Theory and Applications of Categories, Vol. 13, No. 11, 2004, pp. 169–171.

EVERY SMALL Sl-ENRICHED CATEGORY IS MORITA EQUIVALENT TO AN Sl-MONOID

Dedicated to Professor Aurelio Carboni on the occasion of his sixtieth birthday

BACHUKI MESABLISHVILI

Abstract. We show that every small category enriched over Sl—the symmetric monoidal closed category of sup-lattices and sup-preserving morphisms—is Morita equivalent to a Sl-monoid. As a corollary, we obtain a result of Borceux and Vitale [1] asserting that every separable Sl-category is Morita equivalent to a separable Sl-monoid.

1. Theorem

We use [2] as a reference for enriched category theory.

Let Sl = (Sl_{∘}, −⊗−, I) denote the symmetric monoidal closed category of sup-lattices with sup-preserving maps.

It is well-known [3] that Sl_{∘} is a complete, cocomplete pointed category. Moreover, (small) coproducts are biproducts; that is, if {x_{i}} is a (small) family of objects of Sl_{∘}, the unique morphism

δ : ∐ x_{i} [figure] ∏ x_{i}

with components δ_{ii} = 1_{x_{i}} and δ_{ij} = 0 otherwise is an isomorphism. For any small Slcategory A, this property transfers to the Sl-functor category [A^{op}, Sl], since limits and colimits there are computed pointwise.

We will write ⊕x_{i} for the biproduct of the family {x_{i}}.

1.1. Theorem. Every small Sl-category is Morita equivalent to an Sl-monoid.

Proof. We first observe that for each pair of objects c, c′ of an Sl-enriched category C, the set C_{∘}(c, c′) is non-empty because every sup-lattice contains at least the element 0.

Now let A be a small Sl-category. Define the Sl-functor P_{A} : A^{op} [figure] Sl to be ⊕_{a∈A}A(−, a). Since arbitrary (small) coproducts (=products) of representable Sl-functors are in the Cauchy completion \overline{A} of A (see [4]), P_{A} is in \overline{A}. Thus the Sl-functor P_{A} is small projective in [A^{op}, Sl].

The work was partially supported by INTAS-97-31961

Received by the editors 2003-09-15 and, in revised form, 2003-10-13.

Published on 2004-12-05.

2000 Mathematics Subject Classification: 18A25, 18D20.

Key words and phrases: Sup-lattices, Morita equivalence, Separable category .

\begin{matrix}c \\ ◯ \end{matrix} Bachuki Mesablishvili, 2004. Permission to copy for private use granted.

Next, by the observation at the beginning of the proof, no [A^{op}, Sl](A(−, a), A(−, a′)) is empty. Moreover, the small Sl-full subcategory determined by the representable Slfunctors is dense in [A^{op}, Sl]. So that, in view of Proposition 5.22 of [2], the Sl-full subcategory [P_{A}] ⊆ [A^{op}, Sl] generated by the Sl-functor P_{A} is dense, and a fortiori strongly generating, in [A^{op}, Sl]. Thus P_{A} : A^{op} [figure] Sl, besides being small projective in [A^{op}, Sl], is a strong generator for [A^{op}, Sl]. It now follows from Theorem 5.26 of [2] that the Sl-functor

N_{A} : [A^{op}, Sl] [figure] [[P_{A}]^{op}, Sl]

given by

N_{A}(F) = [A^{op}, Sl](P_{A}, F )

is an equivalence of Sl-categories. Whence A is Morita equivalent to the Sl-monoid [A^{op}, Sl](P_{A}, P_{A}).

1.2. Remark. There is precisely an analogous result wherein Sl is replaced by the symmetric monoidal closed category of abelian groups, Ab, and A by an Ab-enriched category with finitely many objects (see, for example, [5]); and still another with Sl replaced by the symmetric monoidal closed category of commutative monoids, once again in the finitely-many-object case.

2. Application

Let V = (V_{∘}, −⊗−, I) be a symmetric monoidal closed category whose underlying ordinary category V_{∘} is locally small, complete and cocomplete. Write Bim(V) for the bicategory of small V-categories, V-bimodules and V-natural transformation between them.

Recall [1] that a small V-category A is separable when the canonical V-functor

A(−, −) : A^{op} ⊗ A [figure] V

is small projective in the V-functor category [A^{op} ⊗ A, V].

2.1. Proposition. Separability is invariant under Morita equivalence.

Proof. Recall (for instance from [2]) that two small V-categories A and B are Morita equivalent if and only if there exist bimodules ϕ : [figure: A; B; ⊙] and ψ : [figure: B; A; ⊙] such that ψ ∘ ϕ ≃ A and ϕ ∘ ψ ≃ B. Then

[A^{op} ⊗ A, V] ≃ Bim(V)(A, A) [figure: −∘ψ; ≃] Bim(V)(B, A) [figure: ϕ∘−; ≃] Bim(V)(B, B) ≃ [B^{op} ⊗ B, V],

and (up to isomorphism) this composite takes the V-functor A(−, −) : A^{op} ⊗ A [figure] V to the V-functor B(−, −) : B^{op} ⊗ B [figure] V. The result now follows from the fact that any equivalence of cocomplete V-categories preserves and reflects the property of objects of being small projective.

EVERY SMALL Sl-ENRICHED CATEGORY MORITA EQUIVALENT TO AN Sl-MONOID171

It follows from Theorem 1.1 and Proposition 2.1 that

2.2. Theorem. [1] Every separable Sl-category is Morita equivalent to a separable Sl-monoid. In particular, every Azumaya Sl-category is Morita equivalent to an Azumaya Sl-monoid.

References

[1] Borceux, F. and Vitale, E.: Azumaya categories, Applied Categorical Structures 10 (2002), 449–467.

[2] Kelly, G. M.: Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Notes Series 20, Cambridge University Press, Cambridge, (1982).

[3] Joyal, A. and Tierney, M.: An Extension of the Galois Theory of Grothendieck, Mem. Amer. Math. Soc. 51, no 309, (1984).

[4] Johnson, S. R.: Small Cauchy Completions, Journal of Pure and Applied Algebra 62 (1989), 35–45.

[5] Mitchell, B.: Rings with Several Objects, Advances in Math. 8 (1972), 1–161.

Mathematical Institute of the Georgian Academy of Sciences, M. Alexidze Str. 1, 380093 Tbilisi, Georgia

Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium.

Email: bachi@rmi.acnet.ge, bachuki@math.ucl.ac.be

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