aa r X i v : . [ m a t h . C T ] O c t TENSOR-RESTRICTION CATEGORIES
CHRIS HEUNEN AND JEAN-SIMON PACAUD LEMAY
Abstract.
Restriction categories were established to handle maps that arepartially defined with respect to composition. Tensor topology realises thatmonoidal categories have an intrinsic notion of space, and deals with objectsand maps that are partially defined with respect to this spatial structure. Weintroduce a construction that turns a firm monoidal category into a restrictioncategory and axiomatise the monoidal restriction categories that arise this way,called tensor-restriction categories. Introduction
The tensor product in a monoidal category encodes a notion of space. Forexample, in the category of sheaves over a topological space, as well as in thecategory of continuous fields of Banach spaces over a topological space, the opensets correspond precisely to so-called subunits : subobjects of the tensor unit thatare idempotent (in a sense made precise below). Under the mild condition thatthe monoidal category is firm , meaning that its subunits are closed under tensorproduct, we may think about the semilattice they form as a base space underlyingthe monoidal category. Moreover, the tensor product provides methods to dealwith partiality, restriction, and support, with respect to this base space.
Tensortopology [7] deals with objects and maps that are partially defined with respect tothe tensor structure.There is another dimension to monoidal categories than the tensor product,namely composition of morphisms.
Restriction categories [6] were established todeal with maps that are partially defined with respect to composition. For re-striction categories, the elegantly simple main technique is to record the domain ofdefinition of a morphism f : A → B in an endomorphism f : A → A . Thus eachobject A has a space O ( A ) = { e : A → A | e = e } underlying it, and there aremethods to deal with partiality, restriction, and support, with respect to this basespace and in terms of composition.In this article, we bring the two notions of partiality on an equal footing. We in-troduce a construction that turns a firm monoidal category C into a restriction cat-egory S [ C ]. We analyse which restriction categories arise this way, and axiomatisethem as tensor-restriction categories . In the other direction, we will show that theknown construction of taking the restriction-total maps turns a tensor-restrictioncategory X back into a firm monoidal category T [ X ]. Indeed, S [ T [ X ]] ≃ X and T [ S [ C ]] ≃ C , and this gives an equivalence of categories.This result has advantages to both tensor topology and restriction category the-ory. On the one hand, restriction categories are relatively more established, and Date : October 15, 2020.We thank Nick Bezhanishvili, Cole Comfort, Richard Garner, Rory Lucyshyn-Wright, ChadNester, and Sean Tull for useful discussions. tensor topology is relatively more recent, so one may hope that techniques fromrestriction category theory can usefully be applied to tensor topology. On the otherhand, tensor topology gives a new source of examples of restriction categories, andthese are in a sense more naturally dealt with as tensor-restriction categories.Another advantage of tensor topology over restriction categories may be thatmonoidal categories afford a graphical calculus [10]. We leave to future work theidea of adapting the graphical calculus to subunits, but make some initial remarksin Section 6, which may lead to appealing visual methods to deal with partialityand restriction in general. More generally, the two dimensions of composition andtensor product are brought together in bicategories. We leave open the question ofwhether our results have a common generalisation in ‘restriction bicategories’, butpoint out that the orthogonal factorisation system of tensor-restriction categories(see Proposition 5.14) strongly resembles the interchange law of bicategories; seealso [5].We start by recalling the basics of tensor topology in Section 2. The requirednotions from restriction category theory are recalled in Section 3. This sectionsimultaneously introduces the S -construction and analyses it to illustrate thesenotions. Section 4 starts to consider categories that have both a tensor productand a restriction structure, and Section 5 finishes the job by axiomatising tensor-restriction categories. We conclude, in Section 6, by discussing possible alternativecharacterisations of tensor-restriction categories. As a matter of terminology, todistinguish between similar notions from tensor topology and restriction categories,we will consistently prefix them, so that a morphism can for example be restriction-total or tensor-total. 2. Subunits
In this section, we recall the notion of a subunit and its properties [7], and drawthem in the graphical calculus of monoidal categories [10]. For a monoidal category C , denote the monoidal product as ⊗ , the monoidal unit as I , and the coherencenatural isomorphisms as λ A : I ⊗ A → A , ρ A : A ⊗ I → A , and α A,B,C : A ⊗ ( B ⊗ C ) → ( A ⊗ B ) ⊗ C . If C is braided, write σ A,B : A ⊗ B → B ⊗ A for the braiding. Wewill often omit subscripts and simply write λ , ρ , α , and σ when there can be noconfusion. We will also abbreviate identity morphisms 1 A : A → A simply as A .First recall that in any category C , a subobject of an object A is an equivalenceclass of monomorphisms s : S A , where s is equivalent to s ′ : S ′ A if there isan isomorphism m : S → S ′ with s ′ ◦ m = s . For subunits, we will use a small letter s to denote a representing monomorphism and a capital letter S for its domain. Definition 2.1. [7, Definition 2.1] A subunit in a monoidal category C is a sub-object s : S I of the monoidal unit such that s ⊗ S : S ⊗ S → S ⊗ I is invertible.We write ISub( C ) for the class of subunits in C .Note that for a subunit s : S I , the composite ρ ◦ ( s ⊗ S ) : S ⊗ S → S is alsoan isomorphism, so S ∼ = S ⊗ S . Furthermore since s is monic and s ⊗ S is invertible,then S ⊗ s is also invertible. The following is a very useful property of subunits. Lemma 2.2. If s : S I is a subunit then λ ◦ ( s ⊗ S ) = ρ ◦ ( S ⊗ s ) .Proof. Note that we have the following equality: s ◦ λ S ◦ ( s ⊗ S ) = λ I ◦ ( I ⊗ s ) ◦ ( s ⊗ S ) (naturality of λ ) ENSOR-RESTRICTION CATEGORIES 3 = λ I ◦ ( s ⊗ s )= ρ I ◦ ( s ⊗ s ) (coherence)= ρ I ◦ ( s ⊗ I ) ◦ ( S ⊗ s )= s ◦ ρ S ◦ ( S ⊗ s ) (naturality of ρ )Since s is monic, it follows that λ ◦ ( s ⊗ S ) = ρ ◦ ( S ⊗ s ). (cid:3) Subunits behave smoothly in a firm monoidal category.
Definition 2.3. [7, Definition 2.5] A braided monoidal category C is firm when s ⊗ T : S ⊗ T → I ⊗ T is a monomorphism for any subunits s : S I and t : T I .The subunits in a firm monoidal categories form a semilattice; as in [7], we willignore the fact that the subunits may form a proper class rather than a set. Proposition 2.4. [7, Proposition 2.9] If C is a firm monoidal category, then ISub( C ) is a semilattice where the meet ∧ of subunits s : S I and t : T I is ( s : S I ) ∧ ( t : T I ) = ( s ∧ t : S ⊗ T I ) where s ∧ t = λ ◦ ( s ⊗ t ) , and withtop element the (equivalence class of the identity on) the monoidal unit I → I .The induced order ≤ on ISub( C ) is precisely the order of subunits: s ≤ t if andonly if s = t ◦ m for some m : S → T . To ease definitions and proofs, we will freely use the graphical calculus formonoidal categories. (We invite readers unfamiliar with string diagrams to see [10]for an introduction). String diagrams in this paper are read from bottom to top.Subunits are determined by their domain: if s, s ′ : S I are subunits, then theyrepresent the same subobject. Therefore we may draw a subunit S I as: S In particular, the equality of Lemma 2.2 is drawn as:(1)
S S = SS We draw the inverse of λ ◦ ( s ⊗ S ) = ρ ◦ ( S ⊗ s ) as: S SS and so in particular:
S S = SS = SS We conclude this section with some running examples of firm monoidal categoriesand their subunits; for more details and other examples see [7, Section 3].
CHRIS HEUNEN AND JEAN-SIMON PACAUD LEMAY
Example 2.5.
Let ( L, ∧ ,
1) be a semilattice. Then L can be regarded as a categorywhose objects are the elements x ∈ L and where there is a unique x → y if x ≤ y .Furthermore, L is a firm monoidal category with x ⊗ y = x ∧ y and unit 1. Everyelement of L is a subunit, and therefore ISub( L ) = L . Example 2.6.
Let X be a locale, with frame of opens O ( X ) and top element ⊤ ,and let Sh( X ) be the category of sheaves over X . For every open U ∈ O ( X ), definethe sheaf χ U : O ( X ) op → Set as follows: χ U ( V ) = ( {∗} if U ≤ V ∅ if U V Now Sh( X ) is a firm monoidal category where the monoidal structure is given byfinite products, so the monoidal product is given by the pointwise cartesian product ⊗ = × and the monoidal unit is the terminal sheaf I = χ ⊤ . The subunits of Sh( X )are given by the opens of X , that is, they are precisely the subterminal sheaves χ U ,so ISub(Sh( X )) ∼ = O ( X ). Example 2.7.
Let R be a commutative unital ring and Mod R be its categoryof R -modules and R -linear morphisms between them. Mod R is a firm monoidalcategory where the monoidal structure is given by the standard tensor product of R -modules and so the monoidal unit is the ring itself I = R . The subunits of Mod R are the idempotent ideals of R , that is, ideals J ⊆ R such that J = J ,where J = { P ni =1 r i r ′ i | r i , r ′ i ∈ J } . Example 2.8.
Recall that a
Boolean ring is a commutative unital ring R such that x = x for all x ∈ R . Every ideal of a boolean ring is idempotent. Therefore, fora Boolean ring R , the subunits of Mod R correspond precisely to the ideals of R .Equivalently, R is a Boolean algebra, and ISub( Mod R ) consists of its order ideals. Example 2.9.
The cateogry
Set of sets and functions is a firm monoidal categoryunder the Cartesian product with monoidal unit a chosen singleton set I = {∗} .There are only two subunits in Set : the empty set ∅ and the singleton {∗} . Example 2.10.
Let R be a commutative semiring, and consider the category SMod R of R -semimodules [8, Chapters 14 and 16]. There are always two subunitsin SMod R : R itself, and the zero object = { } . Similar to the previous example,if these are the only two subunits, then SMod R is a firm monoidal category. Thissituation includes the case where R is a semifield, such as the category Vect k ofvector spaces over a field k , and the category Rel of sets and relations where thesemiring is that of Boolean truth values [10, Definition 0.5].3.
Restriction Categories from Subunits
This section introduce the S [ − ]-construction, that turns a firm monoidal cate-gory C into a restriction category S [ C ] where the restriction structure is determinedby the subunits of C . We will recall various important notions of restriction cate-gories, such as restriction idempotents, restriction-total morphisms, and restrictionisomorphisms, and study them in S [ C ]. (For a more in-depth introduction andmore details on restriction categories, see [6].)We begin with the S [ − ] construction itself. Definition 3.1.
Let C be a firm monoidal category. Define a category S [ C ] asfollows: ENSOR-RESTRICTION CATEGORIES 5 • Objects are the same as in C ; • Morphisms [ s, f ] : A → B in S [ C ] are equivalence classes of pairs ( s, f ) of asubunit s : S I and a morphism f : A ⊗ S → B in C , where pairs ( s, f )and ( s ′ , f ′ ) are identified when s ′ = s ◦ m for an isomorphism m : S ′ → S and f ′ = f ◦ ( A ⊗ m ). We will draw morphisms graphically:[ s, f ] = h s, fA SB i • Identity morphisms are [1 I , ρ ] = [1 I , A ] : A → A • Composition of h s, fA SB i and h t, gB TC i is defined as follows: h t, gB TC i ◦ h s, fA SB i = " s ∧ t, f gA S TC It is straightforward to check that S [ C ] is indeed a well-defined category when C is a firm monoidal category. Our next objective is to explain how S [ C ] is in facta restriction category in such a way that the restriction of h s, fA SB i is determinedby the subunit s . First recall the definition of a restriction category. Definition 3.2. [6, Section 2.1.1] A restriction category is a category X equippedwith a choice of endomorphism f : A → A for each morphism f : A → B satisfying: f ◦ f = f (R1) f ◦ g = g ◦ f if dom( f ) = dom( g )(R2) g ◦ f = g ◦ f if dom( f ) = dom( g )(R3) g ◦ f = f ◦ g ◦ f if cod( f ) = dom( g )(R4)We call f the restriction of f . Example 3.3.
The canonical example of a restriction category is
Par , the categoryof sets and partial functions. The restriction f : X → X of a partial function f : X → Y is f ( x ) = ( x if f ( x ) is defined ↑ otherwisewhere ↑ means “undefined”.For more examples of restriction categories, see [6, Section 2.1.3]. We move tothe restriction structure of S [ C ]. Proposition 3.4. If C is a firm monoidal category, then S [ C ] is a restrictioncategory with: h s, fA SB i = h s, A S i CHRIS HEUNEN AND JEAN-SIMON PACAUD LEMAY
Proof.
We start with verifying (R1): h s, fA SB i ◦ h s, fA SB i = h s, fA SB i ◦ h s, A S i = h s ∧ s, fA SB S i = (cid:20) s, fAB S (cid:21) = h s, fA SB i Next, (R2): h s, fA SB i ◦ h t, gA TC i = h s, A S i ◦ h t, A T i = h s ∧ t, A S T i = h t, gA TC i ◦ h s, fA SB i For (R3): h t, gA TC i ◦ h s, fA SB i = h t, gA TC i ◦ h s, A S i = h t ∧ s, gA TC S i = h s ∧ t, A S T i = h t, gA TC i ◦ h s, fA SB i Finally, (R4): h s, fA SB i ◦ h t, gB TC i ◦ h s, fA SB i = h s, fA SB i ◦ " s ∧ t, f gA S T = h s, fA S i ◦ h s ∧ t, A S T i = h s ∧ s ∧ t, fA S S T i = h s ∧ t, fA S T i = h t, gB TC i ◦ h s, fA SB i So we conclude that S [ C ] is a restriction category. (cid:3) We now apply the S [ − ] construction to our examples of firm monoidal categoriesfrom the previous section and discuss their restriction structure. Most examplesare extensions of the following general principle: S [ C × D ] ≃ S [ C ] × S [ D ] for firmmonoidal categories C and D . Example 3.5.
Regard a semilattice ( L, ∧ ,
1) as a firm monoidal category. Then S [ L ] can be described as follows: • Objects are elements x ∈ L ; ENSOR-RESTRICTION CATEGORIES 7 • Morphisms x → y are elements s ∈ L such that x ∧ s ≤ y ; • Identity morphisms are 1 : x → x ; • Composition of s : x → y and t : y → z is s ∧ t : x → z ; • Restriction of s : x → y is s : x → x .This is a typed version of the known construction of depressing downsets : thehomset S [ L ]( b, a ) is denoted as a ↓ b in [1, Section 2.7]. If L is an implicativesemilattice, also called a Heyting semilattice, that is, if L is closed as a monoidalcategory, then S [ L ]( a, b ) = ↓ ( a ⊸ b ). Note that S [ L ] is enriched over posets. Infact, every homset again has finite meets, but may not have a top element. If L is a so-called pro-Heyting semilattice, then it is implicative if and only if everyhomset has a top element [1, Observation 3.2]. In this sense, the S [ L ] construction“approximates” the construction of the free Heyting algebra on a semilattice L . Example 3.6.
Let X be a locale. If F, G ∈ Sh( X ) are sheaves, then a morphism F → G in S [Sh( X )] consists of an open U ∈ O ( X ) and a natural transformation α : F × χ U ⇒ G . Now let V ∈ O ( X ) be an open such that V U . Then( F × χ U )( V ) = ∅ , and so α V can only be the empty function ∅ → G ( V ). Inother words, α is completely determined by α | U : F | U ⇒ G | U , where we write F | U : O ( U ) op → Set for the restriction of the sheaf F to U given by F | U ( V ) = F ( V )for U ≤ V . Therefore, S [Sh( X )] can be described as: • Objects are sheaves F : O ( X ) op → Set ; • Morphisms [
U, α ] : F → G are pairs of an open U ∈ O ( X ) and a naturaltransformation α : F | U → G | U ; • Identity morphisms are the pairs [ X, F → F ] : F → F (since F = F | X ); • Composition of [
U, α ] : F → G and [ V, β ] : G → H is [ V, β ] ◦ [ U, α ] =[ U ∧ V, β | U ◦ α | V ], where we write α | V for the restriction of α : F | U ⇒ G | U to F | U ∧ V ⇒ G | U ∧ V ; • Restriction of [
U, α ] : F → G is [ U, α ] = [
U, F | U ].In short: for categories of sheaves Sh( X ), the S [ − ]-construction makes the mor-phisms partial in recording an open domain of definition. Example 3.7.
Let R be a unital commutative ring. Then S [ Mod R ] can be de-scribed as follows: • Objects are R -modules A ; • Morphisms [
I, f ] : A → B are pairs of an idempotent ideal I and an R -linearmap f : A ⊗ R I → B ; • Identity morphisms are the pairs [
R, A ⊗ R R ≃ A ]; • Composition of [
I, f ] and [
J, g ] is [
IJ, g ◦ ( f ⊗ R J ) : A ⊗ R I ⊗ R J → C ]; • Restriction of [
I, f ] : A → B is [ I, f ] = [
I, ρ : A ⊗ R I → A ] with ρ ( a ⊗ i ) = ai .Thus, for categories of modules Mod R , the S [ − ]-construction makes the morphismspartial in recording an ideal in the domain where a morphism acts on. This isclosely related to the idea of localisation in algebra. For example, if the ring R issemisimple, then any ideal is generated by an idempotent element, so ISub( Mod R )is the Boolean algebra of idempotent elements. Morphisms in S [ Mod R ] are thuspairs of an idempotent e = e ∈ R and a morphism f : eA → B . Example 3.8.
The previous two examples are related. For example, if R is aBoolean ring, then Mod R is equivalent to the category of sheaves of Z -vector CHRIS HEUNEN AND JEAN-SIMON PACAUD LEMAY spaces over the Stone space of R by Pierce’s representation theorem [11, ChapterV.2]. Ideals of the Boolean ring R correspond to open sets of its Stone space. Example 3.9.
Recall that in
Set , the empty set ∅ is an initial object and so forevery set X there is a unique function ∅ : ∅ → X , called the empty function. Onthe other hand, the only function whose codomain is ∅ is the identity function 1 ∅ .Also X × ∅ = ∅ for any set X . So S [ Set ] is described as follows: • Objects are sets X ; • Morphisms include the usual functions [1 , f ] : X → Y as well as an extramap [ ∅ , ∅ ] : X → Y , so S [ Set ]( X, Y ) ≃ Set ( X, Y ) + 1; • Identity morphisms are the pairs [1 , X × {∗} ≃ X ]; • Composition of [ s, f ] and [ t, g ] can be described in the following three cases:if s = t = 1, then [1 , g ] ◦ [1 , f ] = [1 , g ◦ f ]; if s = ∅ , then f = ∅ and[ t, g ] ◦ [ ∅ , ∅ ] = [ ∅ , ∅ Y ]; if t = ∅ , then g = ∅ and [ ∅ , ∅ ] ◦ [ s, f ] = [0 , • Restriction of [1 , f ] is the identity [1 , f ] = [1 , X × {∗} ≃ X ], while therestriction of [ ∅ , ∅ ] is itself [ ∅ , ∅ ] = [ ∅ , ∅ ].Recall that a category C is said to have zero morphisms if there is a family ofmorphisms z : A → B (for every pair of objects) which are absorbing in the sensethat z ◦ f = z and f ◦ z = z . When they exist, zero morphisms are unique.Now S [ Set ] has zero morphisms [ ∅ , ∅ ]. In fact, S [ Set ] is the free category withzero morphisms over
Set with respect to the functor U : Set → S [ Set ] defined onobjects as U ( A ) = A and on maps as U ( f ) = [1 , f ]. So given a category C with zeromorphisms z and a functor F : Set → C , there exists a unique functor G : S [ Set ] → C which preserves zero morphisms, that is, G ( z ) = [ ∅ , ∅ ], and satisfying G ◦ U = F . Explicitly, G is defined on objects as G ( X ) = F ( X ) and on morphisms as G ([1 , f ]) = F ( f ) and G ([ ∅ , ∅ ]) = z . While ∅ is an initial object in Set , ∅ is nolonger initial in S [ Set ]; it is a terminal object instead.
Example 3.10.
Let R be a commutative semiring, such that only subunits in SMod R are R and . Then similar to the previous example, S [ SMod R ] canbe described as the free category with zero morphisms over SMod R where thezero morphisms are [ , SMod R already had zero morphisms0 : A → B , the morphisms [1 ,
0] : A → B in S [ SMod R ] are no longer zeromorphisms. Similarly, while was a zero object in SMod R , this is no longer the casein S [ SMod R ]; instead becomes a restriction-terminal object [2, Definition 2.16]. Remark 3.11.
The S [ − ]-construction resembles the construction of the free re-striction category on a fibration of semilattices [3]. In our case, the semilattice overeach object is the same, namely ISub( C ). But S [ C ] does not fit neatly in thatframework, for it is not what is called unitary: it is not the case that [ s, f ] = [ t, g ]as soon as s = t and f ⊗ r = g ⊗ r for some r .We return to studying the restriction structure of the S [ − ]-construction, bytaking a closer look at various classes of maps that are important in restrictioncategory theory. Definition 3.12. [6, Section 2.1.1] A restriction idempotent is an endomorphism e : A → A in a restriction category with e = e . Write O ( A ) for the set of allrestriction idempotents of type A → A . Example 3.13. In Par , the restriction idempotents of a set X correspond preciselyto its subsets U ⊆ X . Indeed, for every subset U ⊆ X , define the partial function ENSOR-RESTRICTION CATEGORIES 9 χ U : X → X as follows: χ U ( x ) = ( x if x ∈ U undefined if x / ∈ U Then clearly χ U = χ U . Conversly, given a restriction idempotent e : X → X ,consider the subset U e = { x | e ( x ) = x } ⊆ X . Then χ U e = e because e = e , and so O ( X ) is isomorphic to the powerset P ( X ) of X .Note that f is a restriction idempotent for any morphism f [6, Lemma 2.1].Therefore, e is a restriction idempotent if and only if e = f for some morphism f . Furthermore, O ( A ) is a semilattice where e ∧ e ′ = e ◦ e ′ and the top elementis the identity morphism 1 A : A → A [4, Section 2.1]. So in particular, e ≤ e ′ if e ◦ e ′ = e . In the S [ − ]-construction, restriction idempotents correspond preciselyto the subunits of the base category. Proposition 3.14.
Let C be a firm monoidal category. The restriction idempotentsin S [ C ] are precisely the morphisms of the form (cid:2) s, A S (cid:3) . This gives a semilatticeisomorphism O ( A ) ∼ = ISub( C ) .Proof. As in any restriction category, (cid:2) s, fA SA (cid:3) ∈ O ( A ) if and only if h s, fA SA i = h t, gA TB i for some (cid:2) t, gA TB (cid:3) . Thus (cid:2) s, A S (cid:3) s is a bijection φ A : O ( A ) → ISub( C ). It isclear that φ A preserves the top element. It remains to show that it preserves meets: φ A (cid:16)(cid:2) s, A S (cid:3) ∧ (cid:2) t, A T (cid:3)(cid:17) = φ A (cid:16)(cid:2) s, A S (cid:3) ◦ (cid:2) t, A T (cid:3)(cid:17) = φ A (cid:16)(cid:2) s ∧ t, A S T (cid:3)(cid:17) = (cid:0) s ∧ t : S ⊗ T I (cid:1) = (cid:0) s : S I (cid:1) ∧ (cid:0) t : T I (cid:1) = φ A (cid:16)(cid:2) s, A S (cid:3)(cid:17) ∧ φ A (cid:16)(cid:2) t, A T (cid:3)(cid:17)
Hence O ( A ) is isomorphic to ISub( C ) as semilattices. (cid:3) An important subclass of morphisms of a restriction category is its class ofrestriction-total morphisms. Intuitively, these are the morphisms which are “to-tally defined”.
Definition 3.15. [6, Section 2.1.2] In a restriction category X , a restriction-totalmorphism is a morphism f : A → B such that f = 1 A . The subcategory of allrestriction-total morphisms of X is denoted by T [ X ]. Example 3.16. In Par , the restriction-total morphism are precisely the totalfunctions in the classical sense. So T [ Par ] =
Set .For S [ C ], its subcategory of total morphisms is precisely the base category C . Proposition 3.17.
Let C be a firm monoidal category. A morphism in S [ C ] is restriction-total if and only if it is of the form (cid:2) , fAB (cid:3) for some morphism f : A → B in C . This induces an isomorphism of categories C ∼ = T [ S [ C ]] .Proof. If (cid:2) s, fA SB (cid:3) is a restriction-total morphism, then: (cid:2) s, A S (cid:3) = (cid:2) s, fA SB (cid:3) = [1 , A ]Therefore s = 1, and the restriction-total morphisms are of the form claimed. It isstraightforward that this induces an isomorphism of categories C ∼ = T [ S [ C ]]. (cid:3) The last class of morphisms that we will study are restriction isomorphisms.
Definition 3.18.
In a restriction category X , a restriction isomorphism is a mor-phism f : A → B which has a map f ◦ : B → A , called its restriction inverse , suchthat f ◦ ◦ f = f and f ◦ f ◦ = f ◦ .If a restriction inverse exists, it is unique. Example 3.19. In Par , define the domain and image of f : X → Y as usual: dom ( f ) = { x | f ( x ) = x } ⊆ X im ( f ) = { y | ∃ x ∈ X : f ( x ) = y } ⊆ Y Then f is a restriction isomorphism if and only if the canonical total function dom ( f ) → im ( f ) given by x f ( x ) is a bijection.Restriction isomorphisms in S [ C ] correspond to isomorphisms of a certain typein C . Proposition 3.20.
Let C be a firm monoidal category. A morphism (cid:2) s, fA SB (cid:3) in S [ C ] is a restriction isomorphism if and only if (2) f BA SS is an isomorphism in C .Proof. By definition, (cid:2) t, gB TA (cid:3) is a restriction inverse of (cid:2) s, fA SB (cid:3) if and only (cid:2) s, fA SB (cid:3) = (cid:2) t, gB TA (cid:3) ◦ (cid:2) s, fA SB (cid:3)(cid:2) t, gB TA (cid:3) = (cid:2) s, fA SB (cid:3) ◦ (cid:2) t, gB TA (cid:3) ENSOR-RESTRICTION CATEGORIES 11 In S [ C ], this means t = s and:( ∗ ) f g SAB A = AA S g f
SBA B = BB S
We will show that this is the case if and only if (2) is an isomorphism, with inverse:(3) g AB SS
If (3) inverts (2), then: f g
SAB A = f g SAB A = AA S
Similarly with the composition in the other order. Therefore ( ∗ ) holds.Conversely, if ( ∗ ) holds then: f g SAB A S = f g SAB A S = AA S S = AA SS
The composition in the other order is similar, showing that (2) is invertible. (cid:3)
Recall that an inverse category is a restriction category where every morphismhas a restriction inverse [6, 2.3.2]. The previous proposition immediately gives usa characterisation of when S [ C ] (and hence, jumping ahead slightly, any tensor-restriction category) is an inverse category. Later on, we will show that this givesan instance of the S -construction after Corollary 5.19 below. Corollary 3.21.
Let C be a firm monoidal category. Then S [ C ] is an inversecategory if and only if C is a groupoid.Proof. By definition, S [ C ] is an inverse category if and only if every morphism is arestriction isomorphism. By Proposition 3.20 this happens exactly when any mapof the form (2) is an isomorphism in C for every subunit s . Taking s = 1 implies that C is a groupoid. Conversely, if C is a groupoid, then ISub( C ) = { } , andhence (2) is invertible. (cid:3) We finish this section by exhibiting an important aspect of the S [ − ]-construction:it produces an orthogonal factorisation system, that intuitively separates the restric-tion aspect from the base category aspect. Proposition 3.22. If C is a firm monoidal category, then S [ C ] has an orthogonalfactorisation system given by: E = n restriction isomorphisms of the form (cid:2) s, fA SB S (cid:3)o M = n restriction-total morphisms (cid:2) , fAB (cid:3)o Proof.
Clearly both M and E are closed under composition and contain all iso-morphisms. Furthermore, any morphism (cid:2) s, fA SB (cid:3) factors as the restriction iso-morphism [ s, A S ] : A → A ⊗ S in E followed by the restriction-total morphism (cid:2) , fA SB (cid:3) : A ⊗ S → B in M . A BA ⊗ S h s, fA SB i E ∋ [ s, A S ] (cid:2) , fA SB (cid:3) ∈ M We will show that any commuting square as below has a unique diagonal fill-in:
A B ⊗ SC D (cid:2) s, fA SB (cid:3) ∈ E [ t, h ][ r, g ] [1 I , k ] ∈ M [ t, m ]That the outer square commutes means that r = s ∧ t and: hf DA S T = gk DA S T
ENSOR-RESTRICTION CATEGORIES 13
The fact that h s, fA SB S i ∈ E means that it has a restriction inverse h s, f ◦ B SAS i : ff ◦ SB SB = BB SS f ◦ f SA A = AA S
Define m : B ⊗ S ⊗ T → C as follows: m B TCS = gf ◦ C TSB
Then both triangles commute by Lemma 2.2: mf CA S T = g CA S T mk DB S T = h DB S T
Clearly h t, mB TCS i is the unique map achieving this. (cid:3) Monoidal Restriction Categories
In this section, we will show that S [ − ]-construction results in a firm monoidalcategory whose subunits are precisely those of the base category. Furthermore,we will also explain how this monoidal structure is compatible with the restrictionstructure, which we call a monoidal restriction category. Proposition 4.1. If C is a firm monoidal category, then S [ C ] is a firm monoidalcategory with monoidal unit I and the monoidal product ⊗ defined on objects as A ⊗ B and on morphisms as follows: h s, fA SB i ⊗ h t, gC TD i = (cid:20) s ∧ t, f gA C S TB D (cid:21) Subunits in S [ C ] are exactly the maps (cid:2) , S (cid:3) for subunits s : S I in C . Hence ISub( C ) ∼ = ISub( S [ C ]) as semilattices. Proof.
The coherence isomorphisms α , ρ , λ , and σ of C induce coherence isomor-phisms [1 I , α ], [1 I , ρ ], [1 I , λ ], and [1 I , σ ] for S [ C ], in such a way that the triangle,pentagon, and hexagon equations are satisfied. In particular, we note that theinterchange law comes down to: s ∧ s ′ ∧ t ∧ t ′ , f f ′ g g ′ T T ′ S S ′ = s ∧ t ∧ s ′ ∧ t ′ , f f ′ g g ′ T T ′ S ′ S It is clear that any subunit s in C induces a subunit (cid:2) , S (cid:3) in S [ C ]. Conversely,because monomorphisms are total [6, Lemma 2.2.i], a subunit in S [ C ] is of theform (cid:2) , fA (cid:3) . Since by Proposition 3.17, we have the isomorphism C ∼ = T [ S [ C ]],it follows that f ◦ ρ − must also be monic in C , and that f ⊗ A ⊗ I is invertiblein C . Therefore f ◦ ρ − represents a subunit A → I in C . So we conclude thatISub( C ) ∼ = ISub( S [ C ]). Finally, if (cid:2) , S (cid:3) and (cid:2) , T (cid:3) are subunits in S [ C ], thenby construction so is their tensor product. Hence S [ C ] is firm. (cid:3) The monoidal structure of S [ C ] is compatible with the restriction structure. Definition 4.2. A monoidal restriction category is a monoidal category X that isalso a restriction category where: f ⊗ g = f ⊗ g A firm restriction category is a monoidal restriction category whose underlyingmonoidal category is firm.Because isomorphisms are always total [6, Lemma 2.2.i], the coherence isomor-phisms in a monoidal restriction category are always total. Example 4.3.
Every cartesian restriction category [2, Definition 2.16] is a monoidalrestriction category, just like every cartesian category is a monoidal category. Soin particular,
Par is a monoidal restriction category with monoidal structure givenby the cartesian product of sets, and so f × g = f × g . Furthermore, Par is afirm monoidal restriction category where the subunits are (up to isomorphism) thesingleton {∗} and the empty set ∅ . Corollary 4.4. If C is a firm monoidal category, then S [ C ] is a firm restrictioncategory. Furthermore, the isomorphism C ≃ T [ S [ C ]] of Proposition 3.17 is anisomorphism of firm monoidal categories.Proof. It is clear from the monoidal product defined in Propostion 4.1 that: h s, fA SB i ⊗ h t, gC TD i = h s, fA SB i ⊗ h t, gC TD i So S [ C ] is a firm restriction category. It is clear that the isomorphism C → T [ S [ C ]]of Propostion 4.1 is (strong) monoidal and preserves subunits. (cid:3) We conclude this section by observing that having dual objects or being closed inthe S [ − ]-construction is closely related to the same properties of the base category,and vice-versa. We denote dual objects in a monoidal category C as A ⊣ A ∗ (see [10,Chapter 3] for the full definition), and if C is closed, we write ⊸ for internal homs. ENSOR-RESTRICTION CATEGORIES 15
Lemma 4.5. A ⊣ A ∗ is a duality in C if and only if A ⊣ A ∗ is a duality in S [ C ] .Proof. The functor C → S [ C ] given by A A on objects and f [1 , f ] onmorphisms is (strong) monoidal, and monoidal functors preserve dual objects [10,Theorem 3.14]. So if A ⊣ A ∗ in C , then also A ⊣ A ∗ in S [ C ]. Conversely, supposethat A ⊣ A ∗ in S [ C ]. Then there are [ s, η ] : I → A ∗ ⊗ A and [ t, ε ] : A ⊗ A ∗ → I in S [ C ] satisfying:[1 , A ] = h t, εA ∗ A TA A i ◦ h s, ηA ∗ ASAA i = " s ∧ t, ε ηA A ∗ S AT
But then s = t = 1, and η and ε witness A ⊣ A ∗ in C . (cid:3) Lemma 4.6.
A firm monoidal category C is closed if and only if S [ C ] is closed.Proof. Suppose C is closed. If ε : ( A ⊸ B ) ⊗ A → B is the counit of the adjunction( − ) ⊗ A ⊣ A ⊸ ( − ) in C , then S [ C ]( A, B ⊸ C ) → S [ C ]( A ⊗ B, C ) h s, fA SB i " s, εfB ⊸ C CA SB is a well-defined natural bijection. Hence S [ C ] is closed. Conversely, suppose that S [ C ] is closed. Then there are morphisms:[ s, η B ] : B → A ⊸ ( B ⊗ A ) [ t, ε B ] : ( A ⊸ B ) ⊗ A → B satisfying: [ t, ε A ⊗ B ] ◦ [ s, η B ⊗ A ] = [1 I , A ⊗ B ]. But then s ∧ t = 1 so s = t = 1, andso η and ε are the unit and counit of an adjunction ( − ) ⊗ A ⊣ A ⊸ ( − ) in C . (cid:3) Tensor-restriction categories
The goal of this section is to axiomatise the restriction categories of the form S [ C ] for some firm monoidal category C . We call such restriction categories tensor-restriction categories . Explicitly, we will show that X ≃ S [ T [ X ]] for a tensor-restriction category X . In fact, the category of firm monoidal categories is equiva-lent to the category of tensor-restriction categories.A key concept for this section is the notion of restriction in the sense of [7,Section 4], which can be defined in arbitrary monoidal categories. Definition 5.1. [7, Definition 4.1] In a monoidal category C , a morphism f : A → B tensor-restricts to a subunit s : S I if it factors via B ⊗ s in the sense that thefollowing diagram commutes: A BB ⊗ S B ⊗ IfB ⊗ s ρ If a morphism f : A → B restricts to s , then the map A → B ⊗ S is not necessarilyunique. A special case is when identity morphisms restrict to subunits. Proposition 5.2. [7, Proposition 4.2]
An identity morphism A : A → A in amonoidal category restricts to a subunit s : S I if and only if ρ ◦ ( A ⊗ s ) is anisomorphism A ⊗ S ≃ A . It follows from that, if the identity morphism on A restricts to a subunit s , thenany morphism f : A → B also restricts to s . The converse says that f is totallydefined in the tensor topology sense. Definition 5.3.
A morphism f : A → B in a monoidal category is tensor-total when the identity morphism on A tensor-restricts to a subunit s as soon as f tensor-restricts to s .Equivalently, f : A → B is tensor-total when f and A have the same support (forany support datum) [7, Section 6]. Furthermore, it follows from Proposition 5.2that if f : A → B is tensor-total and tensor-restricts to a subunit s : S I , then A ⊗ S ≃ A . The S [ − ]-construction has a similar property. Lemma 5.4.
Let C be a firm monoidal category. If (cid:2) s, fA SB (cid:3) is tensor-total in S [ C ] , then A ≃ A ⊗ S in C .Proof. Any morphism (cid:2) s, fA SB (cid:3) in S [ C ] restricts to the subunit (cid:2) , S (cid:3) : A BB ⊗ S h s, fA SB i" s, fA SB S I , B S i So if (cid:2) s, fA SB (cid:3) is tensor-total, then A restricts to (cid:2) I , S (cid:3) too. Therefore A ⊗ S ∼ = A in S [ C ] by Proposition 5.2. However, since isomorphisms are total, and because C ≃ T [ S [ C ]] by Proposition 4.1, it follows that A ≃ A ⊗ S in C . (cid:3) The next key ingredient in characterising the S [ − ]-construction is points (in thesense of morphisms I → X ) that are restriction isomorphisms whose restrictioninverse is tensor-total. Definition 5.5. A tensor-restriction point in a firm restriction category X is amorphism d : I → X with a restriction inverse d ◦ : X → I such that d ◦ is tensor-total.In the S [ − ]-construction, tensor-restriction points are characterised by the sub-units of the base category. Lemma 5.6.
Let C be a firm category. The tensor-restriction points in S [ C ] areprecisely the morphisms of the form [ s, S ] : I → S for a subunit s : S I in C .Proof. We will prove that tensor-total restriction isomorphisms into I are preciselythe morphisms in S [ C ] of the form (cid:2) s, S S (cid:3) . Because this has [ s, S ] : I → S asrestriction inverse, and restriction inverses are unique, the claim then follows. ENSOR-RESTRICTION CATEGORIES 17
It is clear that (cid:2) s, S S (cid:3) : S → I is a restriction isomorphism from Proposi-tion 3.20. To see that it is tensor-total: if it restricts to (cid:2) I , T (cid:3) , then s ∧ s = t ◦ m for some m : S ⊗ S → T , whence s = t ◦ m ◦ ( S ⊗ s ) − , and so s ≤ t .For the converse, suppose that (cid:2) s, fA SB (cid:3) : A → I is a tensor-total restrictionisomorphism. Then A ≃ A ⊗ S by Lemma 5.4, and A ⊗ S ≃ S by Proposition 3.20,giving A ≃ S by the following isomorphism m : A → S : f ( A ⊗ s ) − A SA
Define a = s ◦ m : A I . Lemma 2.2 now gives: f ⊗ f = ff SAA S
It follows that (cid:2) s, fA SB (cid:3) = [ a, f ◦ ( A ⊗ m )] = [ a, a ∧ a ]. (cid:3) The last preparation we need to define a tensor-restriction category is to recallthe notion of scalar multiplication in a monoidal category [10, Section 2.1.3]. In amonoidal category C , given a scalar s : I → I and a morphism f : A → B , define the(left) scalar multiplication s • f : A → B as the composite s • f = λ B ◦ ( s ⊗ f ) ◦ λ − B .Now we can axiomatise the types of restriction categories we are interested in,and show that they characterise the S [ − ]-construction. Definition 5.7. A tensor-restriction category is a firm restriction category where:(TR1) any restriction idempotent scalar e = e : I → I factors through a subunit s : S I via a tensor-restriction point d : I → S : I ISe = ed s (TR2) any subunit s : S I has a tensor-restriction point d : I → S as restrictionsection in the sense that the following diagram commutes: I IS sdd (TR3) any restriction idempotent f = f : X → X equals f = e • X for a uniquerestriction idempotent scalar e = e : I → I ; (TR4) any tensor-total morphism f : X → Y equals f = ⌈ f ⌉ ◦ f for a uniquerestriction-total morphism ⌈ f ⌉ : X → Y ;(TR5) tensor-restriction points d : I → X have the left-lifting property againstsubunits s : S I : if s ◦ f = g ◦ d then f = m ◦ d and g = s ◦ m ;(4) I XS Id gf s m (TR6) if d : I → X and d ′ : I → X ′ are tensor-restriction points, then so is theirtensor product ( d ⊗ d ′ ) ◦ λ − : I → X ⊗ X ′ ;(TR7) tensor-restriction points are determined by their codomain: if d, d ′ : I → X are tensor-restriction points, then d ′ = d ◦ m for a unique scalar m : I → I .The seven axioms (TR1)–(TR7) essentially demand that a notion from restrictioncategory theory agrees with the corresponding notion from tensor topology. Proposition 5.8. If C is a firm monoidal category, S [ C ] is a tensor-restrictioncategory.Proof. The previous section already showed that S [ C ] is a firm restriction category.It remains to verify (TR1)–(TR7).For (TR1), recall that by Proposition 3.14, restriction idempotent scalars in S [ C ] are of the form [ s, ] for a subunit s : S I in C . But this morphism factorsthrough the subunit [1 I , s ] as follows: I IS [ s, S ][ s, S ] [1 I , S ]Axiom (TR2) holds similarly.For (TR3), recall by Proposition 3.14 that restriction idempotents on A in S [ C ]are of the form (cid:2) s, A S (cid:3) for a subunit s in C . But this is precisely the scalarmultiplication of the identity on A with the restriction idempotent scalar [ s, ].We turn to (TR4). If (cid:2) s, fA SB (cid:3) is tensor-total in S [ C ], then [1 I , A S ] has aninverse (cid:2) I , gA SA (cid:3) by Lemma 5.4 and [7, Proposition 4.2]. Define: l(cid:2) s, fA SB (cid:3)m = [1 I , f ◦ g ] : A → B Then (cid:2) s, fA SB (cid:3) = l(cid:2) s, fA SB (cid:3)m ◦ (cid:2) s, fA SB (cid:3) , and l(cid:2) s, fA SB (cid:3)m is the unique such map:if (cid:2) s, fA SB (cid:3) = [1 I , h ] ◦ (cid:2) s, fA SB (cid:3) , then f = h ⊗ s , and so h = f ◦ g . ENSOR-RESTRICTION CATEGORIES 19
To see (TR5), notice that (4) in S [ C ] becomes: I RS I [ r, R ] (cid:2) t, gB TC (cid:3) [ r ∧ t, f ] [1 I , s ][ t, m ]There is indeed a unique diagonal fill-in, and it is m = f .Finally, (TR6) and (TR7) follow from Corollary 5.6. (cid:3) We will now show that every tensor-restriction category X is in fact of the form S [ C ] for some firm monoidal category C . To explain this, let us investigate somebasic properties of tensor-restriction categories, specifically regarding its restrictionidempotents. Since ISub( C ) ∼ = ISub( S [ C ]) in S [ C ] and O ( A ) ∼ = ISub( C ) for everyobject A , we have O ( A ) ∼ = O ( I ) ∼ = ISub( S [ C ]). This holds generally: if X is atensor restriction category then O ( X ) ∼ = O ( I ) ∼ = ISub( X ). To set the scene, wefirst consider scalars. Lemma 5.9. If X is a monoidal restriction category, then the semilattice O ( I ) isa retract of the scalar monoid X ( I, I ) . O ( I ) X ( I, I )( − )1 Proof.
Recall that the scalars X ( I, I ) of a monoidal category are always a commu-tative monoid under composition s ◦ t , which equals scalar multiplication s • t = λ ◦ ( s ⊗ t ) ◦ λ − ; see [10, Section 2.1]. Now: s ◦ t = s • t (by [10, Lemma 2.6(b)])= λ ◦ ( s ⊗ t ) ◦ λ − (because f ⊗ g = f ⊗ g )= λ ◦ λ ◦ ( s ⊗ t ) ◦ λ − ( λ is total)= λ ◦ λ ◦ ( s ⊗ t ) ◦ λ − (by [6, Lemma 2.1(iii)])= λ ◦ λ − ◦ λ ◦ ( s ⊗ t ) ◦ λ − (R4)= λ ◦ ( s ⊗ t ) ◦ λ − ( λ is iso)= s ◦ t (by [10, Lemma 2.6(b)])It follows that restriction is a monoid homomorphism X ( I, I ) → O ( I ). Finally, if e ∈ O ( I ), then by definition e = e , making O ( I ) a retract of X ( I, I ), where weregard a semilattice as a commutative idempotent monoid. (cid:3)
Lemma 5.10. If X is a monoidal restriction category, then there is a semilatticemorphism ( − ) • X : O ( I ) → O ( X ) for any object X .Proof. It is a monoid homomorphism by [10, Lemma 2.6]. (cid:3)
Next we use (TR1) to construct a semilattice morphism O ( I ) → ISub( X ). Lemma 5.11. If X is a firm restriction category satisfying (TR1), (TR5), and(TR6), then there is a semilattice morphism O ( I ) → ISub( X ) . Proof. If e = e : I → I , axiom (TR1) provides a least subunit s through which e factors. Say e = s ◦ d ; because s is monic, d is unique. If s and s ′ are both leastsubunits through which e factors, then s ≤ s ′ and s ′ ≤ s so s ′ = s . Thus also s isunique, and we may call these morphisms s e and d e . Hence e s e is a well-definedfunction O ( I ) → ISub( X ). This function preserves top elements, s = 1. Indeed,1 I is the least subunit through which 1 I factors: if 1 I = s ′ ◦ d ′ , then 1 I ≤ s ′ . I IIS ′
11 1 d ′ d ′ s ′ To show that the function also preserves meets, first notice that d e ⊗ d f : I ⊗ I → S e ⊗ S f is a restriction isomorphism, with restriction inverse ( e ◦ s e ) ⊗ ( f ◦ s f ). Now if t ◦ m = e ◦ s e for a subunit t and some m : S e → T , then t ◦ m ◦ d e = e ◦ s e ◦ d e = e ◦ e = e and so s e ≤ t by (TR1). Therefore e ◦ s e is tensor-total. Similarly f ◦ s f is tensor-total. By (TR6) the tensor product d e ⊗ d f is a tensor-restriction point. I I ⊗ I S e ⊗ S f S e ∧ f I ≃ d e ⊗ d f s e ∧ s f d e ∧ f s e ∧ f Now apply (TR5) to find s e ∧ s f = s e ∧ f . So we conclude that we have a semilatticemorphism O ( I ) → ISub( X ). (cid:3) Using (TR2) we can construct a semilattice morphism ISub( X ) → O ( I ) in theother direction. Lemma 5.12. If X is a firm restriction category satisfying (TR2), (TR6), and(TR7), then there is a semilattice morphism ISub( X ) → O ( I ) .Proof. If s is a subunit, axiom (TR2) provides a tensor-restriction point d : I → S such that s ◦ d = d = s ◦ d . By (TR7) this morphism d is unique up to scalar, and wemay call it d s . Thus s e s = s ◦ d s is a well-defined function ISub( X ) → O ( I ). Thisfunction preserves top elements, e = 1 I , as 1 I is a tensor-restriction-point as wellas the unique restriction-section of 1 I : if 1 I ◦ d ′ = d ′ , then d ′ = 1 I ◦ d ′ . This functionalso preserves meets: if r and s are subunits, then d = ( d r ⊗ d s ) ◦ λ − : I → R ⊗ S is a tensor-restriction point by (TR6), and is a restriction-section of r ∧ s because( r ∧ s ) ◦ d = ( r ◦ d r ) ◦ ( s ◦ d s ) = d r • d s = d . We conclude there is a semilatticemorphism ISub( X ) → O ( I ). Hence e r ∧ e s = e r ∧ s . (cid:3) In fact, in the presence of (TR5), the semilattice morphisms in the previous twolemmas are each other’s inverse.
Proposition 5.13. If X is a firm restriction category satisfying (TR1), (TR2),(TR5), (TR6), and (TR7), then there is a semilattice isomorphism O ( I ) ≃ ISub( X ) . ENSOR-RESTRICTION CATEGORIES 21
Proof.
We will prove that the functions of Lemmas 5.11 and 5.12 are inverses. Let s be a subunit, and e = e : I → I be a restriction idempotent scalar.On the one hand, (TR1) and (TR5) guarantee that s e s = s . I IS d s S sd s d d s s d s = s e s d s The dashed morphism m exists by (TR5) and is unique because s is monic. Butthere is also a morphism n in the opposite direction. Because m ◦ n is unique by(TR5) it has to be the identity, and similarly for n ◦ m . Hence m is an isomorphism,and s e s = s .On the other hand, d e and d s e have the same codomain. I II S e s e d e d s e = e s e e = e d s e Hence the dashed morphism exists by (TR7), and similarly there is a unique mor-phism in the opposite direction, showing that e s e = e . (cid:3) It follows from (TR3) that O ( I ) ≃ ISub( X ) for any object X in a tensor-restriction category.From now on, for a restriction idempotent scalar e : I → I , write s e for the leastsubunit through which e factors, and d e for the (unique) mediating map. I IS e e = ed e s e By (TR3) and Lemma 5.10, the notion of restriction from the restriction categoryagrees with the notion of restriction from the firm monoidal category. Write e f forthe restriction idempotent scalar satisfying f = e f • dom( f ), and s f and d f for thecorresponding subunit and mediating map as above. Combining (TR1) and (TR3) then factors any morphism as follows: XX ⊗ IX ⊗ S f YY ⊗ IY ⊗ S f ff ⊗ e f f ⊗ S f λ − λX ⊗ d f Y ⊗ s f (TR4) says that the notion of totality from the restriction category agrees withthe notion of totality from the firm monoidal category as in Definition 5.3, and thatwe may replace the bottom map in the previous diagram by a total one. Proposition 5.14.
Any morphism f : X → Y in a tensor-restriction categoryfactors via the restriction-total morphism T ( f ) = ρ ◦ ( Y ⊗ s f ) ◦⌈ f ⊗ S f ⌉ : X ⊗ S f → Y .Proof. Suppose that f ⊗ S f : X ⊗ S f → Y ⊗ S f restricts to a subunit r , say via g : X ⊗ S f → Y ⊗ S f ⊗ R . Then: f = ρ ◦ ( Y ⊗ ( s f ∧ r )) ◦ g ◦ ( X ⊗ d f ) ◦ ρ − = ( Y ⊗ ( s f ∧ r )) ◦ g ◦ ( X ⊗ d f ) ◦ ρ − (by [6, Lem. 2.1.iii,vi])= ( Y ⊗ ( s f ∧ r )) ◦ g ◦ ( X ⊗ d f ◦ ρ − ) (by [6, Lem. 2.1.iii])= ( Y ⊗ ( s f ∧ r )) ◦ g ◦ ( X ⊗ d f ◦ ρ − )= g ◦ ( X ⊗ d f ) ◦ ρ − (by [6, Lem. 2.1.vi])Therefore: g ◦ ( X ⊗ d f ) ◦ ρ − = ( X ⊗ d f ) ◦ ρ − ◦ g ◦ ( X ⊗ d f ) ◦ ρ − (by R4)= ( X ⊗ d f ) ◦ ρ − ◦ f Because g = e g • ( X ⊗ S f ), now ( X ⊗ s f ) ◦ g ◦ ( X ⊗ d f ) = e g • ( X ⊗ s f ) ◦ ( X ⊗ d f ).It follows that: e f = e f • e g = s g ◦ d g ◦ s f ◦ d f = ( s f ∧ s g ) ◦ (1 ⊗ d g ) ◦ ρ − ◦ d f But s f is the least subunit through which e f factors, so s f ≤ s f ∧ s g . Because g = ( X ⊗ S f ⊗ r ) ◦ g by firmness, similarly e f = e r • e g • e f , so s f ≤ s g ∧ r . Hence X ⊗ S f ⊗ R ≃ X ⊗ S f , so that X ⊗ S f restricts to r [7, Proposition 4.2]. So f ⊗ S f ENSOR-RESTRICTION CATEGORIES 23 is tensor-total, and f factors as: X YX ⊗ I Y ⊗ IX ⊗ S f Y ⊗ S f X ⊗ S f fρ − ρf ⊗ e f X ⊗ d f f ⊗ S f Y ⊗ s f f ⊗ S f ⌈ f ⊗ S f ⌉ As the right vertical morphisms are restriction-total, so is their composition. (cid:3)
Lemma 5.15. If X is a tensor-restriction category, then T [ X ] is a firm monoidalcategory.Proof. First, T [ X ] is a well-defined braided monoidal category: if f and g arerestriction-total, then f ⊗ g = f ⊗ g makes f ⊗ g restriction-total too, and all(coherence) isomorphisms are restriction-total.Let s : S → I be a subunit in T [ X ]. Then S ⊗ s : S ⊗ S → S ⊗ I is invertiblein T [ X ] and hence in X . Suppose that f, g : A → S in X satisfy s ◦ f = g ◦ s .Then f = s ◦ f = s ◦ g = g , and so d f = d g . Observe that s is tensor-total in X ,because if s = t ◦ m for a subunit t in X , then S ⊗ t : S ⊗ T → S is an isomorphismwith inverse ( S ⊗ m ) ◦ ( S ⊗ s ) − . Now s = s ◦ S = s ◦ s , so ⌈ s ⌉ = s in X .Therefore T ( s ) = (1 ⊗ s ) ◦ ⌈ s ⊗ S ⌉ = s ∧ s : S ⊗ S → I . It follows from (TR4)that T ( s ) ◦ T ( f ) = T ( s ◦ f ) = T ( s ◦ g ) = T ( s ) ◦ T ( g ) and so T ( f ) = T ( g ). Thus f = T ( f ) ◦ ( f ⊗ d f ) = T ( g ) ◦ ( g ⊗ d g ) = g . So s is monic and hence a subunit in X ,too. Thus T [ X ] is firm because X is firm. (cid:3) We now state the main result of this section.
Theorem 5.16. If X is a tensor-restriction category, then there is a firm monoidalrestriction category isomorphism X ≃ S [ T [ X ]] .Proof. Define a functor F : X → S [ T [ X ]] by F ( f ) = [ s f , T ( f )], which is well-defined by Proposition 5.14. Because ⌈ X ⌉ = 1 X and the least subunit s X which1 I factors through is 1 I , we have T (1 X ) = ρ and F indeed preserves identities. Tosee that F preserves composition, let f : X → Y and g : Y → Z in X . Then the following diagram commutes by Lemmas 5.11 and 5.12: XX ⊗ I ⊗ IX ⊗ S f ⊗ IX ⊗ S f ⊗ S g X ⊗ S gf YY ⊗ I ⊗ IY ⊗ S f ⊗ I Y ⊗ I ⊗ S g Y ⊗ S gf Y ⊗ S f ⊗ S g ZZ ⊗ I ⊗ IZ ⊗ I ⊗ S g Z ⊗ S f ⊗ S g Z ⊗ S gf f gf ⊗ I g ⊗ If ⊗ S f g ⊗ S g f ⊗ S gf g ⊗ S gf λ − λ − λ − ⊗ d f ⊗ s g ⊗ s f ⊗ s f ⊗ d g ⊗ d g ⊗ d g ⊗ s f ⊗ ⊗ s f ⊗ ⊗ d gf ⊗ s gf ( g ◦ f ) ⊗ S gf Hence s g ◦ f = s g ∧ s f . It follows that gf = e gf • X = e g • e f • X = e g • f . Thereforethe uniqueness of (TR4) guarantees that the following diagram commutes: X ⊗ S gf Z ⊗ S gf ZX ⊗ S f ⊗ S g Z ⊗ S f ⊗ S g Z ⊗ S g Y ⊗ S f ⊗ S g Y ⊗ S g ⌈ ( g ◦ f ) ⊗ ⌉ ⊗ s gf ⊗ s g ⌈ f ⊗ ⌉ ⊗ σ ◦ ( ⌈ g ⊗ ⌉ ⊗ ◦ σ ⊗ s f ⊗ ⊗ s f ⊗ ⌈ g ⊗ ⌉ Thus F ( g ◦ f ) = [ s g ◦ f , T ( g ◦ f )] = [ s g ∧ s f , T ( g ) ◦ ( T ( f ) ⊗ F ( g ) ◦ F ( f ).In the other direction, define G : S [ T [ X ]] → X by G (cid:2) s, fA SB (cid:3) = f ◦ (1 ⊗ d f ).This is a well-defined functor by Lemmas 5.11 and 5.12. Clearly on objects wehave that F ( G ( X )) = X and G ( F ( X )). For morphisms, on the one hand we have G ( F ( f )) = T ( f ) ◦ (1 ⊗ d f ) = f by construction. On the other hand, for restriction-total f : X ⊗ S → Y we have s ⊗ d f = 1, and so: F ( G (cid:2) s, fA SB (cid:3) ) = F ( f ◦ (1 ⊗ d f ))= F ( f ) ◦ F ( X ⊗ d f )= [ s f , T ( f )] ◦ [1 I , d f ]= [ s, T ( f ) ◦ (1 ⊗ d f )]= [ s, T ( f ) ◦ ( f ⊗ d f )]= (cid:2) s, fA SB (cid:3) ENSOR-RESTRICTION CATEGORIES 25
Thus F and G are inverses.It is clear that F and G are strict monoidal functors. It follows directly that F and G preserve subunits, and indeed G (cid:2) I , S (cid:3) = s , and F ( s ) = (cid:2) I , S (cid:3) .Finally, T ( f ) = T ( e f •
1) = T ( e f ) • ⊗ s f , therefore: F ( f ) = [ s f , T ( f )] = [ s f , ⊗ s f ] = [ s f , T ( f )] = F ( f )so F preserves restriction. Similarly, G ( (cid:2) s, fA SB (cid:3) ) = G (cid:2) s, A S (cid:3) = (1 ⊗ s ) ◦ (1 ⊗ d s ) = 1 ⊗ e f = 1 ⊗ d f = f ◦ (1 ⊗ d f ) = G h s, fA SB (cid:3) so G preserves restriction too. So we conclude that it is an isomorphism of tensor-restriction categories. (cid:3) This main result easily lifts to functors. Recall that a morphism of firm monoidalcategories [7, 10.1] is a (strong) monoidal functor that sends subunits to subunits.They form a category
FirmCat . Definition 5.17. A morphism of tensor-restriction categories is a functor F : X → Y that is (strong) monoidal, sends subunits to subunits, and satisfies F ( f ) = F ( f )(i.e. F is a restriction functor [6, Section 2.2.1]). Tensor-restriction categories andtheir morphisms form a category TensRestCat . Theorem 5.18.
There is an equivalence of categories
FirmCat ≃ TensRestCat .Proof.
All that remains to be verified is that S and T are functors. But this is easy:define S [ F ] by A F ( A ) on objects and by [ s, f ] [ F ( s ) , F ( f )] on morphisms,and define T [ G ] by A G ( X ) on objects and as G ( f ) = f on morphisms. (cid:3) This lets us characterise the fixed points of the S -construction. Recall that afirm category is simple when it has no nontrivial subunits [7, Definition 5.3]. Corollary 5.19.
A firm monoidal category C has S [ C ] ≃ C if and only if it issimple.Proof. If ISub( C ) = { } , every map in S [ C ] is total, and so S [ C ] = T [ S [ C ]] ≃ C .Conversely, if S [ C ] ≃ C ≃ T [ S [ C ]], then every map in S [ C ] is total, and soISub( C ) = { } . (cid:3) A particular example of a simple firm monoidal category C is when C is agroupoid. Thus as a consequence of Corollary 3.21, if C is a groupoid, or equiva-lently if S [ C ] is an inverse category, then S [ C ] ≃ C .We conclude this section with examples pointing out that not every firm monoidalcategory is a tensor-restriction category. Example 5.20.
Every category C is trivially a restriction category by setting f = 1, so every map is total and T [ C ] = C . Thus every firm monoidal category istrivially a firm monoidal restriction category. Suppose a tensor-restriction category X had a trivial restriction, so T [ X ] = X . Theorem 5.16 then shows that X ∼ = S [ T [ X ]] = S [ X ]. But Corollary 5.19 then implies that X is simple. Thus everyfirm monoidal category that is not simple (i.e. has a nontrivial subunit), regarded as a firm monoidal restriction category with the trivial restriction, is not a tensor-restriction category. Example 5.21.
There are also many examples of firm monoidal restriction cate-gories with nontrivial restrictions that are not tensor-restriction categories. Con-sider
Par and recall that T [ Par ] =
Set . If it was a tensor-restriction category,Theorem 5.16 would give
Par ≃ S [ T [ Set ]], but this is not the case. Indeed, it isclear that
Par ( X, Y ) Set ( X, Y ) + 1 ∼ S [ Set ]( X, Y ) as in Example 3.9). There-fore
Par
6≃ S [ T [ Set ]]. Thus
Par is not a tensor-restriction category.6.
Alternative axiomatisations
The axiomatisation of tensor-restriction categories of Definition 5.7 has severalfeatures. First, of course, it works, in the sense that Theorem 5.16 holds. Second,it is elementary, in the sense that it is phrased entirely in terms of basic notionsfrom restriction category theory and tensor topology. Third, it is intuitive in thatit conveys the structure of the S -construction. Nevertheless, there is room foralternative axiomatisations. In this section we discuss two aspects: • It is not clear that axioms (TR1)–(TR7) are independent. In fact, theredoes appear to be some redundancy, which we will point out. We will arguethat the “property-based” axiomatisation of Definition 5.7 may be replacedby a “structure-based” one where the tensor-restriction points d are given. • The “uniformity” axiom (TR3) means that S [ C ] is not just a restrictioncategory, but its opposite is too. This may make for a more efficient axioma-tisation. We will make a start towards such an alternative axiomatisationby defining bi-restriction categories and observing that S [ C ] is one.Starting with the first goal, of taking the maps d as structure rather than emer-gent properties, we first make precise the type of these maps. Definition 6.1.
In a firm restriction category X , a restriction-subunit point ofsubunit a s : S I is a morphism d : I → S satisfying d = s ◦ d .The next three lemmas show that these restriction-subunit points have some ofthe properties of axioms (TR1)–(TR7) automatically. Lemma 6.2.
In a firm restriction category, if d : I → S and d ′ : I → T arerestriction-subunit points, then d ∧ d ′ : : I → S ⊗ T , defined as the composite d ∧ d ′ = ( d ⊗ d ′ ) ◦ λ − is a restriction-subunit point of S ⊗ T .Proof. The computation d ∧ d ′ = ( d ⊗ d ′ ) ◦ λ − = λ ◦ ( d ⊗ d ′ ) ◦ λ − = λ ◦ ( s ⊗ t ) ◦ ( d ⊗ d ′ ) ◦ λ − = ( s ∧ t ) ◦ ( d ∧ d ′ )shows that ( d ∧ d ′ ) is a restriction-subunit point of S ⊗ T . (cid:3) Lemma 6.3.
In a firm restriction category, any restriction-subunit point d : I → S is a restriction isomorphism with restriction inverse d ◦ = d ◦ s . ENSOR-RESTRICTION CATEGORIES 27
Proof.
First note that in a monoidal restriction category, a • f = a • f for any scalar a : I → I and any map f : X → Y . Since s is total, d ◦ = d • s = d • s = d • I = d and therefore d ◦ d ◦ = d ◦ d ◦ s = d ◦ s = d = d ◦ .Similarly d ◦ ◦ d = d ◦ s ◦ d = d ◦ d = d , so d is a restriction isomorphism. (cid:3) Lemma 6.4.
Let d : I → S be a restriction-subunit point in a firm restrictioncategory. Suppose that it satisfies the left-lifting property against subunits: if t ◦ f = g ◦ d s for any subunit T and maps f : I → T and g : S → I , then f = m ◦ d and g = t ◦ m for some m : I ST Id gf t m
Then d ◦ : S → I is tensor-total, and d : I → S is a tensor-restriction point.Proof. Suppose d ◦ factors through a subunit T as follows: S IT td ◦ f We need to show that S ⊗ T ≃ S . To do so, we will show that s ≤ t . Observe: s ◦ d = d = d ◦ ◦ d = t ◦ f ◦ d The left lifting property gives m : S → T with: I ST IS d sdf t m
In particular, t ◦ m = s , and therefore s ≤ t . We conclude that S ⊗ T ≃ S , and so d ◦ is tensor-total. Since d ◦ is also a restriction isomorphism, it follows by definitionthat d is a tensor-restriction point. (cid:3) The previous three lemmas seem to exhibit some redundancy in the propertiesthat (TR1)–(TR7) ask of the maps d . However, to progress in obtaining an equiv-alent axiomatisation, we need to demand a universal property of the restriction-subunit points d . Say that a subunit S has a maximal restriction-subunit point ifthere is a restriction subunit point d s : I → S that is initial amongst restriction-subunit points of S . More precisely, every restriction-subunit point d : I → S satisfies d ≤ d s in the restriction category sense, that is d = d s ◦ d . Lemma 6.5.
In a firm restriction category, if subunits S and T have maximalrestriction-subunit point d s : I → S and d t : I → T , then d s ∧ d t = ( d s ⊗ d t ) ◦ λ − : I → S ⊗ T is a maximal restriction-subunit point of S ⊗ T .Proof. Suppose d : I → S ⊗ T is a restriction subunit point of S ⊗ T , that is, d = ( s ∧ t ) ◦ d . We need to show that d ≤ d s ∧ d t . First observe λ ◦ ( s ⊗ T ) ◦ d = d = ( s ∧ t ) ◦ d = λ ◦ ( s ⊗ t ) ◦ d = t ◦ λ ◦ ( s ⊗ T ) ◦ d ,where the first equality holds by totality of λ ◦ ( s ⊗ T ). Therefore, λ ◦ ( s ⊗ T ) ◦ d is a restriction-subunit point of T and so λ ◦ ( s ⊗ T ) ◦ d ≤ d t : λ ◦ ( s ⊗ T ) ◦ d = d t ◦ d Similarly ρ ◦ ( S ⊗ t ) ◦ d is a restriction-subunit point of S and so ρ ◦ ( S ⊗ t ) ◦ d ≤ d s : ρ ◦ ( S ⊗ t ) ◦ d = d s ◦ d Now we can show that ( d s ∧ d t ) ◦ d = d :( d s ∧ d t ) ◦ d = ( d s ∧ d t ) ◦ d ◦ d = ( d s ⊗ d t ) ◦ λ − ◦ d ◦ d = ( d s ⊗ d t ) ◦ ( d ⊗ d ) ◦ λ − = (( λ ◦ ( s ⊗ T ) ◦ d ) ⊗ ( ρ ◦ ( S ⊗ t ) ◦ d )) ◦ λ − = (( λ ◦ ( s ⊗ t ) ◦ d ) ⊗ d ) ◦ λ − = ((( s ∧ t ) ◦ d ) ⊗ d ) ◦ λ − = (cid:0) d ⊗ d (cid:1) ◦ λ − = d ◦ d = d Therefore d ≤ d s ∧ d t . (cid:3) As an intermezzo, we now discuss how to use the tensor-restriction points S ina graphical normal form of maps in tensor-restriction categories. This stronglyresembles a monoidal version of [12] and is a first step towards a general calculushandling restriction graphically. The factorisation system of Proposition 5.14 letsus decompose any morphism f in a tensor-restriction category as follows: f BA = T ( f ) S f A B total partrestriction part
ENSOR-RESTRICTION CATEGORIES 29 In S [ C ], this says (cid:2) s, f (cid:3) = (cid:2) , f (cid:3) ◦ (cid:16)(cid:2) , A (cid:3) ⊗ (cid:2) s, S (cid:3)(cid:17) . Restriction and compo-sition become: f AA = AA S f fg CBA = T ( g ◦ f ) S g S f A C
Finally, we change to the second goal of this section, by observing that the S [ − ]-construction also induces a corestriction category [4, 6]. Definition 6.6. [6, Example 2.1.3.12] A corestriction category is a category X equipped with a choice of endomorphism b f : B → B for each morphism f : A → B satisfying: b f ◦ f = f (CR1) b f ◦ b g = b g ◦ b f if cod( f ) = cod( g )(CR2) [ b g ◦ f = b g ◦ b f if cod( f ) = cod( g )(CR3) g ◦ b f = [ g ◦ f ◦ g if dom( g ) = cod( f )(CR4)We call b f the corestriction of f . Example 6.7.
Let stabLat be the category whose objects are semilattices andwhose morphisms are stable homomorphisms, that is, morphisms between semi-lattices that preserve binary meets but not necessarily the top element. This is acorestriction category, where for a stable homomorphism f : L → L ′ , the corestric-tion is the stable homomorphism b f : L ′ → L ′ given by b f ( x ) = x ∧ f (1).The corestriction structure of S [ C ] is defined similarly to its restriction structure. Proposition 6.8. If C is a firm monoidal category, S [ C ] is a corestriction categorywith: \ (cid:2) s, fA SB (cid:3) = (cid:2) s, B S (cid:3)
Proof.
Completely analogous to Proposition 3.4. (cid:3)
It turns out that this corestriction structure also makes S [ C ] a range category,that is, axiomatises the image of morphisms. Definition 6.9. [6, Definition 2.12]. A range category is a restriction category X which is additionally equipped with a choice of endomorphism b f : B → B for eachmorphism f : A → B satisfying: b f = b f (RR1) b f ◦ f = f (RR2) [ g ◦ f = g ◦ b f if cod( f ) = dom( g )(RR3) [ g ◦ b f = [ g ◦ f if cod( f ) = dom( g )(RR4)We call b f the range of f .Note that not every range category is automatically a corestriction category.Indeed, being a corestriction category does not require restriction structure, whereasbeing a range category does. Here is the paradigmatic example. Example 6.10. Par is a range category where the range of a for a partial function f : X → Y is defined as follows: b f ( y ) = ( y if f ( x ) = y for some x ∈ X undefined otherwiseHowever, this does not make Par a corestriction category since (CR4) fails.If a category is both a restriction category and a corestriction category, and ifthe restriction and corestriction operators are compatible, then in fact the corestric-tion operator is also a range category. We introduce a new notion of birestrictioncategory (which is a stronger version of a bisupport category [6]).
Definition 6.11. A birestriction category is a category equipped with both a re-striction operator and a corestriction operator b satisfying: b f = f (BR1) b f = b f (BR2) Example 6.12. If X is both a restriction category and a dagger category [9], andrestriction idempotents are self-adjoint, that is, f † = f , then X is automatically abirestriction category with b f = f † . Lemma 6.13.
In a birestriction category, the corestriction operator is a rangeoperator for the restriction operator.Proof.
Let be the restriction operator and b be the corestriction operator. Weneed to show that b satisfies (RR1)–(RR4). The first two are immediate since(RR1) is simply (BR2) and (RR2) is (CR1). For (RR3), we use (CR3) and (BR1): [ g ◦ f = [ b g ◦ f = b g ◦ b f = g ◦ b f Finally, (RR4) is the dual of [6, Lemma 2.1.(iii)]. So b is a range operator. (cid:3) The point of introducing birestriction categories is that the S -construction in-duces one, which may lead to an alternative axiomatisation. Proposition 6.14. If C is a firm monoidal category, then S [ C ] is a birestrictioncategory.Proof. It is straightforward to see that both (BR1) and (BR2) hold in S [ C ]. (cid:3) ENSOR-RESTRICTION CATEGORIES 31
References [1] R. N. Ball, A. Pultr, and J. W. Wayland. The Dedekind MacNeille site completion of a meetsemilattice.
Algebra Universalis , 76:183–197, 2016.[2] J. R. B. Cockett, G. S. H. Cruttwell, and J. D. Gallagher. Differential restriction categories.
Theory and Applications of Categories , 25(21):537–613, 2011.[3] J. R. B. Cockett and X. Guo. Stable meet semilattice fibrations and free restriction categories.
Theory and Applications of Categories , 16(15):307–341, 2006.[4] J. R. B. Cockett, X. Guo, and P. Hofstra. Range categories I: General theory.
Theory andApplications of Categories , 26(17):412–452, 2012.[5] J. R. B. Cockett and C. Heunen. Compact inverse categories. 2020.[6] J. R. B. Cockett and S. Lack. Restriction categories I: categories of partial maps.
TheoreticalComputer Science , 270(1–2):223–259, 2002.[7] P. Enrique Moliner, C. Heunen, and S. Tull. Tensor topology.
Journal of Pure and AppliedAlgebra , 224(10):106378, 2020.[8] J. S. Golan.
Semirings and their applications . Kluwer, 1999.[9] C. Heunen and M. Karvonen. Monads on dagger categories.
Theory and Applications ofCategories , 31(35):1016–1043, 2016.[10] C. Heunen and J. Vicary.
Categories for Quantum Theory: an introduction . Oxford Univer-sity Press, 2019.[11] P. T. Johnstone.
Stone spaces . Cambridge University Press, 1982.[12] C. Nester. String diagrams for Cartesian restriction categories. SYCO 5, 2019.
University of Edinburgh, United Kingdom
Email address : [email protected] University of Oxford, United Kingdom
Email address ::