Topological rigidity as a monoidal equivalence

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Communications in Algebra

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Topological rigidity as a monoidal equivalence

Laurent Poinsot

To cite this article: Laurent Poinsot (2019) Topological rigidity as a monoidal equivalence, Communications in Algebra, 47:9, 3457-3480, DOI: 10.1080/00927872.2019.1566957 To link to this article: https://doi.org/10.1080/00927872.2019.1566957

Published online: 04 Mar 2019.

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Topological rigidity as a monoidal equivalence

Laurent Poinsot^a,b

aLIPN-UMR CNRS 7030, University Paris 13, Sorbonne Paris Cite, Villetaneuse, France;^bCREA, French Air Force Academy, Salon-de-Provence, France

ABSTRACT

A topological commutative ring is said to be rigid when for every set X, the topological dual of the X-fold topological product of the ring is iso- morphic to the free module over X. Examples are fields with a ring top- ology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of“topologically free” modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and some coalgebras. In particular, we provide a description of its relationship with the standard duality between algebras and coalgebras, namely finite duality.

ARTICLE HISTORY Received 15 July 2018 Revised 3 December 2018 Communicated by Alberto Facchini

KEYWORDS

Coalgebras; finite duality;

topological dual space;

topological basis MATHEMATICS SUBJECT CLASSIFICATION 13J99; 54H13;

46A20; 19D23

1. Introduction

The main result of [13] states that given a (Hausdorff) topological fieldðk;sÞ, for every setX, the topological dual ððk;sÞ^XÞ⁰ of the X-fold topological product ðk;sÞ^X is isomorphic to the vector spacek^ðXÞof finitely supportedk-valued maps defined onX(i.e., those mapsX!^f k such that for all but finitely many membersxof X, f(x)¼0).

Actually this topological property ofrigidity is shared by more general topological (commuta- tive unital) rings than only topological fields (a fact not noticed in [13]). For instance any discrete ring is rigid in the above sense (see Lemma 15). And even if not all topological rings are rigid (seeSection 4.3for a counterexample), many of them still are (e.g., every real or complex normed commutative algebra).

It is our intention to study in more details some consequences of the property of rigidity for arbitrary commutative rings in particular at the level of some of their topological algebras.¹ So far, for a topological ringðR;sÞ, rigidity reads asððR;sÞ^XÞ⁰’R^ðXÞ for each setX. Suitably topolo- gized (seeSection 3.1), the algebraic dualðR^ðXÞÞ turns out to be isomorphic toðR;sÞ^X.

More appropriately the above correspondence may be upgraded into a dual equivalence of cat- egories between free and topologically free modules, i.e., those topological modules isomorphic to some ðR;sÞ^X (Theorem 45) under the algebraic and topological dual functors. (This extends a similar interpretation from [13] to the realm of arbitrary commutative rigid rings.)

Under the rigidity assumption, the aforementioned dual equivalence enables to provide a topo- logical tensor product ⊛ðR;sÞ for topologically free ðR;sÞ-modules by transporting the algebraic tensor product_R along the dual equivalence. It turns out that ⊛ðR;sÞ is (coherently) associative,

ß2019 Taylor & Francis Group, LLC

CONTACTLaurent Poinsot [email protected] CREA, French Air Force Academy, Base aerienne 701, 13661 Salon-de-Provence, France.

1The results of the present contribution also serve in a subsequent paper under preparation about topological semisimplicity of commutative topological algebras.

https://doi.org/10.1080/00927872.2019.1566957

commutative, and unital, i.e., makes monoidal the category of topologically free modules (Proposition 61). Not too surprisingly the above dual equivalence remains well-behaved, i.e., monoidal, with respect to the (algebraic and topological) tensor products (Theorem 64).

According to the theory of monoidal categories, this in turn provides a dual equivalence between monoids in the tensor category of topologically free modules (some suitably generalized topo- logical algebras) and coalgebras with a free underlying module (Corollary 66). So there are two constructions: a topological dual coalgebra of a monoid (in the tensor category of topologically free modules) and an algebraic dual monoid of a coalgebra, and these constructions are inverse one from the other (up to isomorphism).

There already exists a standard duality theory between algebras and coalgebras, over a field, known as finite duality but contrary to our “topological duality” it is merely an adjunction, not an equivalence. One discusses how these dualities interact (see Section 7) and in particular one proves that the algebraic dual monoid of a coalgebra essentially corresponds to its finite dual (Section 7.2), that over a discrete field, the topological dual coalgebra of a monoid is a subcoalge- bra of the finite dual coalgebra of its underlying algebra and furthermore that they are equal exactly when finite duality provides an equivalence of categories (Theorem 77).

2. Conventions, notations, and basic definitions 2.1. Conventions and notations

One assumes that the reader is familiar with standard notions and notations from category theory ([12]), and some others will be introduced in the text.

Except as otherwise stipulated, all topologies are Hausdorff, and every ring is assumed unital and commutative. Algebras are only assumed associative and commutative.

For a ring R, Rdenotes both its underlying set and the canonical left R-module structure on its underlying additive group, and m_R:RR!R is its bilinear multiplication. Likewise if A is anR-algebra, then A is both its underlying set and its underlying R-module. The unit of a ring R (resp., unital algebra A) is denoted by 1R (resp. 1_A). A ring map (or morphism of rings) is assumed to preserve the units.

A product of topological spaces always has the product topology unless otherwise stated.

When for eachx2X, all ðEx;s_xÞ’s are equal to the same topological spaceðE;sÞ, then theX-fold topological productQ

x2XðEx;sxÞis canonically identified with the setE^Xof all maps fromXto E equipped with the topology of simple convergence, and is denoted byðE;sÞ^X. Under this identifi- cation, the canonical projections ðE;sÞ^X!^pxðE;sÞ are given by pxðfÞ ¼fðxÞ;x2X;f 2E^X. The symbold always represents the discrete topology.

2.2. Basic definitions

Definition 1. Let R be a ring. A (Hausdorff, following our conventions) topology s of (the carrier set of) the ring is called a ring topology when addition, multiplication, and additive inversion of the ring are continuous. By topological ringðR;sÞis meant a ring together with a ring topologyson it². By a field with a ring topology, denotedðk;sÞ, is meant a topological ringðk;sÞwithka field.

Let ðR;sÞbe a topological ring. A pair ðM;rÞ consisting of a (left and unital³) R-module M and a topology r on M which makes continuous the addition, additive inversion, and scalar multiplicationRM!M, is called atopologicalðR;sÞ-module. Such a topology is referred to as a ðR;sÞ-module topology. In particular, when R is a field k, then this provides topological

2In view ofSection 2.1, the multiplication of a topological ring is jointly continuous.

3Unital means that the scalar action of the unit ofRis the identity on the module.

ðk;sÞ-vector spaces. Given topological ðR;sÞ-modules ðM;rÞ;ðN;cÞ, a (continuous) homomorph- ism ðM;rÞ!^f ðN;cÞ is a R-linear map M!^f N which is continuous. Topological ðR;sÞ-modules and these morphisms form a category TopMod_ðR;sÞ, which is denoted TopVect_ðk;sÞ, when k is a field.

A pairðA;rÞ, withAa unitalR-algebra, andr a topology onA, is atopologicalðR;sÞ-algebra, when r is a module topology for the underlying R-module A, and the multiplication of A is a bilinear (jointly) continuous map. Given topological ðR;sÞ-algebras ðA;rÞ;ðB;cÞ, a continuous ðR;sÞ-algebra mapðA;rÞ!^f ðB;cÞis a unit preservingR-algebra mapA!^f Bwhich is also continu- ous. Topological ðR;sÞ-algebras with these morphisms form a category 1TopAlg_ðR;sÞ. One also has the full subcategory_1;cTopAlg_ðR;sÞ of unital and commutative topological algebras.

2.3.X-fold product and finitely supported maps

LetRbe a ring. Mod_R is the category of unital left-R-modules withR-linear maps. WhenRis a fieldkone usesVect_k instead.

LetX be a set. The R-module R^X of all maps from Xto R, equivalently defined as theX-fold power of R in the category Mod_R, is the object component of a functor P from the opposite Set^op of the category of sets to Mod_R, whose action on maps is as follows: given X!^f Y and g2R^Y;PRðfÞðgÞ ¼gf. R^X is merely not just a R-module but, under point-wise multiplication R^XR^X!^MXR^X, a commutative R-algebra, the usual function algebra on X, denoted ARðXÞ, with unit 1_ARðXÞ:¼P

x2Xd^R_x, where d^R_x, or simply d_x, is the member of R^X with d^R_xðxÞ ¼1R;x2R, and fory2X;y6¼x;d^R_xðyÞ ¼0. This actually provides a functor Set^op!^AR_1;cAlg_R, where _1;cAlg_R is the full subcategory spanned by unital and commutative algebras of the category1Alg_R of (asso- ciative) unitalR-algebras with unit preserving algebra maps. (The multiplicationmA of an algebra Athus is aR-bilinear mapAA!^mAA.)

Let f 2R^X. The support of f is the set suppðfÞ:¼ f x2X:fðxÞ 6¼0 g. Let R^ðXÞ be the sub- R-module of R^X consisting of all finitely supported maps (or maps with finite support), i.e., the mapsfsuch thatsupp(f) is finite.

R^ðXÞ is actually the freeR-module overX, and a basis is given byf d^R_x :x2X g. Observe that the mapX!^dRXR^X;x7!dx, is one-to-one if, and only if,Ris nontrivial and jXj>1, orjXj 1 andR is arbitrary (even trivial).

Remark 2. There is the free module functor Set!^FRMod_R which is a left adjoint of the usual for- getful functor Mod_R!^jjSet; FRðXÞ:¼R^ðXÞ, and for X!^f Y;F_RðfÞðd^R_xÞ:¼d^R_fðxÞ;x2X. The map X!^dRXjR^ðXÞjis the component at X of the unit of the adjunctionF_Ra j j:Set!Mod_R.

Let ðR;sÞ be a topological ring and let X be a set. Since for a map X!^f Y;p_xP_ðR;sÞðfÞ ¼ p_fðxÞ;x2X;ðR;sÞ^YP!^RðfÞðR;sÞ^X is continuous, and thus one has a topological power func- torSet^opP!^ðR;sÞTopMod_ðR;sÞ.

Lemma 3.ðR;sÞ^X ðR;sÞ^X!^MXðR;sÞ^X is continuous.

Proof. MX is of course separately continuous in both of its variables. Continuity at zero of m_R almost directly implies that ofMX, and thus its continuity by [19, Theorem 2.14, p. 16]. w

AðR;sÞ:X7!ððR;sÞ^X;MX;1_ARðXÞÞ is a functor too and the diagram below commutes, with the forgetful functors unnamed.

(1)

Notation 4. The underlying topological ring of AðR;sÞðXÞ is denoted ðR;sÞ^X (and is the X-fold product ofðR;sÞin the category of topological rings).

3. Recollection of results about algebraic and topological duals 3.1. Algebraic dual functor

Let Rbe a ring. Let M be a R-module. Let M :¼Mod_RðM;RÞ be thealgebraic (or linear) dual ofM. This is readily aR-module on its own.

WhenðR;sÞis a topological ring, thenM may be topologized with the initial topology ([5, p.

30] or [19, Theorem 2.17, p. 17]) w_ðR;sÞ, called the weak- topology, induced by the family ðM^K!^MðvÞRÞ_v2M ofevaluationsat some points, whereðKMðvÞÞð‘Þ:¼‘ðvÞ. A basis of neighborhoods

of zero in this topology is given by the sets of the form

\v2FKMðvÞ¹ðf Uv gÞ ¼ f ‘2M:8v2F; ‘ðvÞ 2Uv g, where F M is a finite set and for each v2F;02Uv2s. This provides a structure of topological ðR;sÞ-module on M, which is even Hausdorff by [5, Corollary 1, p. 78]. As an initial topology it is characterized by the follow- ing universal property: a linear mapN!^f M, whereðN;rÞis a topological ðR;sÞ-module, is con- tinuous from ðM;rÞ to ðM;w_ðR;sÞÞ if, and only if, KMðvÞ f :ðN;rÞ ! ðR;sÞis continuous for eachv2M.

Moreover given a linear map M!^f N;N!^fM; ‘7!fð‘Þ:¼‘f, is continuous for the above topologies. Consequently, this provides a functor Mod^op_R^Alg!^ðR;sÞTopMod_ðR;sÞ called the algebraic dual functor.

Remark 5. M andf stand respectively forAlg_ðR;sÞðMÞandAlg_ðR;sÞðfÞ.

Up to isomorphism, one recovers the module of allR-valued maps on a setX, with its product topology, as the algebraic dual of the module of all finitely supported maps duly topologized as above.

Lemma 6.For each set X,ðR;sÞ^X’Alg_ðR;sÞðR^ðXÞÞ(inTopMod_ðR;sÞ) under the map q_X:ðR;sÞ^X!ðR^{ð ÞX}Þ;w_ðR;sÞ

given by

q_Xð Þf ð Þp

:¼X

x2X

p xð Þf xð Þ;f 2R^X;p2R^{ð ÞX}:

Proof. Let‘2 ðR^ðXÞÞ. Let us define X!b‘

Rbyb‘ðxÞ:¼‘ðdxÞ;x2X. That the two constructions are linear and inverse one from the other is clear.

It remains to make sure that there are also continuous. Let‘2 ðR^ðXÞÞ, and let x2X. Then, pxðb‘Þ ¼b‘ðxÞ ¼‘ðdxÞ ¼ ðKR^ðXÞðdxÞÞð‘Þ, which ensures continuity of ððR^ðXÞÞ;w_ðR;sÞÞ!^q1XðR;sÞ^X. Let f 2R^X, and p2R^ðXÞ. As ðKR^ðXÞðpÞÞðq_XðfÞÞ ¼ ðq_XðfÞÞðpÞ ¼P

x2XpðxÞfðxÞ ¼P

x2XpxðpÞfðxÞ ¼

Px2XpxðpÞpxðfÞ;KR^ðXÞðpÞ q_X is a finite linear combination of projections, whence is continuous

for the product topology, so isq_X. w

LetM be a free R-module. Let Bbe a basis ofM. This defines a family ofR-linear maps, the coefficient mapsðM!^bRÞ_b2B such that eachv2M is uniquely represented as a finite linear com- binationv¼P

b2BbðvÞb. One denotes FreeMod_R the full subcategory of Mod_Rspanned by the free modules. Whenkis a field,FreeModk is just Vectk itself.

Example 7. For each setX, px¼d_x, wherepx:¼R^ðXÞ,!R^X!^pxR, x2X.

Remark 8. bðdÞ ¼d_bðdÞ;b;d2B.SoðÞ:B!B:¼ f b:b2B gis a bijection.

Given a free R-module, any choice of a basis B provides the initial topology P^sB on M induced byðKMðbÞÞ_b2B. (Of course,P^sB w_ðR;sÞ.)

Lemma 9. Let M be a free R-module. For any basis B of M, P^sB¼w_ðR;sÞ and ðM;w_ðR;sÞÞ ’ ðR;sÞ^B (inTopMod_ðR;sÞ).

Proof. For‘2M;ðKMðvÞÞð‘Þ ¼P

b2BbðvÞ‘ðbÞ ¼P

b2BbðvÞðKMðbÞÞð‘Þ;v2M, thusKMðvÞis a finite linear combination of some KMðbÞ’s, whence is continuous for P^sB, and so w_ðR;sÞ P^sB.

The last assertion is clear. w

3.2. Topological dual functor

Let ðR;sÞ be a topological ring, and let ðM;rÞ be a topological ðR;sÞ-module. Let ðM;rÞ⁰:¼ TopMod_ðR;sÞððM;rÞ;ðR;sÞÞbe thetopological dualof ðM;rÞ, which is aR-submodule ofM. Let ðM;rÞ!^f ðN;cÞ be a continuous homomorphism between topological modules. Let ðN;cÞ⁰!^f0ðM;rÞ⁰ be the R-linear map given by f⁰ð‘Þ:¼‘f. All of this evidently forms a func- torTopMod^op_ðR;sÞ^Top!^ðR;sÞMod_R.

Let ðR;sÞ be a topological ring, and let X be a set. Let R^ðXÞ!^kXðR^XÞ be given byðkXðpÞÞðfÞ:¼P

x2XpðxÞfðxÞ;p2R^ðXÞ;f 2R^X.

Let q_X be the map from Lemma 6. Then, for each p2R^ðXÞ;kXðpÞ ¼KR^ðXÞðpÞ q_X, which ensures continuity of k_XðpÞ, i.e., k_XðpÞ 2 ððR;sÞ^XÞ⁰. Next lemma follows from the equal- itypðxÞ ¼ ðkXðpÞÞðdxÞ;p2R^ðXÞ;x2X.

Lemma 10. kX:R^ðXÞ! ððR;sÞ^XÞ⁰is one-to-one.

4. Rigid rings: definitions and (counter-)examples

The notion of rigidity, recalled at the beginning of the Introduction, was originally but only implicitly introduced in [13, Theorem 5, p. 156] as the main result therein and the possibility that its conclusion could remain valid for more general topological rings than topological division rings was not noticed. Since a large part of this presentation is given for arbitrary rigid rings (Definition 12 below), one here provides a stock of basic examples.

As [13, Lemma 13, p. 158], one has the following fundamental lemma.

Lemma 11. Let ðR;sÞ be a topological ring, and let X be a set. For each f 2R^X;ðfðxÞd_xÞ_x2X is summable inðR;sÞ^Xwith sum f.

Definition 12. Let ðR;sÞ be a topological ring. It is said to be rigid when for each set X, R^ðXÞ!^kXððR;sÞ^XÞ⁰ is an isomorphism in Mod_R, i.e., k_X is onto. In this situation, one sometimes also called rigid a ring topologyssuch thatðR;sÞis rigid.

Lemma 13. Let‘2 ððR;sÞ^XÞ⁰.‘2imðkXÞif, and only if,b‘:X!R given byb‘ðxÞ:¼‘ðdxÞ, belongs to R^ðXÞ. Moreover,ðb-Þ:imðkXÞ !R^ðXÞ; ‘7!b‘¼P

x2X‘ðdxÞdx, is the inverse of k_X.

4.1. Basic stock of examples

Of course, the trivial ring is rigid (under the (in)discrete topology!). The first result below is a slight generalization of [13, Theorem 5, p. 156] since its proof does not use continuity of the inversion.

Lemma 14. Letðk;sÞbe a field with a ring topology. Then,ðk;sÞis rigid.

Lemma 15. For each ringR, the discretely topologized ringðR;dÞis rigid.

Proof. Let ‘2 ððR;dÞ^XÞ⁰. As a consequence of Lemma 11, ð‘ðd_xÞÞ_x is summable in ðR;dÞ, with sum‘ð1_ARðXÞÞ. Sincef 0 gis an open neighborhood of zero inðR;dÞ; ‘ðdxÞ ¼0 for all but finitely manyx2X([19, Theorem 10.5, p. 73]). The conclusion follows by Lemma 13. w

Every normed, complex or real, commutative, and unital algebra (e.g., Banach or C-algebra) is rigid.

Lemma 16. Letk¼R;C. LetðA;jj jjÞbe a commutative normed k-algebra⁴ with a unit. Then, as a topological ring under the topology induced by the norm, it is rigid.

Proof. Let s_jjjj be the topology on A induced by the norm of A, where A is the underlying k-vector space ofA. LetXbe a set. Let ‘2 ððA;s_jjjjÞ^XÞ⁰. Letf 2A^X be given byfðxÞ ¼_jj‘ðd¹_xÞjj1_A if x2suppðb‘Þ and f(x) ¼ 0 for x62suppðb‘Þ. Since by Lemma 11,ðfðxÞdxÞ_x2X is summable with sum f, ðfðxÞ‘ðdxÞÞ_x2X is summable in ðA;s_jjjjÞ with sum ‘ðfÞ. So according to [19, Theorem 10.5, p. 73], for 1>>0, there exists a finite set F X such thatjjfðxÞ‘ðdxÞjj<for allx2XnF. But 1¼ jjfðxÞ‘ðdxÞjjfor allx2suppðb‘Þso thatsuppðb‘Þis finite, andk_Xis onto by Lemma 13. w

4.2. A supplementary example: von Neumann regular rings

A (commutative and unital) ring is said to bevon Neumann regularif for eachx2R, there exists y2Rsuch thatx¼xyx[9, Theorem 4.23, p. 65].

Let us assume thatR is a (commutative) von Neumann regular ring. For eachx2R, there is a uniquex^†2R, called theweak inverseofx, such thatx¼xx^†xandx^†¼x^†xx^†.

Example 17. A field is von Neumann regular with x^†:¼x¹;x6¼0, and 0^†¼0. More generally, letðkiÞ_i2I be a family of fields. LetRbe a ring, and let˚:R!Q

i2Ik_i be a one-to-one ring map.

Assume that for each x2R;˚ðxÞ^†2imð˚Þ, where for ðxiÞ_i2I2Q

i2Ik_i;ðxiÞ^†_i2I:¼ ðx^†_iÞ_i2I. Then, R is von Neumann regular.

4In a normed algebra ðA;jj jjÞ, unital or not, commutative or not, the norm is assumed sub-multiplicative, i.e., jjxyjj jjxjjjjyjj, which ensures that the multiplication of Ais jointly continuous with respect to the topology induced by the norm.

Remark 18.Let Rbe a von Neumann regular ring. For eachx2R;x6¼0 if, and only if,xx^† 6¼0.

Alsoxx^† belongs to the setEðRÞof all idempotents(e²¼e)ofR.

Proposition 19. Let ðR;sÞ be a topological ring with R von Neumann regular. If 062EðRÞ n f 0 g, thenðR;sÞis rigid. In particular, if EðRÞis finite, thenðR;sÞis rigid.

Proof. That the second assertion follows from the first is immediate. Let Xbe a set. Let us assume that 062EðRÞ n f 0 g. Let V2VðR;sÞð0Þ⁵ such that V\ ðEðRÞ n f 0 gÞ ¼ ;. Let ‘2 ððR;sÞ^XÞ⁰. Letf 2R^X be given by fðxÞ:¼‘ðdxÞ^† for each x2X. Since ðfðxÞ‘ðdxÞÞ_x2X is summable in ðR;sÞ with sum ‘ðfÞ, by Cauchy’s condition [19, Definition 10.3, p. 72], there exists a finite setAf;V

X such that for all x62Af;V;fðxÞ‘ðdxÞ 2V. But for x2X;fðxÞ‘ðdxÞ ¼‘ðdxÞ^†‘ðdxÞ 2EðRÞ.

Whence, in view of Remark 18, for all but finitely manyx’s,fðxÞ‘ðdxÞ ¼0, i.e.,‘ðdxÞ ¼0. w

Remark 20.

1. Let us point out that a von Neumann regular ring with only finitely many idempotents is classically semisimple so is (by commutativity) a finite product of fields.

2. Lemma 14 becomes a consequence of Proposition 19.

Now, letðEi;siÞ_i2I be a family of topological spaces. OnQ

i2IEi is defined the box topology [8, p. 107] a basis of open sets of which is given by the“box” Q

i2IVi, where eachVi2s_i;i2I. The product Q

i2IEi together with the box topology is denoted by u_i2IðEi;siÞ. (This topology is Hausdorff as soon as all theðEi;s_iÞ’s are.)

It is not difficult to see that given a familyðRi;s_iÞ_i2I of topological rings, then u_i2IðRi;s_iÞstill is a topological ring (under component wise operations).

Proposition 21. LetðkiÞ_i2I be a family of fields, and for each i2I, lets_ibe a ring topology on k_i. Let R be a ring with a one-to-one ring map ˚:R; ,!Q

i2Ik_i. Let us assume that for each x2 R;˚ðxÞ^†2imð˚Þ(ðxiÞ^†_i as in Example 17). LetRbe topologized with the subspace topologys_˚ inher- ited fromu_i2Iðki;siÞ. Then,ðR;s_˚Þis rigid.

Proof. Naturally ðxiÞ_i2I2EðQ

i2Ik_iÞ if, and only if, xi2 f 0;1ki gfor each i2I. Now, for each i2I, let Ui be an open neighborhood of zero in ðki;s_iÞ such that 1ki 62Ui. Then, Q

iUi is an open neighborhood of zero in ui2Iðki;s_iÞ whose only idempotent member is 0.

Therefore, 062EðQ

i2Ik_iÞ n f 0 g.

Under the assumptions of the statement, an application of Example 17 states thatRis a (com- mutative) von Neumann regular ring. It is also of course a topological ring unders_˚ (since˚is a one-to-one ring map). It is also clear that EðRÞ ’Eð˚ðRÞÞ EðQ

ikiÞ. Furthermore,

˚ðEðRÞ n f 0 gÞ ¼Eð˚ðRÞÞ n f 0 g \˚ðRÞ EðQ

ik_iÞ n f 0 g, and thus 062EðRÞ n f 0 gaccord- ing to the above discussion. Therefore, by Proposition 19,ðR;s_˚Þis rigid. w

4.3. A counterexample

LetðR;sÞbe a topological ring, and let us consider the topological ðR;sÞ^X-module ððR;sÞ^XÞ^X for a given setX. To avoid confusion one denotes byðR^XÞ^X!^PxR^X the canonical projection,x2X.

Let us define a linear map ðR^XÞ^X!^dðR;sÞ^X by setting dðfÞ:x7!ðfðxÞÞðxÞ;f 2 ðR^XÞ^X. d is con- tinuous, and thus belongs to ðððR;sÞ^XÞ^XÞ⁰, since for each x2X;p_xd¼p_xPx. Now, for each

5Given a topological spaceðE;sÞandx2E;VðE;sÞðxÞis the set of all neighborhoods ofx.

x2X;ðdðd^R_x^XÞÞðxÞ ¼ ðd^R_x^XðxÞÞðxÞ ¼1_R^XðxÞ ¼1_R, i.e., dðd^R_x^XÞ ¼d^R_x, so that suppðbdÞ ¼X, whenR6¼ ð0Þ.

Proposition 22. LetðR;sÞbe a nontrivial topological ring, and let X be a set. If X is infinite, then ðR;sÞ^X is not rigid.

The above negative result may be balanced by the following.

Proposition 23. LetðR;sÞbe a rigid ring. If I is finite, thenðR;sÞ^I is rigid too.

Proof. Let ðR;sÞ be a topological ring. For a set I, recall from Notation 4 that ðR;sÞ^I is the underlying ring of A_ðR;sÞðIÞ. Any topological ðR;sÞ^I-module is also a topological ðR;sÞ-module under restriction of scalars along the unit mapðR;sÞ!^gIðR;sÞ^I;g_Ið1_RÞ ¼1_R^I, which of course is a ring map, and is continuous (becauseg_IðaÞ ¼m_R^Iðg_IðaÞ;1_R^IÞ;a2R.)

Let X be a set, and let ‘2 ðððR;sÞ^IÞ^XÞ⁰, i.e., ððR;sÞ^IÞ^X!^‘ðR;sÞ^I is continuous and R^I-linear, and by restriction of scalar along g_I it is also a continuous homomorphism of topological ðR;sÞ-modules. Therefore for each i2I;ððR;sÞ^IÞ^X!^‘ ðR;sÞ^I!^piðR;sÞ belongs to the topological dual space ofððR;sÞ^IÞ^X seen as aðR;sÞ-module.

Let us assume that ðR;sÞ is rigid. Then, by Lemma 13, suppðpdi‘Þ is finite for each i2I.

One also has suppðb‘Þ ¼ [i2Isuppðp_id‘Þ, with X!b‘

R^I;b‘ðxÞ:¼‘ðd^R_x^IÞ;x2X. Whence if I is finite,

thensuppðb‘Þis finite too. w

5. Rigidity as an equivalence of categories

The main result of this section is Theorem 45 which provides a translation of the rigidity condi- tion on a topological ring into a dual equivalence between the category of free modules and that of topologically free modules (see below), provided by the topological dual functor with equiva- lence inverse the (opposite of the) algebraic dual functor, with both functors conveniently co- restricted. The purpose of this section thus is to prove this result.

Topologically free modules. Let ðR;sÞ be a topological ring. Let ðM;rÞ be a topological ðR;sÞ-module. It is said to be a topologically free ðR;sÞ-module if ðM;rÞ ’ ðR;sÞ^X, in TopMod_ðR;sÞ, for some set X. Such topological modules span the full subcategory TopFreeMod_ðR;sÞ ofTopMod_ðR;sÞ. For a fieldðk;sÞwith a ring topology, one defines correspond- ingly the categoryTopFreeVect_ðk;sÞ oftopologically freeðk;sÞ-vector spaces.

Remark 24. The topological power functorSet^opP!^ðR;sÞTopMod_ðR;sÞ factors as indicated below (the co-restriction obtained is still calledP_ðR;sÞ).

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Topologically free modules are characterized by having “topological bases” (see Corollary 28 below) which makes easier a number of calculations and proofs, once such a basis is chosen.

Definition 25.LetðM;rÞbe a topologicalðR;sÞ-module. LetB M. It is said to be a topological basis ofðM;rÞif the following hold.

1. For eachv2M, there exists a unique familyðb⁰ðvÞÞ_b2B, withb⁰ðvÞ 2Rfor eachb2B, such that ðb⁰ðvÞbÞ_b2Bis summable inðM;rÞwith sum v.b⁰ðvÞis referred to as the coefficient ofvatb2B.

2. For each family ða_bÞ_b2B of elements of R, there is a member v of M such that b⁰ðvÞ ¼a_b;b2B. (By the above point such v is unique.)

3. r is equal to the initial topology induced by the (topological) coefficient maps ðM!^b0ðR;sÞÞ_b2B. (According to the two above points, eachb⁰isR-linear.)

Remark 26. It is an immediate consequence of the definition that for a topological basis B of some topological module, 062B and b⁰ðdÞ ¼dbðdÞ;b;d2B (since Pb2BdbðdÞb¼d¼P

b2Bb⁰ðdÞb). In particular,ðÞ⁰:B!B⁰:¼ f b⁰:b2B gis a bijection.

Lemma 27. Let ðM;rÞ and ðN;cÞ be isomorphic topological ðR;sÞ-modules. Let H:ðM;rÞ ’ ðN;cÞbe an isomorphism (in TopMod_ðR;sÞ). Let B be a topological basis of ðM;rÞ. Then, HðBÞ ¼ f HðbÞ:b2B gis a topological basis of ðN;cÞ.

Corollary 28. Let ðM;rÞ be a (Hausdorff) topological ðR;sÞ-module. It admits a topological basis if, and only if, it is topologically free.

Example 29. LetðR;sÞbe a topological ring. For each setX, fd_x:x2X gis a topological basis ofðR;sÞ^X. Moreoverpx¼d⁰_x;x2X.

Let us now take the time to establish a certain number of quite useful properties of topo- logical bases.

Lemma 30. Let ðM;rÞ be a topologically free ðR;sÞ-module with topological basis B. Then, B is R-linearly independent and the linear spanhBiof B is dense inðM;rÞ.

Proof. Concerning the assertion of independence, it suffices to note that 0 may be written as Pb2B0b, and conclude by the uniqueness of the decomposition in a topological basis. Let u2M and let V:¼ f v2M:b⁰ðvÞ 2Ub;b2A g 2VðM;rÞð0Þ, where A is a finite subset of B and Ub2VðR;sÞð0Þ;b2A. Let a_b2Ub;b2A, and v:¼P

b2Aa_bbP

b2BnAb⁰ðuÞb2V. So uþv2 hBi. Thus,uþVmeets hBiandhBiis dense inðM;rÞ. w

Corollary 31. Let ðM;rÞ be a topologically free ðR;sÞ-module, and let ðN;cÞ be a topological ðR;sÞ-module. Let ðM;rÞ!^f;gðN;cÞ be two continuous homomorphisms of topological ðR;sÞ-mod- ules. f¼g if, and only if, for any topological basis B ofðM;rÞ;fðbÞ ¼gðbÞfor each b2B.

Topologically free modules allow for the definition of changes of topological bases (see Proposition 48 for a related construction).

Lemma 32. Let ðM;rÞ and ðN;cÞ be topologically free ðR;sÞ-modules, with respective topological bases B, D. Let f :B!D be a bijection. Then, there is a unique isomorphism g in TopMod_ðR;sÞ such that gðbÞ ¼fðbÞ;b2B.

Proof. The question of uniqueness is settled by Corollary 31, and an isomorphism is given bygðvÞ ¼P

d2Dðf¹ðdÞÞ⁰ðvÞd;v2M. w

Lemma 33. Let M be a free module with basis B. Then,ðM;w_ðR;sÞÞis a topologically free module with topological basis B:¼ f b:b2B g.

Proof. According to Lemma 9, ðM;w_ðR;sÞÞ is a topologically free module. Let h_B:M!R^ðBÞ be the isomorphism given by hBðbÞ ¼db;b2B. Thus, h_B:ðR^ðBÞÞ’M, and h_Bq_B:R^B’

ðR^ðBÞÞ’M is given by h_Bðq_Bðd^R_bÞÞ ¼q_Bðd^R_bÞ hB¼pbhB¼b for b2B (see Example 7 for the definition of q_B). Now, f db:b2B g being a topological basis of R^B, by Lemma 27 this

shows thatB is a topological basis of ðM;w_ðR;sÞÞ. w

Example 34. f px:x2X gis a topological basis ofðR^ðXÞÞ (Example 7).

Remark 35. If B is a basis of a free module M, then B’B under b7!b, because for eachb;d2B;bðdÞ ¼d_bðdÞ.

Corollary 36. LetðR;sÞbe a topological ring. The functor Alg_ðR;sÞ:Mod^op_R !TopMod_ðR;sÞ factors as illustrated in the diagram below.⁶ Moreover the resulting co-restriction of Alg_ðR;sÞ (the bottom arrow of the diagram) is essentially surjective on objects.

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Proof. The first assertion is merely Lemma 33. Regarding the second assertion, let ðM;rÞ be a topologically free module. So, for some set X, ðM;rÞ ’ ðR;sÞ^X. By Lemma

6,ðR;sÞ^X’Alg_ðR;sÞðR^ðXÞÞ. w

Lemma 37. LetðR;sÞ be a rigid ring. Let ðM;rÞ be a topologically free ðR;sÞ-module with topo- logical basis B. Then,ðM;rÞ⁰ is free with basis B⁰:¼ f b⁰:b2B g.

Proof. Let HB:ðM;rÞ ’ ðR;sÞ^B be given byHBðbÞ ¼db. Thus H⁰B:ððR;sÞ^BÞ⁰’ ðM;rÞ⁰, and thus one has an isomorphismH⁰BkB:R^ðBÞ’ ðM;rÞ⁰. Since a module isomorphic to a free module is free, ðM;rÞ⁰ is free. The previous isomorphism acts as: H⁰BðkBðdbÞÞ ¼pbHB¼b⁰ forb2B. It

follows from Lemma 27 thatB⁰ is a basis ofðM;rÞ⁰. w

Example 38. Let ðR;sÞ be a rigid ring. Let ðM;rÞ ¼ ðR;sÞ^X. By Example 29, f d⁰_x:x2X g ¼ f px:x2X gis a linear basis ofððR;sÞ^XÞ⁰.

Corollary 39.LetðR;sÞbe a rigid ring. The functorTopMod^op_ðR;sÞ^TopðR;sÞ!Mod_Rfactors⁷ as indicated by the diagram below.

6Whenkis a field with a ring topologys, then one has the corresponding factorization ofAlg_ðk;sÞ:Vect^op_k !TopVect_ðk;sÞ.

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7Correspondingly for a fieldðk;sÞwith a ring topology,

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The topological dual of the algebraic dual of a free module. Let ðR;sÞ be a topological ring.

Let M be a R-module, and let us consider as in Section 3.1, the R-linear map M!^KMðM;w_ðR;sÞÞ⁰ ðKMðvÞÞð‘Þ ¼‘ðvÞ;v2M; ‘2M.

Lemma 40. Let M be a projective R-module. Then, KM is one-to-one. This holds in particular when M is a freeR-module.

Proof. Let us consider a dual basis forM, i.e., setsB M andf ‘e:e2B g M, such that for all v2M; ‘eðvÞ ¼0 for all but finitely many ‘e 2B and v¼P

e2B‘eðvÞe ([9, p. 23]). Let v2kerKM, i.e.,ðKMðvÞÞð‘Þ ¼‘ðvÞ ¼0 for each ‘2M. Then, in particular,KMðvÞð‘eÞ ¼‘eðvÞ ¼

0 for alle2B, and thusv¼0. w

Let ðR;sÞ (resp. ðk;sÞ) be a rigid ring (resp. field). The functors provided by Corollaries 36 and 39 are still denoted by Alg_ðR;sÞ:FreeMod^op_R !TopFreeMod_ðR;sÞ (resp.

Alg_ðk;sÞ:Vect^op_k !TopFreeVect_ðk;sÞ) and by Top_ðR;sÞ:TopFreeMod^op_ðR;sÞ!FreeMod_R (resp.Top_ðk;sÞ:TopFreeVect^op_ðk;sÞ!Vectk).

Proposition 41. Let us assume that ðR;sÞ is rigid. K:¼ ðKMÞ_M :id)Top_ðR;sÞAlg_ðR;sÞ^op : FreeMod_R!FreeMod_Ris a natural isomorphism.

Proof. Naturality is clear. Let ðR;sÞ be a topological ring. Let M be a free R-module. For each free basis X of M, the following diagram commutes in Mod_R, where M!^hXR^ðXÞ is the canonical isomorphism given by hXðxÞ ¼d^R_x;x2X. Consequently, when ðR;sÞ is rigid KM is an isomorphism.

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w

Corollary 42. Let us assume that ðk;sÞ is a field with a ring topology. Then, K¼ ðKMÞ_M :id) Top_ðk;sÞAlg^op_ðk;sÞ:Vect_k!Vect_kis a natural isomorphism.

The algebraic dual of the topological dual of a topologically-free module.LetðM;rÞbe a topo- logical ðR;sÞ-module. Let us consider the R-linear map CðM;rÞ:M! ððM;rÞ0Þ by set- tingðCðM;rÞðvÞÞð‘Þ:¼‘ðvÞ.

Proposition 43. Let us assume that ðR;sÞ is a rigid ring. Then, C:id)Alg_ðR;sÞTop^op_ðR;sÞ: TopFreeMod_ðR;sÞ!TopFreeMod_ðR;sÞ is a natural isomorphism, withC:¼ ðCðM;rÞÞ_ðM;rÞ.

Proof. Naturality is clear. Let H:ðM;rÞ ’ ðR;sÞ^X be an isomorphism (in TopMod_ðR;sÞ). Since ðR;sÞ is rigid,kX:R^ðXÞ’ ððR;sÞ^XÞ⁰ is an isomorphism. ThereforeR^ðXÞ!^kXððR;sÞ^XÞ⁰!^H0ðM;rÞ⁰ is an isomorphism too in Mod_R. In particular, ðM;rÞ⁰ is free with basis f H⁰ðkXðd^R_xÞÞ:x2X g. By

Lemma 9, the weak- topology on ððM;rÞ0Þ is the same as the initial topology given by the maps ððM;rÞ0Þ^KðM;rÞ0!^ðpxHÞðR;sÞ;x2X, because H⁰ðkXðd^R_xÞÞ ¼H⁰ðpxÞ ¼pxH. Therefore, CðM;rÞ

is continuous if, and only if, for each x2X;KðM;rÞ⁰ðpxHÞ CðM;rÞ¼p_xH is continuous.

Continuity ofCðM;rÞ thus is proved.

ThatCðM;rÞis an isomorphism in TopMod_ðR;sÞ follows from the commutativity of the diagram (inTopMod_ðR;sÞ) below.

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w

Corollary 44. Let us assume thatðk;sÞis a field with a ring topology.C:id)Alg_ðk;sÞTop^op_ðk;sÞ: TopFreeVect_ðk;sÞ!TopFreeVect_ðk;sÞ with C:¼ ðCðM;rÞÞ_ðM;rÞ, is a natural isomorphism.

The equivalence and some of its immediate consequences. Collecting Propositions 41 and 43, one immediately gets the following.

Theorem 45. Let us assume thatðR;sÞis rigid. Top_ðR;sÞ:TopFreeMod^op_ðR;sÞ!FreeMod_Ris an equiva- lence of categories, with equivalence inverse the functor Alg_ðR;^op_sÞ:FreeMod_R!TopFreeMod^op_ðR;sÞ. Corollary 46. Topðk;sÞ:TopFreeVect^op_ðk;sÞ!Vectk is an equivalence of categories, and Alg_ðk;sÞ^op : Vectk!TopFreeVect^op_ðk;sÞ is its equivalence inverse, whenever ðk;sÞ is a field with a ring topology.

Finite-dimensional vector spaces.Let k be a field. LetðM;rÞ be a topologically freeðk;dÞ-vec- tor space with M finite dimensional. Then, r is the discrete topology d on M. It follows that ðM;rÞ⁰¼M, and the equivalence established in Corollary 46 coincides with the classical dual equivalence FinDimVectk’FinDimVect^op_k under the algebraic dual functor, where FinDimVectk is the category of finite dimensionalk-vector spaces.

Linearly compact vector spaces.Let kbe a field. A topologicalðk;dÞ-vector spaceðM;rÞis said to be a linearly compact k-vector space when ðM;rÞ ’ ðk;dÞ^X for some set X (see [4, Proposition 24.4, p. 105]). The full subcategory LCpVectk of TopVect_ðk;dÞ spanned by these spaces is equal toTopFreeVect_ðk;dÞ.

Corollary 47. (of Theorem 45)LetRbe a ring. For each rigid topologies s;r on R, the categories TopFreeMod_ðR;sÞ andTopFreeMod_ðR;rÞ are equivalent. Moreover, for each fieldðk;sÞwith a ring topology,TopFreeVect_ðk;sÞ is equivalent toLCpVectk.

In particular, one recovers the result from [7] thatVect^op_k ’LCpVectk.

The universal property of ðR;sÞ^X. For a ring R, the functor j j:Mod_R!Set (see Remark 2) may be restricted as indicated in the following commutative diagram, and the restriction still is denotedj j:FreeMod_R!Set.

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LikewiseFR:Set!Mod_R (see again Remark 2) may be co-restricted as indicated by the com- mutative diagram below, and the co-restriction is given the same nameF_R:Set!FreeMod_R.

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The adjunction F_Ra j j:Set!Mod_R gives rise to a new one F_Ra j j:Set!FreeMod_R [12, p. 147], and by composition, for each rigid ring ðR;sÞ, there is also the adjunction Alg_ðR;sÞ^op F_Ra j j Top_ðR;sÞ:Set!TopFreeMod^op_ðR;sÞ. Since ðR;sÞ^X’ ððR^ðXÞÞ;w_ðR;sÞÞ (Lemma 6), this may be translated into auniversal propertyofðR;sÞ^X, as explained below, which somehow legitimates the terminologytopologically free.

Proposition 48. Let us assume thatðR;sÞis rigid. Let X be a set. For each topologically free mod- ule ðM;rÞ and any map f :X! jðM;rÞ⁰j, there is a unique continuous homomorphism f^]: ðM;rÞ ! ðR;sÞ^X such thatjðf^]Þ⁰j jkXj d^R_X ¼f (recall thatd^R_XðxÞ ¼d^R_x;x2X).

Proof. There is a uniqueR-linear mapef :R^ðXÞ! ðM;rÞ⁰ such thatjefj d^R_X ¼f. Let us define the continuous linear map ðM;rÞ!^f]ðR;sÞ^X:¼ ðM;rÞ !^CðM;rÞðððM;rÞ0Þ; w_ðR;sÞÞ!^ðefÞ

ððR^ðXÞÞ;w_ðR;sÞÞ!^q1XðR;sÞ^X. One has

j f^{] 0}j jkXj d^R_X ¼ jC⁰ð_M;rÞj j ef 0

j jq¹_X 0

j jkXj d^R_X

¼ jC⁰ð_M;rÞj j ef 0

j jKR^{ð ÞX}j d^R_X because q¹_X 0kX¼KR^{ð ÞX}

¼ jC⁰ðM;rÞj jKð_M;rÞ⁰j jefj d^R_X by naturality of K

ð Þ

¼ jefj d^R_X

triangular identities for an adjunction 12½ ;p: 85

¼ f:

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It remains to check uniqueness of f^]. Let ðM;rÞ!^g ðR;sÞ^X be a continuous linear map such that jg⁰j jkXj d^R_X¼f. Then, g⁰kX¼ef. Thus, k_X ðg0Þ ¼ef¼q_Xf^]C¹_ðM;rÞ. So q_X C¹_ðR;sÞ^X ðg0Þ¼q_Xf^]C¹_ðM;rÞ because C_ðR;sÞ^X ¼ ðk¹_X Þq_X (by direct inspection), and thus C¹_ðR;sÞ^X ðg0Þ¼f^]C¹_ðM;rÞ. Then, by naturality ofC¹;gC¹_ðM;rÞ¼f^]C¹_ðM;rÞ. w

Corollary 49. Let ðR;sÞ be a rigid ring. P^op_ðR;sÞ:Set!TopFreeMod^op_ðR;sÞ is a left adjoint of TopFreeMod^op_ðR;sÞ^Top!^ðR;sÞFreeModR!^jjSet, and thus is naturally equivalent toSet^Alg!

op ðR;sÞF_R

TopFreeMod^op_ðR;sÞ.

Proof. A quick calculation shows that P_ðR;sÞðfÞ ¼ ðjkYj d^R_Y fÞ^] for a set theoretic map f :X!Y. The relation f7!ðkY d^R_YfÞ^] provides a functor from Set^op to TopFreeMod_ðR;sÞ

whose opposite is, by construction, a left adjoint ofjj Top_ðR;sÞ (this is basically the content of

Proposition 48). w

6. Tensor product of topologically free modules

In this section most of the ingredients so far introduced and developed fit together to lift the equivalenceFreeMod’TopFreeMod^op_ðR;sÞ to a duality between some (suitably generalized) topo- logical algebras and some coalgebras.

6.1. Monoidal categories, monoidal functors, and (co-)monoids

Most of the notations and notions from the theory of monoidal categories needed hereafter are taken from [14, Section 2, pp. 4874–4876] and only a few more are introduced below, since they are indispensable.

Monoidal categories.Recall that each natural transformation a:F)G:C!D has an opposite natural transformation a^op:G^op)F^op:C^op!D^op with ða^opÞ_C¼ ðaCÞ^op for each C-object C, where for f 2CðC;DÞ;f^op2C^opðD;CÞ ¼CðC;DÞ is the corresponding C^op-morphism. Recall also that for every categoriesC;D;ðCDÞ^op¼C^opD^op.

IfC¼ ðC; ;I;a;k;qÞ ¼ ðC; ;IÞis a (symmetric) monoidal category, then so is its dual C^op:¼ ðC^op;^op;I;ða¹Þ^op;ð.¹Þ^op;ðk¹Þ^opÞ. (In [14] it is denoted by C^op instead ofC^op.)

Example 50. LetRbe a ring. For eachR-modules M, N, M_RNstands for their (algebraic) ten- sor product, and :MN!M_RN is the universal R-bilinear map.

Mod_R:¼ ðMod_R;_R;RÞ, with the ordinary coherence constraints of associativity, of left and right units and of symmetry, is a symmetric monoidal category ([18, Example 11.2, p. 70]).

1. MonðMod_RÞis isomorphic to the category 1Alg_R of“ordinary” unital R-algebras under the functor O, concrete over Mod_R, such that OðAÞ:¼ ðA;mA;1AÞ, with mAðx;yÞ:¼lðxyÞ;x;y2A, and 1A:¼gð1_RÞ, where A¼ ðA;l_A;g_AÞ is a monoid inMod_R.

2. LikewisecMonðMod_RÞ’1;cAlg_Runder the (co-)restriction of the above functor O.

3. ComonðMod_RÞis the categoryCoalg_Rof counital R-coalgebras ([1,6]), and the category of cocommutative coalgebras_;cocCoalg_RiscocComonðMod_RÞ.

By a(symmetric) monoidal subcategoryof a (symmetric) monoidal categoryC¼ ðC; ;IÞ we mean a subcategory C⁰ of C, closed under tensor products, containing I, and the coherence constraints of C between C⁰-objects. (The last condition is automatically fulfilled when C⁰ is a full subcategory.) The embeddingEC⁰ ofC⁰ into Cthen is a strict monoidal functorE_C⁰ (see e.g., [14, Definition 2, p. 4876]).

For instance, since given free modulesM, Nover a ringR;M_RN is free too, FreeMod_R¼ ðFreeMod_R;_R;RÞ is a symmetric monoidal subcategory of Mod_R. ComonðFreeMod_RÞ (resp., cocComonðFreeMod_RÞ) thus corresponds to the full subcategory of Coalg_R (resp.,

;cocCoalg_R) spanned by the (resp. cocommutative) coalgebras whose underlying module is free.

Monoidal functors and their induced functors. For a (symmetric) monoidal category C;id_C:¼ ðidC;id;idIÞ, or simply id, is a strict (symmetric, [18, p. 86]) monoidal functor from C to itself, which acts as a unit for the usual composition of monoidal functors (see [3, Chapter3, p. 72]). By direct inspection one observes that the composite of strong (resp. symmet- ric) monoidal functors is strong (resp. symmetric) too.