, B) ⟶^{ρ} P (ΦA, ΦA′, B)

∫ ^{A} JA⊗B(ΦA, B) ⟶^{ρ} JB.

We call Φ : A⟶B strong when these arrows ρ are all invertible. In particular, when A, B are monoidal, strong promonoidal amounts to strong monoidal (= tensor-and-unit-preserving up to coherent natural isomorphism).

It may appear that, in the above definitions, we need A to be small and V or C to be cocomplete. We have written this way for ease of reading. Sometimes the necessary weighted (= “indexed”) colimits exist for other reasons.

1 Proposition. If Φ : A⟶B is a strong promonoidal functor then “existential quantification”

∃_{Φ} : [A, C]⟶[B, C],

given by

∃_{Φ}(F)(B) = ∫ ^{A} B(ΦA, B)⊗F A,

has the structure of a strong monoidal functor.

Proof. Starting with the definitions of ∃_{Φ} and ∗, we have the calculation

∃_{Φ}(F ∗ G)(B) = ∫ ^{A} B(ΦA, B)⊗ ∫ ^{A′,A′′} P (A′, A′′, A)⊗(F A′⊗GA′′)

≅ ∫ ^{A′,A′′} ∫ ^{A} B(ΦA, B)⊗P (A′, A′′, A)⊗(F A′⊗GA′′)

by commuting colimits,

≅ ∫ ^{A′,A′′} P (ΦA′, ΦA′′, B)⊗(F A′⊗GA′′)

since Φ is strong,

≅ ∫ ^{A′,A′′} ∫ ^{B′,B′′} B(ΦA′, B′)⊗B(ΦA′′, B′′)⊗P (B′, B′′, B)⊗(F A′⊗GA′′)

by the Yoneda Lemma,

≅ ∫ ^{B′,B′′} P (B′, B′′, B)⊗ ∫ ^{A′} B(ΦA′, B′)⊗F A′⊗ ∫ ^{A′′} B(ΦA′′, B′′)⊗GA′′

by commuting colimits,

≅ (∃_{Φ}(F) ∗ ∃_{Φ}(G)(B) by definitions.

Similarly, we have

∃_{Φ}(J)(B) = ∫ ^{A} B(ΦA, B)⊗J(A) ≅ J(B).

The cartesian monoidal structure on a category with finite products has binary product as tensor product and the terminal object as unit. Dually, a category with finite coproducts has a cocartesian monoidal structure. If A is cocartesian monoidal and C is cartesian monoidal, then convolution on [A, C] is cartesian. Proposition 1 has the corollary that existential quantification ∃_{Φ} along a finite-coproduct-preserving functor Φ preserves finite products; compare [11], Proposition 2.7.

For any promonoidal category A, the Yoneda embedding Y : A⟶[A, V]^{op} is a promonoidal functor (just use the definition and the Yoneda Lemma). The closure in [A, V]^{op} of the representables Y (A) = A(A, −) under tensor products and unit (as in [4]) gives a full monoidal subcategory A′ of [A, V]^{op}, and Y factors through the inclusion via a promonoidal functor N : A⟶A′. This construction has a universal property: to describe it we introduce the ordinary category P Mon(A, B) whose objects are promonoidal functors Φ : A⟶B and whose arrows are promonoidal natural transformations ([2] and [5]); if A, B are both monoidal, we write Mon(A, B) for this same ordinary category. (Later we shall use the ordinary category SP Mon(A, B) of strong promonoidal functors.)

2 Proposition. For each promonoidal category A and each monoidal category B, restriction along N : A⟶A′ provides an equivalence of ordinary categories

Mon(A′, B) ⟶^{∼} P Mon(A, B).

Proof. To see that restriction along N is essentially surjective, take a promonoidal functor Φ : A⟶B. We obtain the following diagram where regions commute up to canonical natural isomorphisms.

[figure: A; B; [A, V]^{op}; [B, V]^{op}; ^{A′}; ^{B′}; N; N; − − −; Φ; Φ′; ^{∃{op \atop Φ}}; Y; Y]

The functor ∃_{Φ}^{op} is monoidal. Thus, so is its restriction Φ′ : A′⟶B′. Since B is monoidal, the functor N : B⟶B′ is an equivalence of monoidal categories. So we obtain a promonoidal functor Ψ : A′⟶B with ΨN ≅ Φ. The remaining details are left to the reader; they will require the reader to know the definition of promonoidal natural transformation.

Suppose A is a small promonoidal category. Observe that a strong promonoidal functor Φ : A⟶C^{op} satisfies the following conditions:

∫ ^{A′′} P (A, A′, A′′)⊗C(B, ΦA′′) ⟶^{≅} C(B, ΦA⊗ΦA′)

∫ ^{A} JA⊗C(B, ΦA) ⟶^{≅} C(B, I).

On tensoring both sides with B and using the Yoneda lemma, we obtain the conditions:

∫ ^{A′′} P (A, A′, A′′)⊗ΦA′′ ⟶^{≅} ΦA⊗ΦA′

∫ ^{A} JA⊗ΦA ⟶^{≅} I.

Let M = SP Mon(A, C^{op})^{op}. There is a forgetful functor M⟶[A^{op}, C]. The transform of a functor F : A⟶V is the functor T (F) : M⟶C given by the coend

T (F)(Φ) = ∫ ^{A} F A⊗ΦA ≅ (∃_{Φ}F)(I).

Notice that this is the colimit of Φ weighted (or indexed) by F. We have defined a functor T : [A, V]⟶[M, C]. As usual, we regard [A, V] as monoidal via convolution, but we regard [M, C] as monoidal via pointwise tensor product in C.

3 PropositionThe transform enriches to a strong monoidal functor

T : [A, V]⟶[M, C].

That is, the transform takes convolution to pointwise tensor product.

Proof. For all F, G : A⟶V, we have the calculations

T (F ∗ G)(Φ) = ∫ ^{A}(F ∗ G)(A)⊗Φ(A)

≅ ∫ ^{AA′A′′} P (A′, A′′, A)⊗F (A′)⊗G(A′′)⊗Φ(A)
≅ ∫ ^{A′,A′′} F (A′)⊗G(A′′)⊗Φ(A′)⊗Φ(A′′)
≅ T (F)(Φ)⊗T (G)(Φ)

≅ (T (F)⊗T (G))(Φ)
T (J)(Φ) = ∫ ^{A} J(A)⊗Φ(A) ≅ I.

In particular, if C is cartesian closed, the transform takes convolution into cartesian product.

Remark One can also trace through the steps in [9] and obtain a generalisation to promonoidal structures using promonoidal functors.

Acknowledgment. We are grateful to Todd Trimble for his assistance in typesetting this paper in LaTEX.

References

[1] B.J. Day. Biclosed bicategories: localisation of convolution. Technical Report 81-0030, Macquarie Math Reports, April 1981.

[2] B.J. Day. Construction of Biclosed Categories. PhD thesis, University of New South Wales, Australia, 1970.

[3] B.J. Day. An embedding of bicategories. Technical Report 262, The University of Sydney, 1976.

[4] B.J. Day. An embedding theorem for closed categories. In Lecture Notes in Mathematics 420, pages 55–64, Springer, 1974.

[5] B.J. Day. Note on monoidal monads. Journal of the Australian Math Society, 23:292– 311, 1977.

[6] B.J. Day. On closed categories of functors. In Lecture Notes in Mathematics 137, pages 1–38, Springer, 1970.

[7] B.J. Day. Promonoidal functor categories. Journal of the Australian Math Society, 23:312–328, 1977.

[8] S. Eilenberg and G.M. Kelly. Closed categories. In Proc. Conf. Categorical Algebra at La Jolla 1965, pages 421–562, Springer, 1966.

[9] G.B. Im and G.M. Kelly. A universal property of the convolution monoidal structure. Journal of Pure and Applied Algebra, 43:75–88, 1986.

[10] G.M. Kelly. Basic Concepts of Enriched Category Theory. Cambridge University Press, Cambridge; New York, 1982.

[11] G.M. Kelly and S. Lack. Finite-product-preserving functors, kan extensions, and strongly-finitary 2-monads. Applied Categorical Structures, 1:85–94, 1993.

Mathematics Department Macquarie University New South Wales 2109 AUSTRALIA

Email: bday@mpce.mq.edu.au and street@mpce.mq.edu.au

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