Theory and Applications of Categories, Vol. 32, No. 36, 2017, pp. 1222–1228.

COEXPONENTIABILITY AND PROJECTIVITY: RIGS, RINGS, AND QUANTALES

S.B. NIEFIELD AND R.J. WOOD

Abstract. We show that a commutative monoid A is coexponentiable in CMon(V) if and only if −⊗A: V [figure] V has a left adjoint, when V is a cocomplete symmetric monoidal closed category with finite biproducts and in which every object is a quotient of a free. Using a general characterization of the latter, we show that an algebra over a rig or ring R is coexponentiable if and only if it is finitely generated and projective as an R-module. Omitting the finiteness condition, the same result (and proof) is obtained for algebras over a quantale.

1. Introduction

Recall that an object A of a category A with finite products is exponentiable if and only if − × A: A [figure] A has a right adjoint. In [Niefield, 1982], the first author showed that for an algebra A over a commutative ring R, the spectrum Spec(A) is exponentiable in the category of affine schemes over Spec(R) if and only if A is finitely generated and projective as an R-module, and later showed in [Niefield, 2016] that essentially the same proof gave a characterization of coexponentiable morphisms of quantales (with the finiteness condition omitted). After a presentation of the latter, Lawvere and Menni asked if this characterization also generalized to rigs and, in particular, idempotent rigs.

A rig (or “ring without negatives”) is another name for a commutative semiring, and an idempotent rig is one in which 1+1 = 1. See, for example [Schanuel, 1991], where the Burnside rig of a distributive category is introduced, [Lawvere/Schanuel] for the study of rigs in “Objective Number Theory,” or more recently [Castiglioni et al]. Note that idempotent rigs are 𝟚-rigs (i.e., rigs under 𝟚), or alternately, commutative monoids in the category of join-semilattices with 0.

COEXPONENTIABILITY AND PROJECTIVITY: RIGS, RINGS, AND QUANTALES 1223

Since rigs, rings, and quantales are commutative monoids in an appropriate monoidal category, such a generalization seemed reasonable. However, the proof for quantales (respectively, rings) used a property of modules that does not appear to hold for modules over a rig, namely, every flat (respectively, finitely presented flat) module is projective (respectively, finitely generated projective). After consulting the vast semiring literature, the ring/quantale approach did not seem feasible for rigs. Then, a 1981 letter from the second author surfaced including an alternate proof that for a module M over a ring R, the endofunctor − ⊗_{R} M has a left adjoint if and only if the canonical morphism

Received by the editors 2016-08-23 and, in final form, 2017-09-09.

Transmitted by Richard Blute. Published on 2017-09-11.

2010 Mathematics Subject Classification: 18A40, 18D15, 13C10, 16Y60.

Key words and phrases: monoidal category, projective module, rig.

\begin{matrix}c \\ ◯ \end{matrix} S.B. Niefield and R.J. Wood, 2017. Permission to copy for private use granted.

M ⊗_{R} Hom_{R}(M, R) [figure] Hom_{R}(M, M)

is an isomorphism, and that this is equivalent to M being finitely generated and projective.

In this paper, we show that the above property (and the construction of coexponentials of rings) generalizes to any cocomplete symmetric monoidal closed category with finite biproducts and in which every object is a quotient of a free, and hence gives a characterization of coexponentiable rigs.

2. Left Adjoints to Tensor

Throughout this section, we assume (V, I, ⊗, [·, ·], …) is a symmetric monoidal closed category in which every object is a quotient of a free, i.e., for every V there is a regular epimorphism of the form ⊕_{α}I [figure] V, where the domain is a coproduct of copies of the unit I indexed by α.

2.1. Proposition. The following are equivalent for an object V of V.

(a) The functor − ⊗ V: V [figure] V has a left adjoint.

(b) The functor − ⊗ V: V [figure] V preserves limits.

(c) The canonical morphism θ_{V}: V ⊗ [V, I] [figure] [V, V] is an isomorphism.

(d) The functor − ⊗ V is left and right adjoint to − ⊗ [V, I].

Proof. Clearly, (d) ⇒ (a) ⇒ (b). To show that (b) ⇒ (c), suppose ⊕_{β}I [figure]⊕_{α} I [figure] V is a coequalizer, and consider the commutative diagram

[figure: [V, V]; [⊕_{α}I, V]; V ⊗ [V, I]; [V, V]; θ_{V}; V ⊗ [V, I]; V ⊗ [⊕_{α}I, I]; V ⊗ [⊕_{α}I, I]; [⊕_{α}I, V]; θ_{α}; V ⊗ [⊕_{β}I, I]; [⊕_{β}I, V]; θ_{β}]

where the rows are equalizers, since [−, W] takes coequalizers to equalizers, for all W, and V ⊗ − preserves equalizers by assumption (b). Since [−, V] takes coproducts to products, the canonical morphism [⊕_{α}I, V] [figure]∏_{α}[I, V] ≅ ∏_{α} V is an isomorphism. Likewise, for [⊕_{α}I, I] [figure]∏_{α} I, and so θ_{α} is an isomorphism, since − ⊗ V preserves products by assumption. Similarly, θ_{β} is an isomorphism, and it follows that θ_{V} is, as well.

For (c) ⇒ (d), we will show that −⊗V is left adjoint to −⊗[V, I]. The other adjunction

follows by symmetry of ⊗. Consider ε_{I}: [V, I] ⊗ V [figure: ε_{V,I}] I, the counit of − ⊗ V ⊣ [V, −], and

η_{I}: I [figure: ι] [V, V] [figure: θ_{V}^{−1}]V ⊗ [V, I], where ι is the transpose of the identity map on V. Tensoring on the left with W, we get

ε_{W}: W ⊗ [V, I] ⊗ V [figure] W and η_{W}: W [figure] W ⊗ V ⊗ [V, I]

To see that the adjunction identities hold, it suffices to show they do when W = I. Since

[figure: V ⊗ [V, I] ⊗ V; [V, V] ⊗ V; θ_{V} ⊗V; V ⊗ [V, I] ⊗ V; V; V ⊗ε_{V,I}; [V, V] ⊗ V; V; ε_{V,V}]

commutes, by definition of θ_{V}, it follows that

[figure: [V, V] ⊗ V; V ⊗ [V, I] ⊗ V; θ_{V}^{−1}⊗V; [V, V] ⊗ V; V; ε_{V,V}; V ⊗ [V, I] ⊗ V; V; V ⊗ε_{V,I}; V; [V, V] ⊗ V; ι⊗V; V; V; id_{V}]

commutes, and so (V ⊗ ε_{I})(η_{I} ⊗ V) = id_{V}. Also,

[figure: [V, I] ⊗ V ⊗ [V, I]; [V, I] ⊗ [V, V]; [V,I]⊗θ_{V}; [V, I] ⊗ V ⊗ [V, I]; [V, I]; ε_{V,I}⊗[V,I]; [V, I] ⊗ [V, V]; [V, I]; ∘]

commutes, by definition of ∘ and θ_{V}. Thus, we get a commutative diagram

[figure: [V, I] ⊗ [V, V]; [V, I] ⊗ V ⊗ [V, I]; [V,I]⊗θ_{V}^{−1}; [V, I] ⊗ [V, V]; [V, I]; ∘; [V, I] ⊗ V ⊗ [V, I]; [V, I]; ε_{V,I}⊗[V,I]; [V, I]; [V, I] ⊗ [V, V]; [V,I]⊗ι; [V, I]; [V, I]; id_{[V,I]}]

and it follows that (ε_{I} ⊗ [V, I])([V, I] ⊗ η_{I}) = id_{[V,I]}, as desired.

Proposition 2.1 applies when V is the category Ab of abelian groups, the category Sup of sup lattices, the category CMon of commutative monoids, and more generally in each case, the category RMod(V) of modules over a commutative monoid R in V (i.e., a ring, quantale, or rig). Taking R = I, we get IMod(V) = V, and so the latter case includes each of the first three. Note that the tensor product ⊗ in CMon is similar to that of Ab and Sup.

Suppose θ_{V}: V ⊗ [V, I] [figure] [V, V] is an isomorphism, and write

θ_{V}^{−1}(id_{V}) = ∑_{α∈S} v_{α} ⊗ φ_{α}

COEXPONENTIABILITY AND PROJECTIVITY: RIGS, RINGS, AND QUANTALES 1225

where S is finite in all cases but V = Sup. Applying θ_{V}, we see that ∑_{α} ϕ_{α}(v)v_{α} = v, for all v ∈ V. Thus, the v_{α} generate V and we get a morphism r : ⊕_{α}I [figure] V. Since ⊕_{α}I ≅ ∏_{α} I, in each case, the ϕ_{α} induce a morphism i: V [figure]⊕_{α} I such that ri = id_{V}, and it follows that V is projective.

Conversely, suppose F [figure: i; r] V is a retraction of F onto V, where F = ⊕_{α}I ≅ ∏_{α} I, indexed by a finite set A in all cases but V = Sup, and consider the diagram

[figure: F ⊗ [F, I]; [F, F]; θ_{F}; V ⊗ [V, I]; [V, V]; θ_{V}]

where the horizontal morphisms are the retractions induced by i and r, and the squares commutes. Since F = ⊕_{α}I ≅ ∏_{α} I, it follows that θ_{F} is an isomorphism, and so θ_{V} is as well.

Thus, we get the following corollaries which are well known when V is Ab and Sup (see [Niefield, 1982], [Joyal/Tierney, 1984]). 2.2. Corollary. Suppose V is CMon or Ab, and R is a commutative monoid in V, i.e., a rig or ring. Then the following are equivalent for an R-module M in V.

(a) The functor − ⊗_{R} M: RMod [figure] RMod has a left adjoint.

(b) M is finitely generated and projective.

(c) The functor − ⊗_{R} M is left and right adjoint to − ⊗_{R} Hom_{R}(M, R). Since the category of 𝟚-modules in CMon is isomorphic to the category Semi of

join-semilattices (with 0), Corollary 2.2 becomes: 2.3. Corollary. The following are equivalent for X in Semi.

(a) The functor − ⊗ X: Semi [figure] Semi has a left adjoint.

(b) X is finitely generated and projective.

(c) The functor − ⊗ X is left and right adjoint to − ⊗ Semi(X, 𝟚). 2.4. Corollary. Suppose Q is a commutative quantale. Then the following are equivalent for a Q-module M.

(a) The functor − ⊗_{Q} M: QMod [figure] QMod has a left adjoint.

(b) M is projective

(c) The functor − ⊗_{Q} M is left and right adjoint to − ⊗_{Q} Hom_{Q}(M, Q).

3. Coexponentiable Commutative Monoids

Throughout this section, we assume V is a cocomplete symmetric monoidal closed category with finite biproducts (in the sense of [Mac Lane, 1971]) and in which every object is a quotient of a free. To simplify notation, let C denote the category of commutative monoids in V. Then C has finite coproducts given by the tensor ⊗ of V, and so one can consider coexponentiable commutative monoids V.

Every object V of V gives rise to two objects of C. In addition to the free commutative monoid SV (see below), there is a monoid structure on I × V defined as follows. Take η: I [figure] I × V to be the morphism whose first projection is the identity on I and second projection is the composite I [figure] 0 [figure] V, where 0 is the initial and terminal object of V, and let μ: (I × V) ⊗ (I × V) [figure] I × V denote the morphism with first projection given by

(I × V) ⊗ (I × V) [figure: π_{1}⊗π_{1}] I ⊗ I ≅ I

and second projection by

(I × V) ⊗ (I × V) [figure: (π_{1}⊗π_{2},π_{2}⊗π_{1})](I ⊗ V) × (V ⊗ I) ≅ (I ⊗ V) ⊕ (V ⊗ I) [figure: ^{\binom{λ}{ρ}}]V

where λ and ρ are the structure isomorphism in V. Then, one can show that I × V is a commutative monoid in V, and every morphism f: V [figure] W of V induces a homomorphism I × f: I × V [figure] I × W. The operation μ can be thought of as a “generalized derivation” since for modules over a ring or rig R, it is defined by (r, v)(s, w) = (rs, rw + sv).

Recall that the free commutative monoid SV (with unit i: V [figure] SV) is defined via the coequalizer

T V ⊗ T V [figure] T V [figure] SV

imposing commutativity on the free monoid T V = ⊕_{n≥0}V^{⊗n}. One can show that the counit ε: SC [figure] C is a regular epimorphism in C, since it is a retraction in V which induces a coequalizer

SC [figure: iε; id_{SC}]SC [figure: ε] C

in V and, since the tensor product of coequalizers of this form is a coequalizer in V, the corresponding diagram

S(SC) [figure] SC [figure: ε] C

is a coequalizer in C.

3.1. Theorem. Let C be a commutative monoid in V. Then C is coexponentiable in C if and only if − ⊗ C: V [figure] V has a left adjoint.

Proof. By Proposition 2.1 it suffices to show that C is coexponentiable in C if and only if − ⊗ C preserves limits in V.

Suppose C is coexponentiable in C. Given a diagram {V_{α}} in V, there is a retraction

I × V_{α} [figure: i_{2}; π_{2}] V_{α} in V, for each α, and thus we get a commutative diagram

[figure: (lim_{α} V_{α}) ⊗ C; lim_{α}(V_{α} ⊗ C); (lim_{α}(I × V_{α})) ⊗ C; lim_{α}((I × V_{α}) ⊗ C); θ]

where the horizontal morphisms are retractions in V. Since C is coexponentiable and θ is a homomorphism, we know that θ is an isomorphism in C, and it follows that − ⊗ C preserves limits in V.

Conversely, suppose − ⊗ C preserves limits in V. Then − ⊗ [C, I] ⊣ − ⊗ C in V, by Proposition 2.1. Then we have the following bijections, natural in V and B,

C(SV, B ⊗ C) ≅ V(V, B ⊗ C) ≅ V(V ⊗ [C, I], B) ≅ C(S(V ⊗ [C, I]), B)

Given A in C, there is a coequalizer S(SA) [figure: f; g] SA [figure] A in C, and so defining L_{C}A to be

the coequalizer in C

S(SA ⊗ [C, I]) [figure] S(A × [C, I]) [figure] L_{C}A

of the morphisms induced by f and g, yields the desired natural bijection

C(A, B ⊗ C) ≅ C(L_{C}A, B)

Theorem 3.1 applies when V = RMod, where R is a commutative ring, rig, or quantale. In each case, C is the category RAlg of (commutative) R-algebras. Thus, by the corollaries to Proposition 2.1 we get:

3.2. Corollary. Suppose R is a commutative ring or rig. Then A is coexponentiable in RAlg if and only if A is finitely generated and projective as an R-module.

3.3. Corollary. Suppose Q is a commutative quantale. Then A is coexponentiable in QAlg if and only if A is projective as a Q-module.

Recall that C is a monoid in V, then CAlg is isomorphic to the category of commutative monoids under C in V, and so these corollaries become the following characterizations of coexponentiable morphisms in C. Note that, since every rig under an idempotent rig R is again idempotent, the category of rigs under such an R is isomorphic to the full subcategory of idempotent ones.

3.4. Corollary. A morphism R [figure] A of commutative rings, rigs, or idempotent rigs is coexponentiable if and only if A is finitely generated and projective as an R-module.

3.5. Corollary. A morphism Q [figure] A of commutative quantales is coexponentiable if and only if A is projective as a Q-module.

References

J.L. Castiglioni, M. Menni, W. J. Zuluaga Botero, A representation theorem for integral rigs and its applications to residuated lattices, preprint (arXiv:1510.06332).

A. Joyal and M. Tierney (1984), An Extension of the Galois Theory of Grothendieck, Amer. Math. Soc. Memoirs 309.

F. W. Lawvere (2008). Core varieties, extensivity, and rig geometry. Theory Appl. Categ., 20(14):497–503.

F. W. Lawvere and S. H. Schanuel, Objective Number Theory, 2015 preprint

S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York, 1971.

S. B. Niefield (1982), Cartesianness: topological spaces, uniform spaces, and affine schemes, J. Pure Appl. Algebra 23, 147–167.

S. B. Niefield (2016), Projectivity, continuity, and adjointness: quantales and their modules, submitted.

S. H. Schanuel (1991) Negative sets have Euler characteristic and dimension. Category Theory, Proc. Int. Conf., Como/Italy 1990, Lect. Notes Math. 1488, 379–385.

Department of Mathematics, Union College Schenectady, NY 12308, USA

Department of Mathematics and Statistics, Dalhousie University Chase Building, Halifax, Nova Scotia, Canada B3H 3J5

Email: niefiels@union.edu
      rjwood@mathstat.dal.ca
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