SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES

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1 Theory and Applications o Categories, Vol. 26, No. 26, 212, pp SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES SUSAN NIEFIELD Abstract. Given a double category D such that D has pushouts, we characterize oplax/lax adjunctions D Cospan(D ) or which the right adjoint is normal and restricts to the identity on D, where Cospan(D ) is the double category on D whose vertical morphisms are cospans. We show that such a pair exists i and only i D has companions, conjoints, and 1-cotabulators. The right adjoints are induced by the companions and conjoints, and the let adjoints by the 1-cotabulators. The notion o a 1-cotabulator is a common generalization o the symmetric algebra o a module and Artin-Wraith glueing o toposes, locales, and topological spaces. 1. Introduction Double categories, irst introduced by Ehresmann [2], provide a setting in which one can simultaneously consider two kinds o morphisms (called horizontal and vertical morphisms). Examples abound in many areas o mathematics. There are double categories whose objects are sets, rings, categories, posets, topological spaces, locales, toposes, quantales, and more. There are general examples, as well. I D is a category with pullbacks, then there is a double category Span(D) whose objects and horizontal morphisms are those o D, and vertical morphisms are spans, i.e., morphisms o D, with vertical compositions via pullback. I D is a category with pushouts, then Cospan(D) is deined dually, in the sense that vertical morphisms are cospans with vertical compositions via pushout. Moreover, i D has both, then pushout o spans and pullback o cospans induce an oplax/lax adjunction (in the sense o Grandis-Paré [5, 12]) Span(D) F G Cospan(D) which restricts to the identity on the horizontal category D. Now, suppose D is a category with pushouts, and we replace Span(D) by a double category D whose horizontal category is also D. Then several questions arise. Under what conditions on D is there an oplax/lax adjunction D Cospan(D) which restricts to the identity on D? In particular, i D has cotabulators, then there is an induced oplax unctor Received by the editors and, in revised orm, Transmitted by Robert Paré. Published on Mathematics Subject Classiication: 18D5, 18A4, 18B25, 18B3, 6D22, 18D1, 15A78. Key words and phrases: double category, lax unctor, (co)span, (co)tabulator, companion, conjoint, symmetric algebra. c Susan Nieield, 212. Permission to copy or private use granted. 729

2 73 SUSAN NIEFIELD F : D Cospan(D), and so one can ask when this unctor F has a right adjoint. Similarly, i D has companions and conjoints, there is a normal lax unctor G: Cospan(D) D which c takes a cospan c 1 c 1 to the composite c 1, and so one can ask when the induced unctor G has a let adjoint. We will see that these questions are related. adjunction D F G Cospan(D) In particular, there is an oplax/lax such that G is normal and restricts to the identity on D precisely when D has 1-cotabulators, conjoints, and companions (where or 1-cotabulators we drop the tetrahedron condition in the deinition o cotabulator). Moreover, i D has pullbacks, then the result dualizes to show that there is oplax/lax adjunctions Span(D) D whose let adjoint is opnormal and restricts to the identity on D precisely when D has 1-tabulators, conjoints, and companions. The double categories mentioned above all have 1-cotabulators, companions, and conjoints, and the unctors F and G are related to amiliar constructions. In the double category o commutative rings (as well as, quantales), the unctor F is given by the symmetric algebra on a bimodule and G is given by restriction o scalars. For categories (and posets), F is the collage construction. In the case o topological spaces, locales, and toposes, the unctor F uses Artin-Wraith glueing. The paper proceeds as ollows. We begin in Section 2 with the double categories under consideration, ollowed by a review o companions and conjoints in Section 3. The notion o 1-tabulators (dually, 1-cotabulators) is then introduced in Section 4. Ater a brie discussion o oplax/lax adjunctions in Section 5, we present our characterization (Theorem 5.5) o those o the orm D Cospan(D) such that right adjoint is normal and restricts to the identity on D. Along the way, we obtain a possibly new characterization (Proposition 5.3) o double categories with companions and conjoints, in the case where the horizontal category D has pushouts, as those or which the identity unctor on D extends to a normal lax unctor Cospan(D) D. We conclude with the dual (Corollary 5.6) classiication o oplax/lax adjunctions Span(D) D whose let adjoint is opnormal and restricts to the identity on D. 2. The Examples o Double Categories Following Grandis-Paré [5, 12] and Shulman [13], we deine a double category D to be a weak internal category D 1 D D 1 c d D 1 D d 1 in CAT. It consists o objects (those o D ), two types o morphisms: horizontal (those o D ) and vertical (objects o D 1 with domain and codomain given by d and d 1 ), and cells

3 (morphisms o D 1 ) denoted by SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES 731 m n ϕ 1 Composition and identity morphisms are given horizontally in D and vertically via c and, respectively. The objects, horizontal morphisms, and special cells (i.e., ones in which the vertical morphisms are identities) orm a 2-category called the horizontal 2-category o D. Since D is a weak internal category in CAT, the associativity and identity axioms or vertical morphisms hold merely up to coherent isomorphism, and so we get an analogous vertical bicategory. When these isomorphisms are identities, we say that D is a strict double category. The ollowing double categories are o interest in this paper Example. Top has topological spaces as objects and continuous maps as horizontal morphisms. Vertical morphisms are inite intersection-preserving maps O( ) O( ) on the open set lattices, and there is a cell o the orm ( ) i and only i 1 1 n m Example. Loc has locales as objects, locale morphisms (in the sense o [8]) as horizontal morphisms, and inite meet-preserving maps as vertical morphisms. There is a cell o the orm ( ) i and only i 1 n m Example. Topos has Grothendieck toposes as objects, geometric morphisms (in the sense o [9]) as horizontal morphisms, inite limit-preserving unctors as vertical morphisms, and natural transormations 1 n m as cells o the orm ( ) Example. Cat has small categories as objects and unctors as horizontal morphisms. Vertical morphisms m: are prounctors (also known as distributors and relators), i.e., unctors m: op Sets, and natural transormations m n(, 1 ) are cells o the orm ( ) Example. Pos has partially-ordered sets as objects and order-preserving maps as horizontal morphisms. Vertical morphisms m: are order ideals m op, and there is a cell o the orm ( ) i and only i (x, x 1 ) m ( (x ), 1 (x 1 )) n Example. For a category D with pullbacks, the span double category Span(D) has objects and horizontal morphisms in D, and vertical morphisms which are spans in D, with composition deined via pullback and the identities id : given by id id. ( )

4 732 SUSAN NIEFIELD The cells m n are commutative diagrams in D o the orm m n m 1 1 n 1 In particular, Span(Sets) is the double category Set considered by Paré in [12], see also [1] Example. Cospan(D) is deined dually, or a category D with pushouts. In particular, Span(Top) is the double category used by Grandis [3, 4] in his study o 2-dimensional topological quantum ield theory Example. For a symmetric monoidal category V with coequalizers, the double category Mod(V) has commutative monoids in V as objects and monoid homomorphisms as horizontal morphisms. Vertical morphisms rom to are (, )-bimodules, with composition via tensor product, and cells are bimodule homomorphisms. Special cases include the double category Ring o commutative rings with identity and the double category Quant o commutative unital quantales. 3. Companions and Conjoints Recall [6] that companions and conjoints in a double category are deined as ollows. Suppose : is a horizontal morphism. A companion or is a vertical morphism : together with cells id id η ε id id whose horizontal and vertical compositions are identity cells. A conjoint or is a vertical morphism morphism : together with cells α id id id β id whose horizontal and vertical compositions are identity cells. We say D has companions and conjoints i every horizontal morphism has a companion and a conjoint. Such a double category is also known as a ramed bicategory [13].

5 SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES 733 I has a companion, then one can show that there is a bijection between cells o the ollowing orm g g m ϕ h n m ψ h n Similarly, i has a conjoint, then there is a bijection between cells m g ϕ h n g m ψ h n There are two other cases o this process (called vertical lipping in [6]) which we do not recall here as they will not be used in the ollowing. All o the double categories mentioned in the previous section have well known companions and conjoints. In Top, Loc, and Topos, the companion and conjoint o are the usual maps denoted by and. For Cat, they are the prounctors deined by (x, y) = (x, y) and (y, x) = (y, x), and analogously, or Pos. I V is a symmetric monoidal category and : is a monoid homomorphism, then becomes an (, )-bimodule and a (, )-bimodule via, and so is both a companion and conjoint or. Finally, or Span(D) (respectively, Cospan(D)) the companion and conjoint o are the span (respectively, cospan) with as one leg and the appropriate identity morphism as the other Tabulators and 1-Cotabulators Tabulators in double categories were deined as ollows in [5] (see also [12]). Suppose D is a double category and m: is a vertical morphism in D. A tabulator o m is an object T together with a cell T τ m

6 734 SUSAN NIEFIELD such that or any cell there exists a unique morphism : tetrahedron o cells there is a unique cell ξ such that ϕ m T such that τ = ϕ, and or any commutative n m n gives the tetrahedron in the obvious way. ξ T τ m Tabulators (and their duals cotabulators) arise in the next section, but we do not use the tetrahedron property in any o our proos or constructions. Thus, we drop this condition in avor o a weaker notion which we call a 1-tabulator (and dually, 1-cotabulator) o a vertical morphism. When the tetrahedron condition holds, we call these tabulators (respectively, cotabulators) strong. It is easy to show that the ollowing proposition gives an alternative deinition in terms o adjoint unctors Proposition. A double category D has 1-tabulators (respectively, 1-cotabulators) i and only i : D D 1 has a right (respectively, let) adjoint which we denote by Σ (respectively, Γ) Corollary. Mod(V) does not have 1-tabulators. Proo. Suppose (V,, I) is a symmetric monoidal category. Then I is an initial object Mod(V), which is the category o commutative monoids in V. Since I is not an initial object o Mod(V) 1, we know does not have a right adjoint, and so the result ollows rom Proposition 4.1. The eight examples under consideration have 1-cotabulators, and we know that 1- tabulators exist in all but one (namely, Mod(V)). In act, we will prove a general proposition that gives the existence o 1-tabulators rom a property o 1-cotabulators (called 2-glueing in [11]) shared by Top, Loc, Topos, Cat, and Pos. We will also show that the 1-cotabulators in Ring are not strong, and so consideration o only strong cotabulators would eliminate this example rom consideration.

7 SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES 735 The cotabulator o m: in Cat (and similarly, Pos), also known as the collage, is the the category over 2 whose ibers over and 1 are and, respectively, and morphism rom objects o to those o are given by m: op Sets, with the obvious cell m id. Cotabulators in Topos, Loc, and Top are constructed using Artin-Wraith glueing (see [9], [1], [11]). In particular, given m: in Top, the points o Γm are given by the disjoint union o and with U open in Γm i and only i U is open in, U 1 is open in, and U 1 m(u ), where U i = U i. c For Cospan(D), the cotabulator o c 1 1 is given by with cell (c, id, c 1 ). s I D has pullbacks and pushouts, then the cotabulator o s 1 in Span(D) is the pushout o s and s 1. The situation in Mod(V) is more complicated. The ollowing example (which generalizes to any V with a symmetric algebra unctor) shows that Ring has 1-cotabulators which do not satisy the tetrahedron condition Example. Suppose M: is a vertical morphism in Ring, i.e., M is an (, )-bimodule. Then M is an -module, and so we can consider the symmetric -algebra SM = M n n together with the canonical inclusion ι: M SM, where M = and M n is the tensor product over o n copies o M. We claim that the corresponding cell M SM ι is a 1-cotabulator o M:. Since is the coproduct o and as commutative rings and S is let adjoint to the orgetul unctor rom -algebras to -modules, we know that any cell M ϕ corresponds to an -module homomorphism ϕ: M, and hence, a unique -algebra homomorphism : SM such that ι = ϕ. Thus, SM has the desired structure o a 1-cotabulator in Ring. To see that the tetrahedron condition does not hold, consider the 1-cotabulator o :, where S = and ι is the unique homomorphism. Taking ι 1 (n) = (n, )

8 736 SUSAN NIEFIELD and ι 2 (n) = (, n), the diagram ι 1 = ι 2 deines a commutative tetrahedron which does not actor as ξ or any homomorphism ξ:. Thus, the 1-cotabulators are not strong in Ring. Now, suppose D has 1-cotabulators and a terminal object 1, and let 2 denote the image under Γ o the vertical identity morphism on 1, where Γ: D 1 D is let adjoint to (see Proposition 4.1). Then Γ induces a unctor D 1 D /2, which we also denote by Γ. I this unctor is an equivalence o categories, then we say D has 2-glueing. In Cat (and similarly, Pos), 2 is the category with two objects and one non-identity morphism. It is the Sierpinski space 2 in Top, the Sierpinski locale O(2) in Loc, and the Sierpinski topos S 2 in Topos. That Cat has 2-glueing is Bénabou s equivalence cited in [14]. For Top, Loc, and Topos, the equivalence ollows rom the glueing construction (see [9], [1], [11]). Note that in each o these cases, 2 is exponentiable in D (see [7, 9]), and so the unctor 2 : D D /2 has a right adjoint, usually denoted by Π Proposition. I D has 2-glueing and 2 is exponentiable in D, then D has 1-tabulators. 2 Proo. Consider the composite F : D D /2 D 1. Since it is not diiculty to show that Γ takes the vertical identity on to the projection 2 2, it ollows that F =, and so has a right adjoint Σ, since 2 does. Thus, D has 1-tabulators by Proposition 4.1. Applying Proposition 4.4, we see that Cat, Pos, Top, Loc, and Topos have 1-tabulators (which can be shown to be strong). Unraveling the construction o Σ given in the proo above, one gets the ollowing descriptions o 1-tabulators in Cat and Top which can be shown to be strong. Given m: in Cat (and similarly Pos), the tabulator is the category o elements o m, i.e., the objects o Σm are o the orm (x, x 1, α), where x is an object o, x 1 is an object o, and α m(x, x 1 ). Morphisms rom (x, x 1, α) to (x, x 1, α ) in Σm are pairs (x x, x 1 x 1) o morphisms, compatible with α and α.

9 SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES 737 The tabulator o m: in Top is the set Σm = {(x, x 1 ) U O( ), x 1 m(u ) x U } with the subspace topology. Note that one can directly see that this is the tabulator o m by showing that (, 1 ): actors through Σm i and only i 1 1 m 1. Although Span(D) and Cospan(D) do not have 2-glueing (since Γ(id 1) = 1), and so Proposition [11] does not apply, the construction o their tabulators is dual to that o their cotabulators. Finally, Ring and Quant do not have 1-tabulators, as they are special cases o Corollary The Adjunction Recall rom [5] that a lax unctor F : D E consists o unctors F i : D i E i, or i =, 1, compatible with d and d 1 ; together with identity and composition comparison cells ρ : id F F (id ) and ρ m,m : F m F m F (m m) or every object and every vertical composite m m o D, respectively; satisying naturality and coherence conditions. I ρ is an isomorphism, or all, we say that F is a normal lax unctor. An oplax unctor is deined dually with comparison cells in the opposite direction. An oplax/lax adjunction consists o an oplax unctor F : D E and a lax double unctor G: E D together with double cells η GF η m F Gn ε n GF F G m GF m η 1 ε F G satisying naturality and coherence conditions, as well as the usual adjunction identities (see [6]) Example. Suppose D is a double category with 1-cotabulators and D has pushouts. Then, by Proposition 4.1, the unctor : D D 1 has a let adjoint (denoted by Γ), and so there is an oplax unctor F : D Cospan(D ) which is the identity on objects and horizontal morphisms, and deined on vertical morphisms and cells by i i ϕ Γm i 1 Γm i m m ε 1 n

10 738 SUSAN NIEFIELD where is induced by the universal property o the 1-cotabulator. The comparison cells F (id ) id F and F (m m) F m F m also arise via the universal property, with the latter given by the horizontal morphism Γ(m m) P corresponding to the diagram m m id m ι m Γm P m ι m Γm id 2 2 where P is a pushout and the large rectangle is the canonical cell. Dually, we get: 5.2. Example. Suppose D is a double category with 1-tabulators and D has pullbacks. Then there is a lax unctor F : D Span(D ) which is the identity on objects and horizontal morphisms and takes m: to the span Σm, where Σ is the right adjoint to Proposition. Suppose D is a double category and D has pushouts. Then D has companions and conjoints i and only the identity unctor on D extends to a normal lax unctor G: Cospan(D ) D. Proo. Suppose D has companions and conjoints. Then it is not diicult to show that there is a normal lax unctor G: Cospan(D ) D which is the identity on objects and horizontal morphisms, and is deined on cells by c c g c 1 c 1 g g 1 1 g c ψ c g c 1 c ψ 1 1 g 1 1 where ψ and ψ 1 arise rom the commutativity o the squares in the cospan cell, and the deinitions o companion and conjoint. Conversely, suppose there is a normal lax unctor G: Cospan(D ) D which is the identity on objects and horizontal morphisms. Then the companion and conjoint o : are deined as ollows. Consider = G( id ): = G( id ):

11 SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES 739 Applying G to the cospan diagrams id id id id id id id id id and composing with ρ and ρ 1, we get cells id id id id id ρ G(id,,) id id G(,idy,id ) id id ρ 1 id which serve as η and ε, respectively, making the companion o. Note that the horizontal and vertical identities or η and ε ollow rom the normality and coherence axioms o G, respectively. Similarly, the cells α and β or arise rom the cospan diagrams id id id id id id id id id and it ollows that D has companions and conjoints Corollary. Suppose D is a double category and D has pullbacks. Then D has companions and conjoints i and only the identity unctor on D extends to an opnormal oplax unctor Span(D ) D. Proo. Apply Proposition 5.3 to D op Theorem. The ollowing are equivalent or a double category D such that D has pushouts: (a) there is an oplax/lax adjunction D to the identity on D ; F G Cospan(D ) such that G is normal and restricts (b) D has companions, conjoints, and 1-cotabulators; (c) D has companions and conjoints, and the induced normal lax unctor Cospan(D ) G D has an oplax let adjoint;

12 74 SUSAN NIEFIELD (d) D has companions, conjoints, and 1-cotabulators, and the induced oplax unctor D F Cospan(D ) is let adjoint to the induced normal lax unctor Cospan(D ) G D; (e) D has 1-cotabulators and the induced oplax unctor D lax right adjoint. F Cospan(D ) has a normal Proo. (a) (b) Given F G such that G is normal and restricts to the identity on D, we know D has companions and conjoints by Proposition 5.3. To see that D has 1-cotabulators, by Proposition 4.1, it suices to show that : D D 1 has a let adjoint. Since actors as D Cospan(D ) G 1 D 1, by normality o G, and both these unctors have let adjoints, the desired result ollows. (b) (c) Suppose D has companions, conjoints, and 1-cotabulators, and consider the induced unctor F : D Cospan(D ). Given m: c and c 1 1, applying the deinition o 1-cotabulator, we know that every cell in Cospan(D ) o the orm c Γm c 1 1 corresponds to a unique cell and hence, a unique cell m c c ϕ c 1 1 id m ψ c by vertical lipping, and it ollows that F G. 1 (c) (d) Suppose D has companions and conjoints, and the induced normal lax unctor G has an oplax let adjoint F. As in the proo o (a) (b), we know that D has 1- cotabulators, and so, it suices to show that F is the induced unctor. We know F takes m: to a cospan o the orm c c Γm 1, since F is the identity on objects and the let adjoint o : D c c 1 Cospan(D ) takes Γm 1 to, and so the desired result easily ollows. c 1

13 (d) (e) is clear. SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES 741 (e) (a) Suppose D has 1-cotabulators and the induced oplax unctor F has a normal lax right adjoint G. Since F restricts to the identity on D, then so does G, and the proo is complete. Note that this proo shows that i there is an oplax/lax adjunction D F G Cospan(D ) such that G is normal and restricts to the identity on D, then F is the induced by 1-cotabulators and G by companions and conjoints. Since the double categories in Examples all have companions, conjoints, and 1-cotabulators, it ollows that they each admits a unique (up to equivalence) oplax/lax adjunction o this orm and it is induced in this manner. Applying Theorem 5.5 to D op, we get: 5.6. Corollary. The ollowing are equivalent or a double category D such that D has pullbacks: (a) there is an oplax/lax adjunction Span(D ) G D such that G is opnormal and restricts F to the identity on D ; (b) D has companions, conjoints, and 1-tabulators; (c) D has companions and conjoints, and the induced opnormal oplax unctor (d) D Span(D ) G D has a lax right adjoint; has companions, conjoints, and 1-tabulators, and the induced lax unctor D F Span(D ) is right adjoint to the induced opnormal oplax unctor Span(D ) G D; (e) D has 1-tabulators and the induced lax unctor D F Span(D ) has an opnormal oplax let adjoint. As in the cospan case, i there is an oplax/lax adjunction Span(D ) G F such that G is opnormal and restricts to the identity on D, then F is the induced by 1-tabulators and G is by companions and conjoints. Since the double categories in Examples (i.e., all but Mod(V) ) have companions, conjoints, and 1-tabulators, it ollows they each admits is a unique (up to equivalence) oplax/lax adjunction o this orm, and it is induced by companions, conjoints, and 1-tabulators. D

14 742 SUSAN NIEFIELD Reerences [1] R. J. MACG. Dawson, R. Paré, amd D. A. Pronk, The span construction, Theory Appl. Categ. 24 (21), [2] Charles Ehresmann. Catégories structuées, Ann. Sci. Ecole Norm. Sup. 8 (1963) [3] Marco Grandis, Higher cospans and weak cubical categories (cospans in algebraic topology, I), Theory Appl. Categ. 18 (27), [4] Marco Grandis, Collared cospans, cohomotopy and TQFT (cospans in algebraic topology, II), Theory Appl. Categ. 18 (27), [5] Marco Grandis and Robert Paré, Limits in double categories, Cahiers de Top. et Géom. Di. Catég. 4 (1999), [6] Marco Grandis and Robert Paré, Adjoint or double categories, Cahiers de Top. et Géom. Di. Catég. 45 (24), [7] J. M. E. Hyland, Function spaces in the category o locales, Springer Lecture Notes in Math. 871 (1981) [8] P. T. Johnstone, Stone Spaces, Cambridge University Press, [9] P. T. Johnstone, Topos Theory, Academic Press, [1] S. B. Nieield, Cartesian inclusions: locales and toposes, Comm. in Alg. 9(16) (1981), [11] S. B. Nieield, The glueing construction and double categories, J. Pure Appl. Alg. 216 (212) [12] Robert Paré, oneda theory or double categories, Theory Appl. Categ. 25 (211), [13] M. Shulman, Framed bicategories and monoidal ibrations, Theory Appl. Categ. 2 (28), [14] R. Street, Powerul unctors, expository note: street/pow.un.pd (21). Union College Department o Mathematics Schenectady, N238 nieiels@union.edu This article may be accessed at or by anonymous tp at tp://tp.tac.mta.ca/pub/tac/html/volumes/26/26/26-26.{dvi,ps,pd}

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