Polymorphic Automorphisms and the Picard Group
PPolymorphic Automorphisms and the PicardGroup
Pieter Hofstra
Dept. of Mathematics & Statistics, University of Ottawa, Ottawa, Ontario, Canada [email protected]
Jason Parker
Department of Mathematics & Computer Science, Brandon University, Brandon, Manitoba, Canada [email protected]
Philip J. Scott Dept. of Mathematics & Statistics, University of Ottawa, Ottawa, Ontario, Canada [email protected]
Abstract
We investigate the concept of definable, or inner, automorphism in the logical setting of partialHorn theories. The central technical result extends a syntactical characterization of the group ofsuch automorphisms (called the covariant isotropy group) associated with an algebraic theory to thewider class of quasi-equational theories. We apply this characterization to prove that the isotropygroup of a strict monoidal category is precisely its Picard group of invertible objects. Furthermore,we obtain an explicit description of the covariant isotropy group of a presheaf category.
Theory of computation → Equational logic and rewriting, Theoryof computation → Categorical Semantics
Keywords and phrases
Partial Horn Theories, Monoidal Categories, Definable Automorphisms,Polymorphism, Indeterminates, Normal Forms
Digital Object Identifier
Funding
Pieter Hofstra : Research funded by an NSERC Discovery Grant
Jason Parker : Postdoctoral research funded by NSERC grant of R. Lucyshyn-Wright (Brandon)
Philip J. Scott : Research funded by an NSERC Discovery Grant
Acknowledgements
Pieter Hofstra would like to acknowledge illuminating discussions with MarttiKarvonen and Eugenia Cheng.
In algebra, model theory, and computer science, one encounters the notion of definableautomorphism (the nomenclature varies by discipline). In first-order logic for example (seee.g. [10]), an automorphism α of a model M is called definable (with parameters in M ) whenthere is a formula φ ( x, y ) in the ambient language (possibly containing constants from M )such that for all a, b ∈ M we have α ( a ) = b ⇐⇒ M | = ϕ ( a, b ) . The case of groups is instructive: for a group M , consider the formula ϕ ( x, y ) given as ϕ ( x, y ) : y = c − xc corresponding author © Pieter Hofstra, Jason Parker, and Philip J. Scott;licensed under Creative Commons License CC-BY 4.042nd Conference on Very Important Topics (CVIT 2016).Editors: John Q. Open and Joan R. Access; Article No. 23; pp. 23:1–23:16Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany a r X i v : . [ c s . L O ] F e b for some c ∈ M . This defines an (inner) automorphism of M . Note that in this case theautomorphism is also determined by a term t ( x ) := c − xc via a t ( a ).These definable automorphisms have various interesting aspects: first of all, they are insome sense polymorphic or uniform. This means roughly that the same term t , possibly afterreplacing constants from M , can also define an automorphism of another model N . Secondly,the definable automorphisms can also provide a generalized notion of inner automorphism ,even for theories where it does not make sense to speak of group-theoretic conjugation.Indeed, Bergman [1, Theorem 1] shows that in the category of groups, the definable groupautomorphisms, i.e. the inner automorphisms given by conjugation, can be characterizedpurely categorically by the fact that they extend naturally along any homomorphism. Thatis: an automorphism α : G ∼ −→ G is inner precisely when for any homomorphism m : G → H there is an extension α m : H ∼ −→ H making diagram (a) commute and also making( a ) G α (cid:15) (cid:15) m (cid:47) (cid:47) H α m (cid:15) (cid:15) G m (cid:47) (cid:47) H ( b ) H α m (cid:15) (cid:15) n (cid:47) (cid:47) K α nm (cid:15) (cid:15) H n (cid:47) (cid:47) K diagram (b) commute for any further homomorphism n : H → K , so that in particular α = α id G by diagram (a). If α is conjugation by g ∈ G , then α m is conjugation by m ( g ) ∈ H .Conversely, given any system of group automorphisms { α m : H ∼ −→ H | m : G → H } with α = α id G that makes diagrams (a) and (b) commute, Bergman shows that there is a uniqueelement s ∈ G such that α is given by conjugation with s . Bergman therefore refers to sucha system { α m | m : G → H } as an extended inner automorphism of G .In categorical logic, we have a canonical method for studying this phenomenon. To anycategory C , we may associate the functor Z C : C → Grp ; Z C ( C ) := Aut ( π : C/ C → C ) . (1)Let us unpack this. We have the co-slice category C/ C whose objects are maps C → D andwhose arrows are commutative triangles. The projection functor π : C/ C → C sends C → D to D . We then consider the group of natural automorphisms of this projection functor, i.e.the group of invertible natural transformations α : π ⇒ π . To give such an α is equivalentto giving, for each object m : C → D of C/ C , an automorphism α m : D ∼ −→ D , subject tothe naturality condition that for any composable pair m : C → D, n : D → E in C , we have α nm n = nα m as in diagram (b) above. Thus, in Bergman’s terminology, Z C ( C ) is the groupof extended inner automorphisms of C . We call Z C the (covariant) isotropy group (functor) of C . Another useful way of thinking about this group is by noticing that the assignment C Aut ( C ) is generally not functorial, unless C is a groupoid. The isotropy group offers aremedy: the assignment C
7→ Z C ( C ) is functorial, as is straightforward to check, and foreach C there is a comparison homomorphism θ C : Z C ( C ) → Aut ( C ) ; α α id C (2)that sends an extended inner automorphism α to its component at the identity of C . Wecan then turn Bergman’s aforementioned result for the category
Grp into a definition for an P. Freyd [2] studied a somewhat similar notion while modelling Reynolds’ parametricity for parametricpolymorphism. As a special case, his work leads to a monoid of natural endomorphisms of the projectionfunctor, whereas in our case, we would obtain the subgroup of invertible elements in this monoid. . Hofstra, J. Parker, and P. J. Scott 23:3 arbitrary category C , by defining an automorphism f : C ∼ −→ C of an object C ∈ C to be inner just if f is in the image of θ C : Z C ( C ) → Aut ( C ). Less precisely, the automorphism f : C ∼ −→ C is inner if it can be coherently extended along any arrow out of C .(For readers familiar with topos theory and/or earlier papers on the subject of isotropygroups, we point out that in [4, 3] we consider instead the contravariant isotropy groups Aut ( π : C /C → C ). Now if T is a suitable logical theory with classifying topos B ( T ), then (arestriction of) the contravariant isotropy group of B ( T ) coincides with the covariant isotropygroup of the category fp T mod of finitely presented T -models. Moreover, calculation of thelatter group generally also yields a description of the covariant isotropy group of the largercategory T mod of all T -models, which is our focus in the present paper.)In [6], the case where C is the category of models of an equational theory is analysed.Among other things, a complete syntactic characterization of covariant isotropy for such a C is obtained, recovering not only Bergman’s result for C = Grp but also characterizing thedefinable automorphisms of other common algebraic structures such as monoids and rings.In applying the general characterization in specific instances, one typically needs to analysethe result of adjoining one or more indeterminates to a given model, and this in turn leadsone to consider the word problem for such models.The present paper, which is based on the PhD research [8] of the second author, isconcerned with the analysis of the notion of isotropy or definable automorphism for (strict)monoidal categories and related structures. It hardly needs arguing that monoidal categoriesplay various important roles in mathematics and theoretical computer science, both asobjects of study in their own right, as models of logical theories, and as basic tools forstudying other phenomena. However, we should point out here an observation by RichardGarner [5, Proposition 3] to the effect that both
Cat and
Grpd , the categories of smallcategories and small groupoids respectively, have trivial covariant isotropy, in the sense thatfor any category/groupoid C we have Z ( C ) = 1, the trivial group. As such, it is perhapssurprising that the category of strict monoidal categories has non-trivial isotropy. In fact,and this is the central result of the present paper, the isotropy group of a strict monoidalcategory is precisely its Picard group (its group of invertible objects).Since the theory of strict monoidal categories is not a purely equational theory, we cannotdirectly use results from [6]. Instead, we need to work in the setting of quasi-equationaltheories . These are multi-sorted theories in which the operations can be partial ; equivalently,they are finite-limit theories. These include the theories of categories, groupoids, strictmonoidal categories, symmetric/braided/balanced monoidal categories, and crossed modules.They also include what one might call diagram theories , which are theories describingdiagrams of a specified shape in a category of models. As a special case, one obtains theorieswhose categories of models are presheaf categories. Our first main contribution of the paperis then a generalization of the syntactic characterization of isotropy from equational theoriesto this wider class of quasi-equational theories.While we indicated why the non-trivial isotropy of strict monoidal categories is perhapssurprising, there is also a sense in which it is to be expected. Indeed, since strict monoidalcategories are monoids internal to
Cat , we expect that the isotropy of strict monoidalcategories is closely related to that of monoids. Since the isotropy of a monoid M is itssubgroup of invertible elements, the conjecture that the isotropy of a strict monoidal categoryis its group of invertible objects is not unreasonable. However, it is not at all immediate Not to be confused with the so-called theories of presheaf type , which are theories whose classifyingtopos happens to be a presheaf topos.
C V I T 2 0 1 6 that the isotropy of a strict monoidal category should be determined completely by its set ofobjects; the recognition that this is the case is the second main contribution of this paper. A priori , one can try to establish this result in a variety of ways. First of all, it canbe approached purely syntactically, making careful analysis of the word problem for strictmonoidal categories. However, several aspects of this analysis can also be cast in moreconceptual terms, giving rise to a categorical way of deriving the isotropy of strict monoidalcategories from that of monoids. We thus also include a more categorical viewpoint, whichapplies to several other theories of categorical structures, including crossed modules.
We begin by reviewing the relevant notions from categorical logic. For more details concerningquasi-equational theories and partial Horn logic, we refer to [7]. For a general treatment ofcategorical logic, see [9]. ▶ Definition 1 ( Signatures, Terms, Horn Formulas, Horn Sequents, Quasi-EquationalTheories ) . A signature Σ is a pair of sets Σ = (Σ
Sort , Σ Fun ), where Σ
Sort is the set of sorts of Σ andΣ
Fun is the set of function/operation symbols of Σ. Each element f ∈ Σ Fun comes equippedwith a finite tuple of sorts ( A , . . . , A n , A ), and we write f : A × . . . × A n → A .Given a signature Σ, we assume that we have a countably infinite set of variables of eachsort A . Then one can recursively define the set Term (Σ) of terms of Σ in the usual way,so that each term will have a uniquely defined sort. We write
Term c (Σ) for the set of closed terms of Σ, i.e. terms containing no variables.Given a signature Σ, one can recursively define the set Horn (Σ) of
Horn formulas of Σ inthe usual way, where a Horn formula is a finite conjunction of equations between elementsof
Term (Σ). We write ⊤ for the empty conjunction.A Horn sequent over a signature Σ is an expression of the form φ ⊢ ⃗x ψ , where φ, ψ ∈ Horn (Σ) and have variables among ⃗x .A quasi-equational theory T over a signature Σ is a set of Horn sequents over Σ, whichwe call the axioms of T . ◀ One can set up a deduction system of partial Horn logic (PHL) for quasi-equational theories,axiomatizing the notion of a provable sequent φ ⊢ ⃗x ψ . Accordingly, for a theory T we havethe notion of a T -provable sequent; moreover, if ⊤ ⊢ ⃗x φ is T -provable, then we simply saythat T proves φ , and write T ⊢ ⃗x φ .We refer the reader to [7, Definition 1] for the logical axioms and inference rules of PHL.The distinguishing feature of this deduction system is that equality of terms is not assumedto be reflexive, i.e. if t ( ⃗x ) is a term over a given signature, then ⊤ ⊢ ⃗x t ( ⃗x ) = t ( ⃗x ) is not alogical axiom of partial Horn logic, unless t is a variable. In other words, if we abbreviatethe equation t = t by t ↓ (read: t is defined ), then unless t is a variable, the sequent ⊤ ⊢ ⃗x t ↓ is not a logical axiom of PHL. Furthermore, the logical inference rule of term substitution isthen only formulated for defined terms. ▶ Example 2.
We have the following examples of quasi-equational theories:Every single-sorted algebraic theory is a quasi-equational theory; this includes the usualalgebraic theories of (commutative) monoids, (abelian) groups, (commutative) unitalrings, etc.The theories of (small) categories, groupoids, categories with a (chosen) terminal object,categories with (chosen) finite products, categories with (chosen) finite limits, locally . Hofstra, J. Parker, and P. J. Scott 23:5 cartesian closed categories, and elementary toposes, can all be axiomatized as quasi-equational theories over a two-sorted signature (with one sort O for objects and one sort A for arrows). For details see [7, Example 4 and Section 6]. The theory of (small) strictmonoidal categories can also be axiomatized as a quasi-equational theory (see Section 4below).If T is any quasi-equational theory and J is any small category, then one can axiomatizethe functor category T mod J by a quasi-equational theory T J , see [8, Chapter 5]. ◀ In the remainder of the paper, by theory we shall mean quasi-equational theory , unlessexplicitly stated otherwise.We now review the set-theoretic semantics of PHL. This follows the standard patternof algebraic theories, with the key difference being that function symbols are now onlyinterpreted as partial functions. We write f : A ⇁ B for a partial function from A to B ,which is by definition a total function f : dom ( f ) → B for some subset dom ( f ) ⊆ A . IfΣ is a signature, then a Σ -structure M is a family of sets M C indexed by the sorts C ofΣ, together with interpretations of the function symbols f : A × · · · × A k → A as partialfunctions f M : M A × · · · × M A k ⇁ M A . By induction on the structure of a term t invariable context x : A , . . . , x k : A k , we obtain its interpretation as a partial function t M : M A × · · · × M A k ⇁ M A in a Σ-structure M , while a Horn formula φ ( x , . . . , x k ) isinterpreted as a subset φ ( x , . . . , x k ) M ⊆ M A × . . . × M A k .A Σ-structure M satisfies a Horn sequent φ ⊢ ⃗x ψ if φ ( x , . . . , x k ) M ⊆ ψ ( x , . . . , x k ) M .When T is a theory, then a Σ-structure M is a T - model when it satisfies all the T -axioms,and hence all the T -provable sequents (by soundness of partial Horn logic). ▶ Definition 3.
Let Σ be a signature and
M, N
Σ-structures. A homomorphism h : M → N is a family of total functions h = ( h A : M A → N A ) A : Sort with the property that if f : A × . . . × A n → A is any function symbol of Σ and ( a , . . . , a n ) ∈ dom (cid:0) f M (cid:1) , then( h A ( a ) , . . . , h A n ( a n )) ∈ dom (cid:0) f N (cid:1) and h A (cid:0) f M ( a , . . . , a n ) (cid:1) = f N ( h A ( a ) , . . . , h A n ( a n )).The homomorphism h reflects definedness if moreover ( h A ( a ) , . . . , h A n ( a n )) ∈ dom (cid:0) f N (cid:1) always implies ( a , . . . , a n ) ∈ dom (cid:0) f M (cid:1) . ◀ When working with homomorphisms we often suppress the sort subscripts. The T -modelsand their homomorphisms then form a category T mod , which is complete and cocomplete. ▶ Definition 4. A morphism of theories ρ : T → S consists of a mapping A ρ ( A ) fromthe sorts of T to the sorts of S and a mapping f ρ ( f ) from the function symbols of T tothe terms of S that preserves both typing and provability. ◀ When ρ : T → S is a morphism of theories, we have an induced functor ρ ∗ : S mod → T mod by [7, Proposition 28]. This functor ρ ∗ sends an S -model M to the T -model ρ ∗ M with( ρ ∗ M ) A := M ρ ( A ) for each sort A of T and f ρ ∗ M := ρ ( f ) M for each function symbol f of T .In particular, for every sort A of T there is a forgetful functor U A : T mod → Set sendinga model M to the carrier set M A (induced by the theory morphism from the single-sortedempty theory to T that sends the unique sort of the former theory to the sort A ). Eachsuch functor also has a left adjoint F A (see e.g. [7, Theorem 29]), giving for a set X the free T -model F A ( X ) generated by X : F A ⊣ U A : Set ⇄ T mod . ▶ Definition 5.
For a T -model M , we can form the extension T ( M ), the diagram theory of M , adapted from ordinary model theory. It is the extension of T byA constant a : A and an axiom ⊤ ⊢ a ↓ for every element a ∈ M A (for every sort A ).An axiom ⊤ ⊢ f ( a , . . . , a k ) = f ( a , . . . , a k ) for every function symbol f : A × · · · × A k → A and tuple ( a , . . . , a k ) ∈ dom (cid:0) f M (cid:1) . ◀ C V I T 2 0 1 6
For better readability, we will generally omit the bar notation on constants of M . Clearly M is a model of T ( M ), and in fact it is the initial model: T ( M ) mod ≃ M/ T mod (see [8,Lemma 2.2.4] for a proof). The obvious theory morphism T → T ( M ) corresponds to theforgetful functor M/ T mod → T mod .One of the central constructions in the present paper is that of adjoining an indeterminate to a model. Given a T -model M and a sort A of T , we form a new model M ⟨ x A ⟩ whichis the result of freely adjoining a new element x A of sort A to M . Formally, one candefine M ⟨ x A ⟩ as M + F A (1), where F A (1) is the free T -model on one generator of sort A .Consequently, homomorphisms M ⟨ x A ⟩ → N are in natural bijective correspondence withpairs ( h, n ) consisting of a homomorphism h : M → N and an element n ∈ N A . We willwrite T ( M, x A ) for the theory extending the diagram theory T ( M ) by a new constant x A : A and a new axiom ⊤ ⊢ x A ↓ . One can then equivalently define the T -model M ⟨ x A ⟩ as theinitial model of T ( M, x A ). For a sequence of (not necessarily distinct) sorts A , . . . , A k , wewill also write T ( M, x , . . . , x k ) for the theory extending T ( M ) by new, pairwise distinctconstants x i : A i and axioms ⊤ ⊢ x i ↓ for each 1 ≤ i ≤ k .Finally, we note that for a T -model M , an indeterminate x A of sort A , and an arbitrarysort B , we have M ⟨ x A ⟩ B = { t ∈ Term c ( T ( M ) , x A ) | t : B and T ( M, x A ) ⊢ t ↓} / = , (3)i.e. the carrier set M ⟨ x A ⟩ B is the quotient of the set of provably defined closed terms of sort B , possibly containing x A and constants from M , modulo the partial congruence relation of T ( M, x A )-provable equality. For more details, see [8, Remark 2.2.7]. We now embark on the syntactic description of the covariant isotropy group of a theory.First, let us briefly review the simpler situation of a single-sorted equational theory T . Thatis, we describe the isotropy group of a T -model M (details are in [6]). The elements of themodel M ⟨ x ⟩ (for x an indeterminate) can be described explicitly as congruence classes ofterms t ( x ), built from the indeterminate x , constants from M , and the operation symbols of T . Two such terms are congruent if they are T ( M, x )-provably equal. For example, if T isthe theory of monoids and M is a monoid with m , m , m ∈ M , unit e , and m m = m ,then the terms t = x m x m m x and x em e x em x are congruent.For a set-theoretic T -model M , each congruence class [ t ] ∈ M ⟨ x ⟩ can be interpreted as afunction t M : M → M , via substitution into the indeterminate x . We thus have a mapping M ⟨ x ⟩ → [ M, M ] ; [ t ] t M where [ M, M ] is the set of functions from M to itself (well-definedness follows from soundnessof the set-theoretic semantics of equational logic). Moreover, this mapping is a homomorphismof monoids, where the monoid structure on M ⟨ x ⟩ is given by substitution: [ t ] · [ s ] := [ t [ s/ x ]],the unit being [ x ]. We then restrict on both sides to the invertible elements, obtaining agroup homomorphism Inv ( M ⟨ x ⟩ ) → Perm ( M ) from the group of substitutionally invertible(congruence classes of) terms to the permutation group of the set M . However, we do notwish to just consider arbitrary permutations of the set M , but rather automorphisms of the T -model M ; in fact, we want to consider inner automorphisms, i.e. automorphisms thatextend naturally along any homomorphism M → N . On the level of terms [ t ] ∈ M ⟨ x ⟩ , thisis achieved by the following definition: [ t ] is said to commute generically with a function . Hofstra, J. Parker, and P. J. Scott 23:7 symbol f : A n → A ( A being the unique sort of T ) if T ( M, x , . . . , x n ) ⊢ t [ f ( x , . . . , x n ) / x ] = f ( t [ x / x ] , . . . , t [ x n / x ]) . We then form the subgroup
DefInn ( M ) of Inv ( M ⟨ x ⟩ ) on those [ t ] that commute genericallywith all function symbols of the theory. This ensures that such a [ t ] induces an automorphism of the T -model M and not merely a permutation of its underlying set, thus yielding amapping ( − ) M : DefInn ( M ) → Aut ( M ). However, it turns out that such an automorphisminduced by an element of DefInn ( M ) is also inner . Indeed, given h : M → N , we obtaina homomorphism h ⟨ x ⟩ : M ⟨ x ⟩ → N ⟨ x ⟩ of the substitution monoids, which restricts to agroup homomorphism DefInn ( M ) → DefInn ( N ). It can then be shown that the subgroup DefInn ( M ) is isomorphic to the covariant isotropy group of M , where θ M : Z ( M ) → Aut ( M )is the comparison homomorphism (2): DefInn ( M ) ( − ) M (cid:15) (cid:15) ⊆ (cid:47) (cid:47) Inv ( M ⟨ x ⟩ ) ( − ) M (cid:15) (cid:15) Z ( M ) ∼ = (cid:57) (cid:57) (cid:114)(cid:114)(cid:114)(cid:114)(cid:114) θ M (cid:47) (cid:47) Aut ( M ) ⊆ (cid:47) (cid:47) Perm ( M )We now explain how to extend this result to a (multi-sorted) quasi-equational theory T . Firstof all, in order to accommodate multi-sortedness, we need to consider, for a T -model M ,the model M ⟨ x A ⟩ obtained by adjoining an indeterminate x A of sort A for any sort A of T .From the fact that under the interpretation mapping t t M substitution corresponds tocomposition, it follows that M ⟨ x A ⟩ A carries a monoid structure, defined as before in termsof substitution into the indeterminate x A . We now write M ⟨ ¯ x ⟩ := Y A : Sort M ⟨ x A ⟩ A for the sort-indexed product monoid of these substitution monoids. An element of M ⟨ ¯ x ⟩ istherefore a sort-indexed family of congruence classes of terms [ s A ] A , where s A ∈ Term c ( T ( M ) , x A )is of sort A and T ( M, x A ) ⊢ s A ↓ . Given such a tuple [ s A ] A , its interpretation gives us, ateach sort A , a total function s MA : M A → M A (because s A is provably defined in T ( M, x A )),defined via substitution into the indeterminate x A (cf. [8, Remark 2.2.12]). The centraldefinitions towards characterizing those [ s A ] A ∈ M ⟨ ¯ x ⟩ that induce elements of isotropy for M are then as follows: ▶ Definition 6.
Let M be a T -model and [ s C ] C ∈ M ⟨ ¯ x ⟩ .If f : A × . . . × A n → A is a function symbol of Σ, then we say that ([ s C ]) C commutesgenerically with f if the Horn sequent f ( x , . . . , x n ) ↓ ⊢ s A [ f ( x , . . . , x n ) / x A ] = f ( s A [ x / x A ] , . . . , s A n [ x n / x A n ])is provable in T ( M, x , . . . , x n ).We say that ([ s C ]) C is invertible if for each sort A there is some (cid:2) s − A (cid:3) ∈ M ⟨ x A ⟩ A with T ( M, x A ) ⊢ s A (cid:2) s − A / x A (cid:3) = x A = s − A [ s A / x A ] . We say that ([ s C ]) C reflects definedness if for every function symbol f : A × . . . × A n → A in Σ with n ≥
1, the sequent f ( s A [ x / x A ] , . . . , s A n [ x n / x A n ]) ↓ ⊢ f ( x , . . . , x n ) ↓ is provable in T ( M, x , . . . , x n ). ◀ C V I T 2 0 1 6
The condition that [ s C ] C commutes generically with the function symbols of T then en-sures that [ s C ] C induces not just an endofunction of each carrier set M C but in fact anendo morphism of the T -model M . Invertibility of [ s C ] C then ensures that these endomor-phisms are bijective. However, due to the fact that function symbols are interpreted aspartial maps, a (sortwise) bijective homomorphism is not in general an isomorphism in T mod : a bijective homomorphism is an isomorphism precisely when it reflects definedness(cf. [8, Lemma 2.2.33]). Thus, the third condition ensures that the inverses (cid:2) s − A (cid:3) also induceendomorphisms.Let us write DefInn ( M ) again for the subgroup of the product monoid M ⟨ ¯ x ⟩ consist-ing of those elements satisfying the three conditions above. We then have the followingcharacterization, of which detailed proofs can be found in [8, Theorems 2.2.41, 2.2.53] ▶ Theorem 7.
Let T be a quasi-equational theory. Then for any M ∈ T mod we have Z ( M ) ∼ = DefInn ( M ) = (cid:26) [ s C ] C ∈ M ⟨ ¯ x ⟩ [ s C ] C is invertible, commutes generically withall operations, and reflects definedness. (cid:27) . ◀ With this description of the isotropy group of an arbitrary quasi-equational theory, we nowturn to the specific example of strict monoidal categories. We can axiomatize these using thefollowing signature Σ (where the first two ingredients comprise the signature for categories):two sorts O and A (for objects and arrows);function symbols dom , cod : A → O , id : O → A , and ◦ : A × A → A ;function symbols ⊗ O : O × O → O , ⊗ A : A × A → A ;constant symbols I O : O and I A : A .Whenever reasonable, we omit the subscripts on ⊗ and I . As axioms, we take those forcategories and add (omitting the hypothesis ⊤ ): x ⊗ y ↓ , I ↓ , x ⊗ ( y ⊗ z ) = ( x ⊗ y ) ⊗ z, x ⊗ I = x = I ⊗ x , dom ( f ⊗ g ) = dom ( f ) ⊗ dom ( g ) , cod ( f ⊗ g ) = cod ( f ) ⊗ cod ( g ),( f ⊗ g ) ◦ ( h ⊗ k ) = ( f ◦ h ) ⊗ ( g ◦ k ), id ( x ⊗ y ) = id ( x ) ⊗ id ( y ) , id ( I O ) = I A ,where the penultimate axiom of course requires the hypotheses that f ◦ h and g ◦ k aredefined. Note that in this fragment of logic, we need to include an axiom forcing the tensorproducts and unit object and arrow to be total operations. Because of strict associativity,we may omit brackets when dealing with nested expressions involving tensor products. Weshall henceforth denote this theory by T , and write StrMonCat for its category of models,whose objects are small strict monoidal categories and whose morphisms are strict monoidalfunctors. Our goal is now to prove the following: ▶ Theorem 8.
The covariant isotropy group Z ( C ) of a strict monoidal category C isisomorphic to the Picard group of C , i.e. the group of invertible elements in the monoid ofobjects of C . ◀ Because a strict monoidal category is a monoid object in
Cat , we have two functorsOb , Arr :
Cat ( Mon ) =
StrMonCat ⇒ Mon . . Hofstra, J. Parker, and P. J. Scott 23:9 We shall ultimately prove that the diagram
StrMonCat Ob (cid:47) (cid:47) Z (cid:37) (cid:37) (cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75) Mon Z Mon (cid:124) (cid:124) (cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)
Grp (4)commutes up to isomorphism, showing that the covariant isotropy functor of
StrMonCat is completely determined by the covariant isotropy functor of
Mon . Since we have provedin [6, Example 4.3] that the latter sends a monoid M to its subgroup of invertible elements,Theorem 8 then follows. In this section we analyse the process of adjoining an indeterminate to a strict monoidalcategory. Let us first describe explicitly the result of adjoining an indeterminate to a monoid. ▶ Definition 9.
Let M be a monoid, and X a set of symbols disjoint from M .A word over M ⟨ X ⟩ is formal string of symbols from the alphabet M ∪ X .A word w is in (expanded) normal form when it has the form w = m x m x · · · x n − m n for m i ∈ M and x j ∈ X . In other words, w is in expanded normal form if it contains notwo consecutive elements of M , and if every occurrence of some x ∈ X in w is flanked onboth sides by an element of M . ◀ We then have (by taking an arbitrary word, multiplying adjacent elements from M andinserting the unit of M whenever necessary): ▶ Lemma 10.
When M = ( M, e, · ) is a monoid, every element w of the monoid M ⟨ x ⟩ has acanonical representative w = m x m x · · · x m n in expanded normal form.Moreover, the unit of M ⟨ x ⟩ is represented as the word e and multiplication is given by ( m x m x · · · x m j ) · ( m ′ x m ′ x · · · x m ′ k ) = m x m x · · · x ( m j · m ′ ) x m ′ · · · x m ′ k . ◀ We now turn turn to the process of adjoining an indeterminate object x O , i.e. anindeterminate of sort O , to a strict monoidal category C . In order to determine the objectsof C ⟨ x O ⟩ , we note that the functor Ob : StrMonCat → Mon has both adjoints:
StrMonCat Ob ⊥⊥ (cid:47) (cid:47) Mon ∆ (cid:114) (cid:114) ∇ (cid:108) (cid:108) Here ∆ sends a monoid M to the discrete strict monoidal category on M and ∇ sends M to the indiscrete strict monoidal category on M . In fact, if E is any category with finitelimits, then the forgetful functor Ob : Cat ( E ) → E has both adjoints (for a proof, mimic theargument for E = Set ). As such, Ob :
StrMonCat → Mon preserves all limits and colimits.Now by definition C ⟨ x O ⟩ ∼ = C + F , where F is the free strict monoidal category on a singleobject; moreover, the latter is easily seen to be isomorphic to ∆( F F C ⟨ x O ⟩ ) ∼ = Ob( C + F ) ∼ = Ob( C ) + Ob( F ) = Ob( C ) + F ∼ = Ob( C ) ⟨ x ⟩ . For a general functor F : E → F it is not the case that Z E ∼ = Z F ◦ F . In fact, in [3] it is explained thatin general the relationship between Z E and Z F ◦ F takes the form of a span . The commutativity of (4)may thus be expressed by saying that both legs of the span associated with Ob are isomorphisms. C V I T 2 0 1 6
This shows that the object forgetful functor preserves the process of adjoining an indeterminateof sort O . We now describe the monoid of arrows of C ⟨ x O ⟩ . It is not true that Arr : StrMonCat → Mon preserves binary coproducts. However, it does preserve the coproduct C + F . ▶ Lemma 11. If C ∈ StrMonCat , we have
Arr( C ⟨ x O ⟩ ) ∼ = Arr( C ) ⟨ x ⟩ . Proof.
We sketch a syntactic proof, noting that the result can also be deduced categoricallyfrom the fact that the endofunctor − + F Mon → Mon preserves pullbacks.An element of Arr( C ⟨ x O ⟩ ) is a congruence class of terms t built up from the operationsof T , arrows of C , and the term id ( x O ). One shows by induction that every such term t iscongruent to one of the form t = f ⊗ id ( x O ) ⊗ f ⊗ id ( x O ) ⊗ · · · ⊗ id ( x O ) ⊗ f n where each f i isan arrow of C . Thus, the monoid Arr( C ⟨ x O ⟩ ) is isomorphic, by Lemma 10, to Arr( C ) ⟨ x ⟩ . ◀ In fact, we may describe the relationship between the functor ( − ) + F adjoining anindeterminate object to a strict monoidal category and the functor ( − ) + F ▶ Proposition 12.
The functor ( − ) + F : Cat ( Mon ) → Cat ( Mon ) is naturally isomorphicto Cat ( − + F . ◀ We thus obtain the following explicit description of the strict monoidal category C ⟨ x O ⟩ : Objects:
Words a x a x · · · x a n where each a i is an object of C . Morphisms:
Words f x f x · · · x f n where each f i is an arrow of C . Domain: dom ( f x · · · x f n ) = dom ( f ) x · · · xdom ( f n ). Codomain: cod ( f x · · · x f n ) = cod ( f ) x · · · xcod ( f n ). Identities: id ( a x · · · x a n ) = id ( a ) x · · · xid ( a n ). Composition: ( f x · · · x f n ) ◦ ( g x · · · x g n ) = f g x · · · x f n g n . Tensors: ( a x · · · x a n ) ⊗ ( b x · · · x b m ) = a x · · · x ( a n ⊗ b ) x · · · x b m . Tensor units: I O , I A (tensor units of C regarded as one-letter words). ◀ Next, we address the issue of adjoining an indeterminate arrow x A to C . Here we cannotinvoke a simple categorical fact about coproducts, because Arr : StrMonCat → Mon doesnot preserve coproducts of the relevant kind (which, to be explicit, is coproducts with thefree strict monoidal category F , where is the free-living arrow). We are thus forced tocarry out a direct syntactic analysis of the objects and arrows of C ⟨ x A ⟩ . Note that these aregenerated, under the operations of domain, codomain, identities, composition, and tensorproduct, from the objects and arrows of C , together with the new arrow x A . In particular,there will be two new objects dom ( x A ) and cod ( x A ), and corresponding identity arrows id ( dom ( x A )), id ( cod ( x A )). ▶ Definition 13.
Let C ∈ StrMonCat . A closed term t ∈ Term c ( C , x A ) of sort O is in normalform when it is of the form t = a ⊗ x ⊗ · · · ⊗ x k − ⊗ a k , where each a i is an object of C and each x i ∈ { dom ( x A ) , cod ( x A ) } . A closed term t ∈ Term c ( C , x A ) of sort A is in normalform when it is of the form t = f ⊗ x ⊗ · · · ⊗ x k − ⊗ f k , where each f i is an arrow of C andeach x i ∈ { x A , id ( dom ( x A )) , id ( cod ( x A )) } . ◀ Note that for a functor ρ ∗ : S mod → T mod induced by a theory morphism ρ : T → S it is not in generalthe case that ρ ∗ ( M ⟨ x ⟩ ) ∼ = ( ρ ∗ M ) ⟨ x ⟩ . . Hofstra, J. Parker, and P. J. Scott 23:11 We may now describe C ⟨ x A ⟩ in terms of normal forms. It is straightforward to prove,by directly verifying the universal property, that the category described below is indeedisomorphic to C ⟨ x A ⟩ . Alternatively, one can endow the set { t ∈ Term c ( C , x A ) | t ↓} with arewriting system and show that each term has a unique normal form. Objects: closed terms of sort O in normal form. Arrows: closed terms of sort A in normal form. Domain: dom ( f ⊗ x ⊗ · · · ⊗ x k − ⊗ f k ) = dom ( f ) ⊗ y ⊗ · · · ⊗ y k − ⊗ dom ( f k ) where y i = dom ( x A ) when x i = x A or x i = id ( dom ( x A )), and y i = cod ( x A ) otherwise. Codomain: cod ( f ⊗ x ⊗ · · · ⊗ x k − ⊗ f k ) = cod ( f ) ⊗ y ⊗ · · · ⊗ y k − ⊗ cod ( f k ) where y i = cod ( x A ) when x i = x A or x i = id ( cod ( x A )), and y i = dom ( x A ) otherwise. Identities: id ( a ⊗ x ⊗ · · · ⊗ x k − ⊗ a k ) = id ( a ) ⊗ id ( x ) ⊗ · · · ⊗ id ( x k − ) ⊗ id ( a k ). Composition:
For t = f ⊗ x ⊗ · · · ⊗ x k − ⊗ f k and s = g ⊗ x ′ ⊗ · · · ⊗ x ′ k − ⊗ g k with cod ( t ) = dom ( s ), define s ◦ t = ( g f ) ⊗ z ⊗ · · · ⊗ · · · ⊗ z k − ⊗ ( g k f k ), where z i is definedfrom x i and x ′ i in the evident way. Tensors: ( a ⊗ x ⊗ · · · ⊗ x n − ⊗ a n ) ⊗ ( b ⊗ y ⊗ · · · ⊗ y m − ⊗ b m ) = a ⊗ x ⊗ · · · ⊗ x n − ⊗ ( a n ⊗ b ) ⊗ y ⊗ · · · ⊗ y m − ⊗ b m . Tensor units: I O , I A (tensor units of C regarded as one-letter words). We are now in a position to analyse the isotropy group of a strict monoidal category. Bythe results of the previous section, we know that an element of isotropy of a strict monoidalcategory C may be taken to be of the form ( s O , s A ), where s O and s A are closed terms innormal form of sort O and A respectively.The first observation is that elements of isotropy of the monoid Ob( C ) induce elementsof isotropy of C . (As we shall see in the next section, this is not specific to strict monoidalcategories.) In what follows, we write Z ( C ) for the isotropy group of a strict monoidalcategory C , and Z Mon ( M ) for the isotropy group of a monoid M (which is the group ofinvertible elements of M by [6, Example 4.3]). ▶ Lemma 14.
Let C ∈ StrMonCat . When a is an invertible object in the monoid Ob( C ) with inverse b , the pair ( a ⊗ x O ⊗ b, id ( a ) ⊗ x A ⊗ id ( b )) is an element of Z ( C ) . Proof.
To show that ( a ⊗ x O ⊗ b, id ( a ) ⊗ x A ⊗ id ( b )) is an element of isotropy, one canstraightforwardly verify that it is invertible, commutes generically with all operations of T ,and reflects definedness (for details, see [8, Proposition 3.9.35]). However, it is less work toshow directly that given a strict monoidal functor F : C → D , we obtain an automorphism α F of D as follows. On objects we set α F ( d ) = F a ⊗ d ⊗ F b , while on arrows we set α F ( f ) = id ( F a ) ⊗ f ⊗ id ( F b ). It is routine to check that this defines an automorphism andthat the family α F is natural in F . ◀ The above lemma gives us a mapping θ C : Z Mon (Ob( C )) → Z ( C ). It is easily verifiedthat this is in fact a group homomorphism, natural in C .Next, we define a retraction σ of θ . This is done categorically using the right adjoint ∇ to Ob. Concretely, given an element of isotropy α ∈ Z ( C ), we define an element σ C ( α ) ∈ Z Mon (Ob( C )) as follows: consider a monoid homomorphism h : Ob( C ) → N . Thiscorresponds by the adjunction Ob ⊣ ∇ to a strict monoidal functor ˜ h : C → ∇ ( N ); thecomponent of α at ˜ h is an automorphism of ∇ ( N ), whence Ob ( α ˜ h ) is an automorphism of N (using the fact that Ob ◦ ∇ = 1). This leads to: C V I T 2 0 1 6 ▶ Lemma 15. If C ∈ StrMonCat , the map σ C : Z ( C ) → Z Mon (Ob( C )) is a group homomor-phism. ◀ Interpreting this syntactically, we find that if ( s O , s A ) ∈ Z ( C ), then s O ∈ Z Mon (Ob( C )),and hence s O = a ⊗ x O ⊗ b for an invertible object a with inverse b . We also see that σ C is aretraction of θ C , i.e. that σ C ◦ θ C = 1.Since θ C is a section, it now remains to show that θ C is an epimorphism of groups, i.e. issurjective. So we must show for any element of isotropy ( s O , s A ) = ( a ⊗ x O ⊗ b, s A ) ∈ Z ( C )(with invertible object a and inverse b ) that we have s A = id ( a ) ⊗ x A ⊗ id ( b ). To this end, wefirst note that since ( s O , s A ) commutes generically with the operations dom and cod we get a ⊗ dom ( x A ) ⊗ b = s O [ dom ( x A ) / x O ] = dom ( s A )and likewise a ⊗ cod ( x A ) ⊗ b = s O [ cod ( x A ) / x O ] = cod ( s A ) . Thus, by uniqueness of normal forms, s A must have the form f ⊗ x A ⊗ g for some morphisms f : a → a and g : b → b of C . So we must now show that f = id ( a ) and g = id ( b ), and forthat we use the fact that ( s O , s A ) commutes generically with id , giving f ⊗ id ( x O ) ⊗ g = s A [ id ( x O ) / x A ] = id ( s O ) = id ( a ⊗ x O ⊗ b ) = id ( a ) ⊗ id ( x O ) ⊗ id ( b ) . We now get the desired equalities f = id ( a ) and g = id ( b ) by appealing to the uniqueness ofnormal forms. This concludes the proof of Theorem 8. In this section we briefly explore some further theories of interest, and indicate the extent towhich the analysis of the case of strict monoidal categories can be generalized.
The analysis of strict monoidal categories reveals that it is profitable, at least for the purposesof understanding isotropy, to regard strict monoidal categories as internal categories in thecategory
Mon of monoids. This naturally raises the following question: are there otheralgebraic theories T for which the forgetful functor Ob : Cat ( T mod ) → T mod induces anisomorphism on the level of isotropy groups?Let us first state which of the ideas from the case of monoids carry over to a generalalgebraic theory T . First of all, we still have a string of adjunctions Cat ( T mod ) Ob ⊥⊥ (cid:47) (cid:47) T mod ∆ (cid:114) (cid:114) ∇ (cid:108) (cid:108) with Ob ◦ ∇ ∼ = 1 ∼ = Ob ◦ ∆. This allows us to deduce the existence of a pair of naturalcomparison homomorphisms θ C : Z T (Ob( C )) → Z ( C ) ; σ C : Z ( C ) → Z T (Ob( C ))with σ ◦ θ = 1 (here Z denotes the isotropy of Cat ( T mod ) and Z T that of T mod ). We thushave: . Hofstra, J. Parker, and P. J. Scott 23:13 ▶ Lemma 16.
Let T be any algebraic theory and C any internal category in T mod . Then Z T (Ob( C )) is a retract of Z ( C ) , naturally in C . In the case of strict monoidal categories, we were able to prove syntactically that theembedding-retraction pair ( θ, σ ) is an isomorphism. The same proof can also be applied intwo other cases of interest: ▶ Proposition 17.
The isotropy group of a crossed module A → G is isomorphic to G . ◀ Proof.
When composing the functor Ob :
Cat ( Grp ) → Grp with the equivalence
XMod ∼ −→ Cat ( Grp ), one obtains the forgetful functor which sends a crossed module A → G to G .Moreover, the isotropy group of a group G is G itself by [6, Example 4.1]. ◀▶ Proposition 18.
The isotropy group of a strict symmetric monoidal category is trivial.
Proof.
The isotropy group of commutative monoids is trivial by [6, Example 4.4]. ◀ Using Theorem 7, we can also compute the covariant isotropy of any presheaf category
Set J for a small category J . We first axiomatize Set J as a quasi-equational theory. ▶ Definition 19 ( Presheaf Theory ) . Let J be a small category. We define the signatureΣ J to have one sort X i for each i ∈ Ob( J ) and one function symbol α f : X i → X j for eacharrow f : i → j in J .We define the presheaf theory T J to be the quasi-equational theory over the signatureΣ J with the following axioms: ⊤ ⊢ x : X i α f ( x ) ↓ for any f : i → j in J (i.e. each α f is total ). ⊤ ⊢ x : X i α id i ( x ) = x for every i ∈ Ob J (i.e. each α id i acts as an identity). ⊤ ⊢ x : X i α g ( α f ( x )) = α g ◦ f ( x ) for any composable pair i f −→ j g −→ k in J . ◀ We will lighten notation and write i instead of X i and f instead of α f . We write x i for anindeterminate of sort i . It is completely straightforward to verify that we have an isomorphismof categories T J mod ∼ = Set J (for details, see [8, Proposition 5.1.8]). So to compute thecovariant isotropy group Z Set J : Set J → Grp of the category
Set J , it is equivalent to computethe covariant isotropy group Z T J : T J mod → Grp of the theory T J .According to Theorem 7, we have for a T J -model (i.e. functor) F : J →
Set that Z ( F ) ∼ = ( [ s i ] i ∈ Y i ∈J F ⟨ x i ⟩ i | [ s i ] i is invertible and commutes gen. with all f : i → j ) . Note that since all terms are provably total in T J , the condition that [ s i ] i reflects definednesscan be omitted. We now require the following preparatory lemma. ▶ Lemma 20.
Let M ∈ T J mod . If f, f ′ : i → j are parallel arrows in J and T J ( M, x i ) ⊢ f ( x i ) = f ′ ( x i ) , then f = f ′ . Proof.
Note that the assumption T J ( M, x i ) ⊢ f ( x i ) = f ′ ( x i ) implies that for any homomor-phism (i.e. natural transformation) η : M → N we have N ( f ) = N ( f ′ ), since given any a ∈ N i there is a homomorphism [ η, a ] : M ⟨ x i ⟩ → N sending x i to a (cf. also [8, Lemma3.1.2]). We take N : J →
Set to be N := M + J ( i, − ) and η to be the coproduct inclusion.Then f = f ◦ id ( i ) = N ( f )( id ( i )) = N ( f ′ )( id ( i )) = f ′ ◦ id ( i ) = f ′ as required. ◀ C V I T 2 0 1 6
As a consequence of this lemma, we find that any term congruence class [ t ] ∈ T J ( M, x i ) hasa unique representation as t ≡ a for some a ∈ M j or t ≡ f ( x i ) for some f with domain i ,depending on whether the indeterminate x i occurs in t .Let Aut ( Id J ) be the group of natural automorphisms of the identity functor Id J : J → J of a small category J . This group is sometimes called the center of J . We now have: ▶ Proposition 21.
Let J be a small category. For any M ∈ T J mod we have Z ( M ) = ( ([ ψ i ( x i )]) i ∈ Y i ∈J M ⟨ x i ⟩ i : ψ ∈ Aut ( Id J ) ) . Proof.
It is straightforward to prove the right-to-left inclusion using the assumption that ψ is anatural automorphism of Id J , so let us turn to the less obvious converse inclusion. So supposethat ([ s i ]) i ∈J ∈ Z T J ( M ) ⊆ Q i M ⟨ x i ⟩ i . By the lemma, as well as the fact that invertibleterms must contain the indeterminate, we may represent s i = ψ i ( x i ), where ψ i : i → i is a mapin J . We show that ψ := ( ψ i ) i ∈J is a natural automorphism of Id J . First, each ψ i : i → i is an isomorphism: take the inverse ([ t i ]) i of ([ s i ]) i , and represent this inverse as χ i ( x i ) for χ i : i → i . Since T J ( M, x i ) proves the equations ( ψ i ◦ χ i )( x i ) = ψ i ( χ i ( x i )) = x i = id i ( x i ) and( χ i ◦ ψ i )( x i ) = id i ( x i ), it follows by Lemma 20 that ψ i is the inverse of χ i .To show that ψ is natural, let f : j → k be any arrow in J , and let us show that ψ k ◦ f = f ◦ ψ j . We know that ([ ψ i ( x i )]) i = [ s i ] i commutes generically with the functionsymbol f : X j → X k of Σ J , which implies that T J ( M, x j ) ⊢ ( ψ k ◦ f )( x j ) = ( f ◦ ψ j )( x j ), fromwhich we obtain the required ψ k ◦ f = f ◦ ψ j again by Lemma 20. Thus ψ : Id J ∼ −→ Id J isindeed a natural automorphism with ([ s i ]) i = ([ ψ i ( x i )]) i . ◀▶ Corollary 22.
Let J be a small category. For any functor F : J →
Set we have Z ( F ) ∼ = Aut ( Id J ) , and hence the covariant isotropy group functor of Set J is constanton the automorphism group of Id J . Proof.
Given ([ s i ]) i ∈J ∈ Z T J ( F ), we know by Proposition 21 that there is some ψ ∈ Aut ( Id J )with [ s i ] i = [ ψ i ( x i )] i . We now show that this assignment ([ s i ]) i ψ is a well-definedgroup isomorphism Z T J ( F ) ∼ −→ Aut ( Id J ). It is well-defined, because if there is also some χ ∈ Aut ( Id J ) with [ s i ] i = [ ψ i ( x i )] i = [ χ i ( x i )] i , then from Lemma 20 we obtain ψ = χ . It isclearly injective, it is surjective by Proposition 21, and it is readily seen to preserve groupmultiplication, so that it is indeed a group isomorphism. ◀ We can now use Corollary 22 to characterize the covariant isotropy groups of certain presheafcategories of interest. ▶ Proposition 23. If M is a monoid, then the covariant isotropy group Z : Set M → Grp of the category of M -sets and M -equivariant maps is constant on Inv ( Z ( M )) , the subgroupof invertible elements of the centre of M . In particular, if G is a group, then the covariantisotropy group Z : Set G → Grp is constant on Z ( G ) . Proof.
The result follows immediately from Corollary 22 and the observation that theautomorphism group of the identity functor on the monoid M , regarded as a one-objectcategory, is isomorphic to Inv ( Z ( M )). ◀▶ Proposition 24.
Let J be a rigid category, i.e. a category whose objects have no non-identity automorphisms (e.g. J could be a preorder or poset). Then the covariant isotropygroup Z : Set J → Grp is trivial. ◀ . Hofstra, J. Parker, and P. J. Scott 23:15 We point out that Corollary 22 illustrates an important difference between covariantisotropy
Set J → Grp and contravariant isotropy (cid:16)
Set J (cid:17) op → Grp . Indeed, the latter isgenerally not constant, but is a representable functor F Set J [ F, Z ] for a suitable presheafof groups Z , that is, an internal group object in Set J . The connection between covariantand contravariant isotropy is then as follows: the group of global sections of Z is isomorphicto the group Aut ( Id J ):Γ( Z ) = Set J (1 , Z ) ∼ = Z ( F ) for F : J →
Set . We have shown how a syntactic description of polymorphic automorphisms can be fruitfullyapplied to characterize the covariant isotropy of several kinds of structures of relevancein logic, algebra, and computer science. Most notably, we have shown that the covariantisotropy group of a strict monoidal category coincides with its Picard group of invertibleobjects. We have also shown that the covariant isotropy group of a presheaf category
Set J behaves quite differently from the contravariant one, in that it is the constant group withvalue Aut ( Id J ).There are several open questions and possible lines for further inquiry: The generalization from algebraic to quasi-equational theories presented in this paper isthe first step on a path upwards through the various fragments of logic. In particular,we hope to generalize some of the techniques to determine the isotropy groups of somegeometric theories of interest. We have shown how to determine the covariant isotropy groups of presheaf categories,but we have left open the question of how to determine the isotropy of sheaf toposes.In particular, it would be of interest to determine the covariant isotropy of the topos ofnominal sets (also known as the Schanuel topos). For a theory T and diagram category J , there is a theory S = S ( T , J ) with S mod ∼ = T mod J (in Section 5.2 we considered the special case where T is the trivial theory, i.e. thetheory of sets). In [8, Chapter 5] the second author has obtained, under mild assumptionson T , a description of the covariant isotropy group of T J mod in terms of Aut ( Id J ) andthe isotropy group of T . We have not yet investigated in detail how isotropy behaves with respect to morphismsof theories ρ : T → S . (We have seen a rather special case in Section 4 with Ob : StrMonCat → Mon , but the general case is more involved.) One possible perspective on the theory of strict monoidal categories is that it is a tensorproduct of the theory of categories with that of monoids. This leads to the questionwhether, under suitable conditions on the theories T and S , we can describe the isotropyof T ⊗ S in terms of that of T and S . One can define, for a 2-category E and object X ∈ E , the group of natural auto-equivalencesof X/ E → E . This leads to a 2-dimensional version of isotropy, taking values in 2-groups.It is then possible to show that the 2-isotropy group of a (non-strict) monoidal category(regarded as an object in the 2-category of monoidal categories and strong monoidalfunctors) is the Picard 2-group. This will be presented in forthcoming work.
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