A logic for categories

A LOGIC F OR C A TEGORIES CLA UDIO PISANI A BSTRACT . W e present a do ctrinal approach to catego ry theory , obtained by abstra ct- ing from the indexed inclusion (via discrete fibratio ns and opfibra tions) of left a nd of right actions of X ∈ Cat in catego r ies o v er X . Namely , a “weak tempo ral do ctrine” consists essentially of t w o indexed functors with the same co domain such that the in- duced functors hav e b oth left a nd right adjoints satisfying some e xactness conditions, in the spir it of ca tegorical logic. The derived logical rules include some adjunction-like laws inv olving the truth-v alues- enriched hom and tensor functor s, which condense several basic ca tegorical pro perties and display a nice symmetr y . The symmetr y bec omes more apparent in the slig h tly stronger c o n text of “tempo r al do ctrines”, whic h we initially treat and which include as an instance the inclusion of lo w er and upper sets in the pa rts of a pos e t, as w ell a s the inclusion of left and right a ctions of a graph in the graphs o v er it. Conten ts 1 In tro duction 1 2 Enric hing adjunctions 5 3 The logic of h yperdo ctrines 7 4 T empora l do ctrines 9 5 Basic prop erties 12 6 F unctors v alued in truth v a lues 14 7 Limits, colimits and Y oneda prop erties 16 8 Exploiting comprehension 18 9 The sup and inf reflections 20 10 The logic of categories 21 1. Intro duction Let X b e a set endo w ed with an equiv alence relation ∼ , a nd let V X b e the p oset of closed par t s, that is those subsets V of X suc h that x ∈ V and x ∼ y implies y ∈ V . A part P ∈ P X has b oth a “closure” ♦ P a nd an “in terior ” ⊓ ⊔ P , that is the inclusion i : V X → P X has b ot h a left and a rig h t a djoin t: ♦ ⊣ i ⊣ ⊓ ⊔ : P X → V X 2000 Ma thema tics Sub ject Classifica tion: 1 8A05, 18A30, 1 8A40, 18D99. Key words and phrases: T empo r al do c trine, internal hom and tenso r , F rob enius law, adjunction- lik e laws, in ternal limits and colimits, quan tification formulas. c  Cla udio Pisani, 2010. Permission to co p y for pr iv a te use grant ed. 1 2 Th us the (co)reflection maps (inclusions) ε P : i ⊓ ⊔ P → P and η P : P → i ♦ P induce bijections (b et w een 0-elemen ts or 1- elemen ts sets): P X ( iV , i ⊓ ⊔ P ) P X ( iV , P ) ; P X ( i ♦ P , iV ) P X ( P , iV ) By taking ⊓ ⊔ ( P ⇒ Q ) as a V X -enric hment of P X ( P , Q ), it turns out that the ab o ve adjunctions are also enric hed in V X giving isomorphisms: ⊓ ⊔ ( iV ⇒ i ⊓ ⊔ P ) ⊓ ⊔ ( iV ⇒ P ) ; ⊓ ⊔ ( i ♦ P ⇒ iV ) ⊓ ⊔ ( P ⇒ iV ) W e also hav e the related laws: i ♦ ( iV × iW ) iV × iW ; i ⊓ ⊔ ( iV ⇒ iW ) iV ⇒ iW ; ♦ ( i ♦ P × iV ) ♦ ( P × iV ) the first tw o of them sa ying roughly that closed par t s are closed with resp ect to pro duct (in t ersec tion) and exponentiation (implication). Giv en a group oid X , the same law s hold for the inclusion of the actions of X in the g r oup o ids ov er X (via “co v ering group oids”). The ab o v e situat io n will b e placed in the prop er general con text in sections 2 a nd 3, where we dev elop e some tec hnical to ols concerning enric hed adjunctions and apply them to h yperdo ctrines [Lawv ere, 19 70]. No w, let us drop the symmetry condition on ∼ , that is sup p ose that X is a p oset; then w e ha v e t he p oset of lo w er-closed parts D X and that of upp er-closed pa rts U X . Again, the inclusions i : D X → P X and i ′ : U X → P X hav e b oth a left and a right adjo int: ♦ ⊣ i ⊣ ⊓ ⊔ : P X → D X ; ♦ ′ ⊣ i ′ ⊣ ⊓ ⊔ ′ : P X → U X While some of the a b ov e laws still hold “on eac h side”: ⊓ ⊔ ( iV ⇒ i ⊓ ⊔ P ) ⊓ ⊔ ( iV ⇒ P ) ; ⊓ ⊔ ′ ( i ′ V ⇒ i ′ ⊓ ⊔ ′ P ) ⊓ ⊔ ′ ( i ′ V ⇒ P ) (1) i ♦ ( iV × iW ) iV × iW ; i ′ ♦ ′ ( i ′ V × i ′ W ) i ′ V × i ′ W (2) the other ones hold only in a mixed wa y: ⊓ ⊔ ′ ( i ♦ P ⇒ iV ) ⊓ ⊔ ′ ( P ⇒ iV ) ; ⊓ ⊔ ( i ′ ♦ ′ P ⇒ i ′ V ) ⊓ ⊔ ( P ⇒ i ′ V ) (3) 3 i ⊓ ⊔ ( i ′ V ⇒ iW ) i ′ V ⇒ iW ; i ′ ⊓ ⊔ ′ ( iV ⇒ i ′ W ) iV ⇒ i ′ W (4) ♦ ( i ♦ P × i ′ V ) ♦ ( P × i ′ V ) ; ♦ ′ ( i ′ ♦ ′ P × iV ) ♦ ′ ( P × iV ) (5) The law s (1) through (5) hold also for the inclusion o f t he left and the right actions of a category X in categories ov er X (via discrete fibrations and opfibrations): i : Set X op → Cat /X ; i ′ : Set X → Cat /X and, when they mak e sense, also fo r the inclusion of op en and closed parts in the parts of a top ological space (or, more generally , of lo cal homeomorphisms and prop er maps to a space X in spaces ov er X ; see [Pisani, 20 09]). Abstracting from these situations, w e may define a “temp oral algebra” as a cartesian closed category with tw o reflectiv e and coreflectiv e full sub catego ries satisfying the ab o v e la ws (in fact, it is enough to assume either (3) or (4) or (5)). A “tempor al do ctrine” is then esse n tia lly an indexed temp oral algebra h i X : M X → P X ← M ′ X : i ′ X ; X ∈ C i suc h tha t the inclusions i 1 and i ′ 1 o v er the terminal o b ject 1 ∈ C are isomorphic. T e mp oral do ctrines and their basic prop erties are presen ted in sections 4 and 5. In Section 6 we sho w how the “truth-v alues” M 1 ∼ = M ′ 1 serv e as v alues for an enric hing of P X , M X and M ′ X in whic h the adjunctions Σ f ⊣ f ∗ ⊣ Π f : P X → P Y ♦ X ⊣ i X ⊣ ⊓ ⊔ X : P X → M X ; ♦ ′ X ⊣ i ′ X ⊣ ⊓ ⊔ ′ X : P X → M ′ X ∃ f ⊣ f · ⊣ ∀ f : M X → M Y ; ∃ ′ f ⊣ f ′ · ⊣ ∀ ′ f : M ′ X → M ′ Y (6) are also enriche d (where, for f : X → Y in C , ∃ f M ∼ = ♦ Y Σ f i X M ). F or example, the temp oral do ctrine of p osets is tw o-v alued while that of reflexiv e graphs is Set -v alued, by iden tifying sets with discrete graphs. If C = Cat , the functors Π f are not alw ays av ailable, and the ab ov e men tioned enric h- men t is only part ia lly defined. This w eak er situation will be axiomatized in Section 10, where w e will see that ( 6) still can b e enric hed giving: nat X ( L, f · M ) nat Y ( ∃ f L, M ) ; nat X ( f · M , L ) nat Y ( M , ∀ f L ) (7) (and similarly for “righ t actions” or “right closed parts” in M ′ ) where nat X ( L, M ) := end X ( i X L ⇒ i X M ) := ∀ X ⊓ ⊔ X ( i X L ⇒ i X M ) 4 In a somewhat dual w ay one a lso obtains: ten X ( N , f · M ) ten Y ( ∃ ′ f N , M ) ; ten X ( f ′ · N , L ) ten Y ( N , ∃ f L ) (8) where one defines the tensor pro duct b y ten X ( N , M ) := co end X ( i ′ X N × i X M ) := ∃ X ♦ X ( i ′ X N × i X M ) In Section 7 w e show how the laws (7) and (8) allow one to deriv e in an effectiv e and transparen t w a y sev eral basic facts of category theory , in particular concerning (co)limits, the Y oneda lemma, Kan extensions and final functors. In sec tion 8 and 9 other “classic al” prop erties are obtained exploiting also a “comprehension” axiom, relating P X and C /X . This approac h also offers a new p ersp ectiv e on dualit y: w e do not assume the coex- istence of a (generalized) category X and its dual X op (whic h in fact is not so obvious as it may seem at a first sigh t). Rather, w e capture the interpla y b etw een left and righ t “actions” or “parts” of a category or “space” X b y the ab ov e-sk etc hed axioms concerning the inclusion of b oth of them in a category o f more general “lab ellings” or “parts”. It is remark able that while (4) is equiv alen t t o (3) and to (5), they underlie seem- ingly unrelated items. On the one hand, for a truth v alue V in M 1 ∼ = M ′ 1 the “ V − complemen t” of M ∈ M X ¬ ( M , V ) := i X M ⇒ i ′ X X ′ · V “is v alued” in M ′ X , that is factors b y (4) through i ′ X (and con v ersely). This generalizes the op en-closed dualit y via complemen ta tion in top olo g y (and in pa r ticular the upp er- lo w er-sets dua lity for a p oset) whic h is given b y ¬ ( M , false ). On the other hand, if we denote by { x } := Σ x 1 the “part” in Cat /X corresp onding to the ob ject x : 1 → X , then ♦ X { x } = X/x corresp onds to the presheaf represen ted by x and for N ∈ M ′ X we can pro v e t ha t ten X ( N , X/x ) ∼ = x ′ · N using (5) as f o llo ws: ten X ( N , X/x ) ∼ = ∃ X ♦ X ( i ′ X N × X i X ♦ X { x } ) ∼ = ∃ X ♦ X ( i ′ X N × X { x } ) ∼ = ∼ = ♦ 1 Σ X ( i ′ X N × X Σ x 1) ∼ = ♦ 1 Σ X Σ x ( x ∗ i ′ X N × 1) ∼ = ♦ 1 i 1 ( x ′ · N ) ∼ = x ′ · N While in any temp oral do ctrine w e can similarly deriv e hom X ( X/x, M ) ∼ = x · M usin g (3), in Cat such a pro of of the Y oneda lemma stum bles against the la ck o f Π x and the related non-exp onen tiability of { x } (whenev er x is a non-trivial retract in X ). In this case, or in an y (w eak) temp oral do ctrine, one can use directly the first o f ( 7 ): nat X ( X/x, M ) ∼ = nat X ( ∃ x 1 , M ) ∼ = nat 1 (1 , x · M ) ∼ = x · M Similarly , using (8) o ne gets again: ten X ( N , X/x ) ∼ = ten X ( N , ∃ x 1) ∼ = ten 1 ( x ′ · N , 1) ∼ = x ′ · N The presen t pap er is a dev elopmen t of previous works on “balanced category theory” (see in particular [Pisani, 200 8] and [Pisani, 20 0 9]); the do ctrinal approach adopted here emphasizes the logical asp ects and suggests a wider ra nge of applications. 5 2. Enr ic h i ng adj unctions In this section, w e mak e some remarks that will b e used in the sequel. Along with ordinary adjunctions, Kan defined and studied what a re no w kno wn as adjunctions with para meter and enriche d adjunctions. In particular, w e will use the follow ing r esult from [Kan, 1958]: 2.1. Lemma. Given functors F , F ′ : C × P → V and R, R ′ : P op × V → C such that ther e ar e adjunctions (with p ar ameter) F ( X, P ) ⊣ R ( P , V ) ; F ′ ( X , P ) ⊣ R ′ ( P , V ) the natur al tr ansformations F → F ′ c orr esp ond bije ctively to the o n es R ′ → R , and this c orr esp ondenc e r estricts to natur al isomorp h isms. In p articular F ∼ = F ′ iff R ∼ = R ′ . The next r emark roughly sa ys that a geometric morphism is naturally enriche d in its co domain: 2.2. Proposition. L et F ⊣ R : C → V b e an adjunction b etwe en c artesia n close d c ate gories, with F left exact. T hen C is enriche d in V via hom V C ( X , Y ) := R ( X ⇒ C Y ) (9) and ther e ar e natur al isomo rp h isms: hom V C ( F V , X ) ∼ = hom V ( V , RX ) (wher e hom V ( V , W ) := V ⇒ V W is the internal hom of V ) that is the adjunction F ⊣ R is itself enrich e d in V . F urthermor e, the na tur al tr an sformations given by the arr ow mappings of F and R ar e also enriche d: hom V ( V , W ) → hom V C ( F V , F W ) ; hom V C ( X , Y ) → hom V ( RX , RY ) Proo f. F or the first part , we ha ve V (1 , R ( X ⇒ Y )) ∼ = C ( F 1 , X ⇒ Y ) ∼ = C (1 , X ⇒ Y ) ∼ = C ( X , Y ) (In fa ct, more generally , R transfers an y enric hing in C to an enric hing in V .) F or the second part, since F ( V × W ) ∼ = F V × F W , w e can apply Lemma 2.1 to the adjunctions: F ( V × W ) ⊣ W ⇒ RX ; F V × F W ⊣ R ( F W ⇒ X ) F or the third part, the c hain of natural transfor ma t io ns: V ( U, V ⇒ W )) ∼ = V ( U × V , W ) → C ( F ( U × V ) , F W ) ∼ = ∼ = C ( F U × F V , F W ) ∼ = C ( F U, F V ⇒ F W ) ∼ = V ( U, R ( F V ⇒ F W )) 6 yields the desired natural tr ansformation, whic h is easily seen to enric h the a r r o w mapping of F . F or R w e similarly hav e: V ( U, R ( X ⇒ Y )) ∼ = C ( F U, X ⇒ Y ) ∼ = C ( F U × X , Y ) → → C ( F ( U × RX ) , Y ) ∼ = V ( U × RX , RY ) ∼ = V ( U, RX ⇒ RY ) where the non-isomorphic step is induced by the canonical h F p, εF q i : F ( U × R X ) → F U × X (10) Th us, R is fully faithful, also as an enric hed functor, iff (10) is an iso, that is F ⊣ R satisfies the F rob enius law . Since here w e ha ve not used the fa ct that F preserv es a ll finite pro ducts, but only the terminal ob ject (in o rder to obtain an eric hmen t of R ) w e get in particular a pro o f of Corolla ry 1 .5 .9 (i) in [Johnstone, 2002]. If F has a further left adjoin t L : C → V , then it is left exact a nd the ab ov e prop osition applies. W e no w sho w tha t in this case the adjunction L ⊣ F is also enric hed in V iff it satisfies the F rob enius recipro cit y law : 2.3. Proposition. Supp ose that C and V ar e c artesian close d and that L ⊣ F ⊣ R : C → V Then the existenc e of the fol lowing natur al isomorp hisms ar e e quivalent: 1. LX × V V ∼ = L ( X × C F V ) 2. F ( V ⇒ V W ) ∼ = F V ⇒ C F W 3. hom V C ( X , F V ) ∼ = hom V ( LX, V ) Proo f. As b efore, w e apply Lemma 2.1 to the adjunctions: LX × V ⊣ F ( V ⇒ W ) ; L ( X × F V ) ⊣ F V ⇒ RW getting the equiv alence of 1) and 2) (whic h is well-kno wn; see e.g. [Lawv ere, 1970]), and to the adjunctions: LX × V ⊣ LX ⇒ W ; L ( X × F V ) ⊣ R ( X ⇒ F W ) getting the equiv alence of 1) and 3). 7 Note that the same functor C × V → V has tw o differen t rig ht adjoints, dep ending on the parameter c hosen. 2.4. Remark. It is w ell known that giv en a djunctions L ⊣ F ⊣ R : C → V , with F fully faithful, if C is cartesian closed then so is also V ; in fact, pro ducts in V can b e defined b y 1 V := R 1 C ; V × V W := R ( F V × C F W ) or also by 1 V := L 1 C ; V × V W := L ( F V × C F W ) and exp onen tials b y V ⇒ V W := R ( F V ⇒ C F W ) (11) Note that, follow ing Prop osition 2.2, (11) indicates that F is f ully faithful as an enric hed functor, and w e get hom V C ( F V , X ) ∼ = hom V C ( F V , F RX ) (12) Note also that, in this case, the equiv alen t conditions of Prop osition 2.3 can b e rewritten as follows : L ( X × F V ) ∼ = L ( F LX × F V ) (13) F V ⇒ F W ∼ = F R ( F V ⇒ F W ) (14) hom V C ( X , F V ) ∼ = hom V C ( F LX , F V ) (15) where the isomorphisms are induced by the unit of L ⊣ F (the first and the third ones) and b y the counit of F ⊣ R (the second one). 3. Th e l ogic of hyper do ctrines W e now sho w ho w some of t he results of Section 2 apply to hyperdo ctrines [Lawv ere, 1970], giving in teresting consequences. R ecall that a h yp erdo ctrine is an indexed category hP X ; X ∈ C i suc h that C and all the categories P X ar e cartesian closed, and suc h that eac h substitution functor f ∗ : P Y → P X has b oth a left and a righ t adjoin t Σ f ⊣ f ∗ ⊣ Π f for an y f : X → Y in C . The logical significance o f hyperdo ctrines, and in particular the role of the adjo in ts to the substitution functors as existen tial and univ ersal quan tification, and that of P 1 as “sen tences” or “truth v alues”, are clearly illustrated in [La wv ere, 1970] and in other pap ers by t he same aut ho r. Here w e also assume that the a dj unctions Σ f ⊣ f ∗ satisfy t he F rob enius la w. On the other hand, w e do not need to assume that C is cartesian closed but only t ha t it has a terminal ob ject. 8 3.1. Coro llar y. L et hP X ; X ∈ C i b e a hyp er do ctrine and define hom X ( P , Q ) := Π X ( P ⇒ Q ) : ( P X ) op × P X → P 1 meets X ( P , Q ) := Σ X ( P × Q ) : P X × P X → P 1 wher e the quantific a tion indexes denote the ma p X → 1 . Then hom X enriches P X i n P 1 and, for any map f : X → Y , the fol lowi n g adjunction-like laws hold: hom X ( f ∗ Q, P ) ∼ = hom Y ( Q, Π f P ) ; hom X ( P , f ∗ Q ) ∼ = hom Y (Σ f P , Q ) meets X ( f ∗ Q, P ) ∼ = meets Y ( Q, Σ f P ) ; meets X ( P , f ∗ Q ) ∼ = meets Y (Σ f P , Q ) Proo f. Prop ositions 2.2 and 2 .3 , and the F rob enius la w itself, give : Π X ( f ∗ Q ⇒ P ) ∼ = Π Y Π f ( f ∗ Q ⇒ P ) ∼ = Π Y ( Q ⇒ Π f P ) Π X ( P ⇒ f ∗ Q ) ∼ = Π Y Π f ( P ⇒ f ∗ Q ) ∼ = Π Y (Σ f P ⇒ Q ) Σ X ( P × f ∗ Q ) ∼ = Σ Y Σ f ( P × f ∗ Q ) ∼ = Σ Y (Σ f P × Q ) Sa y that a map f : X → Y is “surjectiv e” if Σ f ⊤ X ∼ = ⊤ Y , where ⊤ X is a terminal ob j ect of P X . 3.2. Coro llar y. If f : X → Y is surje ctive map then Π X ( f ∗ Q ) ∼ = Π Y Q ; Σ X ( f ∗ Q ) ∼ = Σ Y Q Proo f. F or the first one w e hav e: Π X ( f ∗ Q ) ∼ = hom X ( ⊤ X , f ∗ Q ) ∼ = hom Y (Σ f ⊤ X , Q ) ∼ = hom Y ( ⊤ Y , Q ) ∼ = Π Y Q The pro of of the second one follo ws the same pattern: Σ X ( f ∗ Q ) ∼ = meets X ( ⊤ X , f ∗ Q ) ∼ = meets Y (Σ f ⊤ X , Q ) ∼ = meets Y ( ⊤ Y , Q ) ∼ = Σ Y Q Note that for hP X ; X ∈ Set i , Corollary 3.2 b ecomes the fact that the inv erse imag e functor along a surjectiv e mapping f : X → Y preserv es non-empt yness and reflects maximalit y: if P ⊆ Y is non-empt y so it is f − 1 P and if f − 1 P = X then P = Y . There are three canonical w ay s to get a “truth v alue” in P 1 from P ∈ P X , namely quan tificatio ns along X : X → 1 and ev aluatio n at a p oin t x : 1 → X : Π X P ; Σ X P ; x ∗ P In the ab o v e prop osition, w e ha v e used the fact that quan tifications alo ng X are “repre- sen ted” (b y ⊤ X ): Π X P ∼ = hom X ( ⊤ X , P ) ; Σ X P ∼ = meets X ( ⊤ X , P ) No w w e show that the same is true for ev aluation; namely , ev alua t ion at x is “represen ted” b y the “singleton”: { x } := Σ x ⊤ 1 9 3.3. Coro llar y. Given a p oint x : 1 → X ther e ar e isomorphisms x ∗ P ∼ = hom X ( { x } , P ) ; x ∗ P ∼ = meets X ( { x } , P ) natur al in P ∈ P X . Proo f. hom X (Σ x ⊤ 1 , P ) ∼ = hom 1 ( ⊤ 1 , x ∗ P ) ∼ = Π 1 ( x ∗ P ) ∼ = x ∗ P meets X (Σ x ⊤ 1 , P ) ∼ = meets 1 ( ⊤ 1 , x ∗ P ) ∼ = Σ 1 ( x ∗ P ) ∼ = x ∗ P (Note that the last index 1 is the iden tit y o n 1 ∈ C .) 3.4. Remark. Supp ose that C has pullbac ks, so that w e also hav e the do ctrine hC /X ; X ∈ C i , with f ! ⊣ f − 1 : C / Y → C /X for f : X → Y . Supp ose also t ha t hP X ; X ∈ C i satisfies the comprehension axiom [Lawv ere, 1 9 70] c X ⊣ k X : P X → C /X . Then the set-v alued “external ev aluation” of P ∈ P X at x : 1 → X can b e expressed in v arious w a ys: P 1( ⊤ 1 , x − 1 P ) ∼ = P X ( { x } , P ) ∼ = P X ( c X x, P ) ∼ = ∼ = C /X ( x, k X P ) ∼ = C /X ( x ! 1 , k X P ) ∼ = C (1 , x − 1 k X P ) 3.5. Coro llar y. [form ulas fo r quan tifications] Give n P ∈ P X , a map f : X → Y and a p oint y : 1 → Y , ther e ar e isomorphism s y ∗ Π f P ∼ = hom X ( f ∗ { x } , P ) ; y ∗ Σ f P ∼ = meets X ( f ∗ { x } , P ) natur al in P ∈ P X . Proo f. y ∗ Π f P ∼ = hom Y ( { y } , Π f P ) ∼ = hom X ( f ∗ { y } , P ) y ∗ Σ f P ∼ = meets Y ( { y } , Σ f P ) ∼ = meets X ( f ∗ { y } , P ) Note that for h P X ; X ∈ Set i , Corollary 3 .5 giv es the classical formula for the coim- age of a part a long a mapping f , and a ( less classical) formu la for the image: y is in the image Σ f P iff its in v erse image meets P . 4. T emp oral do ct rines A temp oral do ctrine h i X : M X → P X ← M ′ X : i ′ X ; X ∈ C i consis ts of t w o indexed functors with the same co domain, satisfying the axioms listed b elo w. W e denote the substitution functors along a map f : X → Y in C b y f · : M Y → M X ; f ′ · : M ′ Y → M ′ X ; f ∗ : P Y → P X Th us w e hav e (coheren t) isomorphisms: ( g f ) · ∼ = f · g · ; ( g f ) ′ · ∼ = f ′ · g ′ · ; ( g f ) ∗ ∼ = f ∗ g ∗ 10 (and similarly for iden tities) and also i X f · ∼ = f ∗ i Y ; i ′ X f ′ · ∼ = f ∗ i ′ Y W e denote b y B X the indexed pullbac k M X × P X M ′ X , b y j X and j ′ X its indexed pro- jections to M X and M ′ X resp ectiv ely , and b X := i X j X = i ′ X j ′ X : B X → P X The first group of a xioms r equires the existence of some adjo in t f unctors: 1. The indexing category has a terminal ob ject: 1 ∈ C . 2. The categories of P X are cartesian closed. Th us, for an y X ∈ C , w e ha ve a terminal ob j ect 1 X ∈ P X , pro ducts P × X Q and exp onen tials P ⇒ X Q . 3. The substitution functors f ∗ : P Y → P X ha v e b oth left and right adjoin ts: Σ f ⊣ f ∗ ⊣ Π f 4. The functors i X : M X → P X and i ′ X : M ′ X → P X ha ve b o th left and righ t adjoin ts: ♦ X ⊣ i X ⊣ ⊓ ⊔ X ; ♦ ′ X ⊣ i ′ X ⊣ ⊓ ⊔ ′ X 5. The do ctrine P X satisfies the comprehension axiom [Lawv ere, 1970]: the canonical functors c X : C /X → P X (sending f : T → X to Σ f 1 T ) hav e right adjoints: c X ⊣ k X : P X → C /X The second group o f a xioms imp oses some exactness condition o n these functors: 1. The functors i X and i ′ X are fully faithful: ♦ X i X ∼ = id M X ; ⊓ ⊔ X i X ∼ = id M X (and similarly for i ′ X ). 2. The do ctrine P X satisfies the F rob enius law: Σ f P × Y Q ∼ = Σ f ( P × X f ∗ Q ) (16) for an y f : X → Y (naturally in P ∈ P X and Q ∈ P Y ). 3. The adjunctions ♦ X ⊣ i X and ♦ ′ X ⊣ i ′ X satisfy the “mixed F rob enius law s”, that is their units induce isomorphisms ♦ X ( P × X i ′ X N ) ∼ = ♦ X ( i X ♦ X P × X i ′ X N ) (17) ♦ ′ X ( P × X i X M ) ∼ = ♦ ′ X ( i ′ X ♦ ′ X P × X i X M ) (18) (natural in P ∈ P X , N ∈ M ′ X and M ∈ M X ) . 11 4. The pro jections j 1 : B 1 → M 1 and j ′ 1 : B 1 → M ′ 1 are isomorphisms. 5. The comprehension f unctors k X : P X → C /X are fully faithful: c X k X P = Σ k X P 1 X ! ( k X P ) ∼ = P (19) (where we use t he notations of Remark 3 .4, so that X ! is the do main pro jection C /X → C . Note that the index k X P of Σ is an ob ject of C /X , so that it should b e more exactly b e replaced b y X ! ( k X P ), where now k X P denote the map to the terminal in C /X ). 4.1. Examples. 1. An y hyperdo ctrine hP X ; X ∈ C i (see Section 3) with a fully faithful comprehension functor giv es rise to a (rat her trivial) temp oral do ctrine: h id : P X → P X ← P X : id ; X ∈ C i Th us the results of Section 3 can b e seen as particular cases of those w e will obtain for temp oral do ctrines. 2. h i X : D X → P X ← U X : i ′ X ; X ∈ P os i , where h P X ; X ∈ P os i is the do ctrine of all the parts of a p oset, while D X and U X are the sub do ctrines o f low er-closed and upp er-closed parts of X . 3. h i X : M X → Grph /X ← M ′ X : i ′ X ; X ∈ Grph i , where Grph is the category of reflexiv e graphs, while M X and M ′ X are the categories of left and rig ht actions of X (o r of the free category generated b y it). 4. Group oids or sets endo we d with an equiv alence relation give rise to “symmetrical” temp oral do ctrines: all the pro jections j X and j ′ X are isomorphisms . Note that, since the axioms are symmetrical, eac h temp oral do ctrine has a dual obtained b y exc hanging the left and the rig h t side (that is i and i ′ ); while a symmetrical temp oral do ctrine is clearly self-dual (that is isomorphic to its o wn dual) the same is true for Grph , via the “o pp osite” functor Grph → Grph . 5. An y stro ng balanced factorization categor y hC ; E , Mi [Pisani, 2008, Pisani, 2009] suc h that C is lo cally cartesian closed and M /X and M ′ /X are coreflectiv e in C / X , giv es rise to the temp oral do ctrine h i X : M /X → C /X ← M ′ /X : i ′ X ; X ∈ C i 6. Giv en a tempora l doctrine on a category C and an y subcategory C ′ of C such that 1 ∈ C ′ , one gets b y restriction another temp oral do ctrine on C ′ . 12 4.2. Remark. The name “temp oral do ctrine” is clearly suggested by the functors ♦ , ⊓ ⊔ , ♦ ′ and ⊓ ⊔ ′ , whic h can b e seen as mo da l op erato rs acting in the t w o directions of time. A catego rical approac h to mo dal and tense logic was dev elop ed in the eigh t ies b y Ghilardi and Meloni a nd indipenden tly b y R eyes et al. Not b eing here sp ecifically concerned with these logics, w e just note that the temp oral do ctrine of p osets men- tioned in the examples ab ov e is also an instance of tempo r al do ctrine in the sense of [Ghilardi & Meloni, 1991]. Let me also ac know eldge that it was prof. Giancarlo Meloni, the sup ervisor of m y phd thesis, who introduced me to categorical logic showin g in particular ho w adjunctions can b e an effectiv e to ol for doing calculations. 5. Ba sic prop erti es 5.1. terminology. Since a (w eak er form of ) temp oral do ctrine is mainly intende d to mo del the situation h Set X op → Cat /X ← Set X ; X ∈ Cat i , the ob jects o f C should b e though t o f as generalized categories. In fact in the sequel w e will freely b orrow terminology from category theory , whenev er opp ortune. How ev er, the in terior ⊓ ⊔ X and closure ♦ X op erators suggest that it also mak e sense to consider the ob jects of C as a sort of spaces, so that w e will also b orrow some terminology from top ology; in fact, the links with that sub ject can b e take n quite seriously as ske tc hed in [Pisani, 2 009], where it is discussed also the significance of the “closure” r eflection in “op en part s” (or “lo cal homeomorphisms”). An ywa y , if X is a top ological space and i X and i ′ X are the inclusion of op en and closed parts resp ectiv ely in P X , the mixed F r o b enius laws (and their equiv alent ones) hold true when they mak e sense, that is when only the o p era t o rs ⊓ ⊔ X and ♦ ′ X are in v olved. Th us w e sometimes refer to ob jects a nd arro ws of C a s “spaces” and “maps”; to the ob j ects of P X as “parts” of X a nd to those of M X and M ′ X as left closed and righ t closed parts of X , r esp ectiv ely . The reflections ♦ X and ♦ ′ X are the left and righ t “closure” op erators resp ectiv ely , while ⊓ ⊔ X and ⊓ ⊔ ′ X are the left and rig h t “ in terior” op erator s. Apart from t he axioms concerning the comprehension a dj unctions, a temp oral do c- trine is a h yperdo ctrine P X (in the sense of Section 3) with tw o reflectiv e and coreflectiv e indexed subcategories M X and M ′ X suc h that M 1 and M ′ 1 a r e isomorphic a s sub- categories of P 1; furthermore, and most imp orta n tly , w e assume the mixed F rob enius la ws (17) and (18), whic h are ric h of imp ortant consequences. The reason of their name follo ws b y Remark 2.4: they lo ok lik e the F rob enius la ws for ♦ X ⊣ i X and ♦ ′ X ⊣ i ′ X , except that i X and i ′ X are exc hanged in the second factors. In fact, w e ha ve the corresp onding mixed equiv alen t conditions: 5.2. Proposition. The fol lowing laws hold in a temp or al do ctrine, and e a c h o them c an b e use d in the definition in plac e o f (17 ) and (18) : i ′ X N ⇒ i X M ∼ = i X ⊓ ⊔ X ( i ′ X N ⇒ i X M ) ; i X M ⇒ i ′ X N ∼ = i ′ X ⊓ ⊔ ′ X ( i X M ⇒ i ′ X N ) (20) ⊓ ⊔ ′ X ( P ⇒ i X M ) ∼ = ⊓ ⊔ ′ X ( i X ♦ X P ⇒ i X M ) ; ⊓ ⊔ X ( P ⇒ i ′ X N ) ∼ = ⊓ ⊔ X ( i ′ X ♦ ′ X P ⇒ i ′ X N ) (21) 13 F urthermor e ⊓ ⊔ X ( i X M ⇒ P ) ∼ = ⊓ ⊔ X ( i X M ⇒ i X ⊓ ⊔ X P ) ; ⊓ ⊔ ′ X ( i ′ X N ⇒ P ) ∼ = ⊓ ⊔ ′ X ( i ′ X N ⇒ i ′ X ⊓ ⊔ ′ X P ) Proo f. As in Prop osition 2.3, b oth the members of (17) a nd of (18) ha v e t w o right adjoin ts, one for eac h parameter considered, giving the conditions ab o ve. F or the last statemen t, recall (12). F rom (20) w e immediatley get: 5.3. Coro llar y. If the p art P ∈ P X is left close d and Q ∈ P X is rig h t close d (that is P ∼ = i X M and Q ∼ = i ′ X N ) then P ⇒ Q is itself right close d. As already men tioned in the Intro duction, w e so hav e an “ explanation” of the fact that the complemen t o f a n upp er-closed par t of a p oset is low er-closed (and con ve rsely). 5.4. Coro llar y. The c ate gories B X ar e themselves c artesian close d, with the “same” exp onential of P X : b X B ⇒ b X C ∼ = b X ( B ⇒ B X C ) 5.5. Proposition. hM X ; X ∈ C i and hM ′ X ; X ∈ C i ar e themsel v es hyp er do c- trines, with a ful ly faithful c ompr ehen sion adjoint. Proo f. 1. As in Remark 2.4, the categories M X and M ′ X are cartesian closed, with expo- nen tials giv en by ⊓ ⊔ X ( i X L ⇒ i X M ) ; ⊓ ⊔ ′ X ( i ′ X N ⇒ i ′ X O ) W e denote pro ducts in M X and M ′ X by ⊤ X ; L ∧ X M ; ⊤ ′ X ; N ∧ ′ X O 2. The substitution functors for left and righ t closed part s ha v e b oth left and righ t adjoin ts: ∃ f ⊣ f · ⊣ ∀ f ; ∃ ′ f ⊣ f ′ · ⊣ ∀ ′ f where ∃ f ∼ = ♦ Y Σ f i X ; ∀ f ∼ = ⊓ ⊔ Y Π f i X (22) (and similarly for ∃ ′ f and ∀ ′ f ). Note that these satisfy: ∃ f ♦ X ∼ = ♦ Y Σ f ; ∀ f ⊓ ⊔ X ∼ = ⊓ ⊔ Y Π f (23) (and similarly for ∃ ′ f and ∀ ′ f ). 14 3. The canonical functors C /X → M X send f : T → X to ∃ f ⊤ T ∼ = ♦ X Σ f i X ⊤ T ∼ = ♦ X Σ f 1 T ∼ = ♦ X c X f that is factor through the corresp onding ones for P X . Thu s, they ha v e the functors k X i X : M X → C / X as fully faithful righ t adjoints (and similarly fo r M ′ X ; w e leav e it to the reader to c hec k the ab o v e fa ctorization for the arrow mapping). The fact that the adjunctions ♦ X ⊣ i X and ♦ ′ X ⊣ i ′ X satisfy the mixed F rob enius laws implies a restricted form of the F rob enius law fo r each of them and also f or ∃ f ⊣ f · and ∃ ′ f ⊣ f ′ · , whic h will b e used in the sequel: 5.6. Proposition. [restricted F rob enius laws] F or any X ∈ C , ther e ar e natur al isomorphisms: ♦ X ( P × X i X j X B ) ∼ = ♦ X P ∧ X j X B F or any f : X → Y in C , ther e ar e n atur al is o m orphisms: ∃ f ( M ∧ X f · j Y B ) ∼ = ∃ f M ∧ X j Y B ; ∃ ′ f ( N ∧ ′ X f ′ · j ′ Y B ) ∼ = ∃ ′ f N ∧ ′ X j ′ Y B Proo f. F or the first one, b y the mixed F rob enius law w e get: ♦ X ( P × X i X j X B ) ∼ = ♦ X ( P × X i ′ X j ′ X B ) ∼ = ♦ X ( i X ♦ X P × X i ′ X j ′ X B ) ∼ = ∼ = ♦ X ( i X ♦ X P × X i X j X B ) ∼ = ♦ X i X ( ♦ X P ∧ X j X B ) ∼ = ♦ X P ∧ X j X B F or the second one, we then hav e: ∃ f ( M ∧ X f · j Y B ) ∼ = ♦ Y Σ f i X ( M ∧ X f · j Y B ) ∼ = ∼ = ♦ Y Σ f ( i X M × X i X f · j Y B ) ∼ = ♦ Y Σ f ( i X M × X f ∗ i Y j Y B ) ∼ = ∼ = ♦ Y (Σ f i X M × Y i Y j Y B ) ∼ = ( ♦ Y Σ f i X M ) ∧ Y j Y B ∼ = ∃ f M ∧ Y j Y B 6. F unctors v al ued in trut h v alues In the sequel, a ma jor role will b e play ed b y the “truth v a lues” category B 1. W e denote b y true its terminal o b ject, so that j 1 true ∼ = ⊤ 1 ; j ′ 1 true ∼ = ⊤ ′ 1 The functors X ∗ b 1 : B 1 → P X can b e facto r ized in v arious w a ys: i X X · j 1 = X ∗ i 1 j 1 = X ∗ i ′ 1 j ′ 1 = i ′ X X ′ · j ′ 1 15 (where X denotes also the map X → 1). Th us their left and righ t adjoints can b e factorized as: j − 1 1 ∃ X ♦ X ∼ = j − 1 1 ♦ 1 Σ X ∼ = j ′ 1 − 1 ♦ ′ 1 Σ X ∼ = j ′ 1 − 1 ∃ ′ X ♦ ′ X (24) j − 1 1 ∀ X ⊓ ⊔ X ∼ = j − 1 1 ⊓ ⊔ 1 Π X ∼ = j ′ 1 − 1 ⊓ ⊔ ′ 1 Π X ∼ = j ′ 1 − 1 ∀ ′ X ⊓ ⊔ ′ X (25) W e refer to (any one of ) these a s the “co end” and “end” functors a t X , resp ectiv ely: co end X ⊣ X ∗ b 1 ⊣ end X : P X → B 1 This terminolog y is justified b y the fact that, for a bifunctor H : X op × X → Set , one can easily construct an ob ject h of Cat /X suc h that end X h giv es the usual end of H , while co end X h g iv es the co end of H in the sense of strong dinaturality , whic h in most relev an t cases reduces to the usual one as w ell (see [Pisani, 2007]). Next w e define the functors meets X : P X × P X → B 1 ; hom X : ( P X ) op × P X → B 1 meets X ( P , Q ) := coend X ( P × Q ) ; hom X ( P , Q ) := end X ( P ⇒ Q ) and their restrictions ten X : M ′ X × M X → B 1 nat X : ( M X ) op × M X → B 1 ; na t ′ X : ( M ′ X ) op × M ′ X → B 1 ten X ( N , M ) := meets X ( i ′ X N , i X M ) nat X ( L, M ) := hom X ( i X L, i X M ) ; nat ′ X ( N , O ) := hom X ( i ′ X N , i ′ X O ) F or instance, in the temp oral do ctrine of p osets B 1 ∼ = P 1 ∼ = { true , false } and meets X ( P , Q ) = true iff P and Q ha v e a non-empty in tersection (and similarly for ten X ( N , M )). Of course, hom X ( P , Q ) = true iff P ⊆ Q (and similarly for nat X ( L, M ) and nat ′ X ( N , O )). In the temp oral do ctrine of reflexiv e g r aphs, P 1 ∼ = Grph while B 1 ∼ = Set . Note that hom X and ten X are v alued in B 1 rather than in P 1 as in Section 3 , so that the notation is in fact consisten t only for the first example in 4.1. 6.1. The e nriched “adjunction” la ws. In the follow ing pro po sition, w e show that the adjunctions whic h define a temp oral do ctrine can b e in ternalized, t hat is they are enric hed in the truth v alues catego ry B 1. F urthermore, some of them ha v e an exact coun terpart in a similar la w, with the “meets” or “tensor” functors in place of the “hom” or “nat” functors; the pro o fs a re also nicely symmetrical. 16 6.2. Proposition. The functors hom X , nat X and nat ′ X enrich P X , M X a nd M ′ X r esp e ctively in B 1 and, for any s p ac e X ∈ C or map f : X → Y , ther e ar e natur al isomorphisms: hom X ( f ∗ Q, P ) ∼ = hom Y ( Q, Π f P ) ; hom X ( P , f ∗ Q ) ∼ = hom Y (Σ f P , Q ) (26) meets X ( f ∗ Q, P ) ∼ = meets Y ( Q, Σ f P ) ; meets X ( P , f ∗ Q ) ∼ = meets Y (Σ f P , Q ) (27) nat X ( M , ⊓ ⊔ X P ) ∼ = hom X ( i X M , P ) ; nat ′ X ( N , ⊓ ⊔ ′ X P ) ∼ = hom X ( i ′ X N , P ) (28) nat X ( ♦ X P , M ) ∼ = hom X ( P , i X M ) ; nat ′ X ( ♦ ′ X P , N ) ∼ = hom X ( P , i ′ X N ) (29) ten X ( N , ♦ X P ) ∼ = meets X ( i ′ X N , P ) ; ten X ( ♦ ′ X P , M ) ∼ = meets X ( P , i X M ) (30) nat X ( f · M , L ) ∼ = nat Y ( M , ∀ f L ) ; nat ′ X ( f ′ · O , N ) ∼ = nat ′ Y ( O , ∀ ′ f N ) (31) nat X ( L, f · M ) ∼ = nat Y ( ∃ f L, M ) ; nat ′ X ( N , f ′ · O ) ∼ = nat ′ Y ( ∃ ′ f N , O ) (32) ten X ( f ′ · N , L ) ∼ = ten Y ( N , ∃ f L ) ; ten X ( N , f · M ) ∼ = ten Y ( ∃ ′ f N , M ) (33) Proo f. F or the first part , see Prop osition 2.2 . F or (26) and (27 ), recall Corollar y 3.1 and note that the presen t “hom” and “meets” functors factor through the ones there. Equations (28), (29) and (30) follo w from Prop osition 5.2 and the other factorizations of the co end and the end f unctor s in (24) and (25). Recalling (22), w e obtain the remaining ones by comp osition of (enric hed) a djoin ts. Alternativ ely , one can explicitly derive them as w e exemplify for (32) : j − 1 1 ⊓ ⊔ 1 Π X ( i X L ⇒ i X f · M ) ∼ = j − 1 1 ⊓ ⊔ 1 Π Y Π f ( i X L ⇒ f ∗ i Y M ) ∼ = j − 1 1 ⊓ ⊔ 1 Π Y (Σ f i X L ⇒ i Y M ) ∼ = j ′ − 1 1 ∀ ′ Y ⊓ ⊔ ′ Y (Σ f i X L ⇒ i Y M ) ∼ = j ′ − 1 1 ∀ ′ Y ⊓ ⊔ ′ Y ( i Y ♦ Y Σ f i X L ⇒ i Y M ) 7. Li m its, colim i ts and Y oneda pr op erties As w e will see in Section 10, most of the la ws in Prop osition 6.2 (namely those not contain- ing hom ) still hold for w eak temp oral do ctrines, whic h include the motiv ating instance h Set X op → Cat /X ← Set X ; X ∈ Cat i . Th us, with t he same tec hnique exp loited in Section 3, w e b egin to dra w some consequences whic h in fact hold in the w eaker con text as w ell. Accordingly , we mainly main tain the p olicy of using terms whic h reflect the case C = Cat j ust men tioned. W e define the (in ternal) “limit” and “colimit” functors by restricting the end and t he co end functors to (left or rig h t) closed parts: lim X := end X i X ∼ = j − 1 1 ∀ X : M X → B 1 lim ′ X := end X i ′ X ∼ = j ′ 1 − 1 ∀ ′ X : M ′ X → B 1 colim X := co end X i X ∼ = j − 1 1 ∃ X : M X → B 1 colim ′ X := co end X i ′ X ∼ = j ′ 1 − 1 ∃ ′ X : M ′ X → B 1 17 7.1. Coro llar y. [final maps preserv e limits] L et f : X → Y b e a fina l map, that is ∃ f ⊤ X ∼ = ⊤ Y . Then lim X ( f · M ) ∼ = lim Y M ; colim ′ X ( f ′ · N ) ∼ = colim ′ Y N (34) Dual ly, if f : X → Y is initial, that is ∃ ′ f ⊤ ′ X ∼ = ⊤ ′ Y , then colim X ( f · M ) ∼ = colim Y M ; lim ′ X ( f ′ · N ) ∼ = lim ′ Y N Proo f. F or the first one of (34), using (32) we hav e: lim X ( f · M ) ∼ = nat X ( ⊤ X , f · M ) ∼ = nat Y ( ∃ f ⊤ X , M ) ∼ = nat Y ( ⊤ Y , M ) ∼ = lim Y M F or the second one of (34), w e follo w exactly the same pattern using (33) instead: colim ′ X ( f ′ · N ) ∼ = ten X ( f ′ · N , ⊤ X ) ∼ = ten Y ( N , ∃ f ⊤ X ) ∼ = ten Y ( N , ⊤ Y ) ∼ = colim ′ Y N The other ones a re prov ed in the same w ay . There are three canonical wa ys of obtaining a truth v a lue in B 1 from a closed part, namely the limit or the colimit functors and “ev aluation” at a p oint x : 1 → X : lim X M ; colim X M ; j − 1 1 ( x · M ) lim ′ X N ; colim ′ X N ; j ′ 1 − 1 ( x ′ · N ) In the ab ov e prop osition, w e hav e used the fact that limits and colimits ov er X are “represen ted” (by ⊤ X or ⊤ ′ X ). Now w e sho w that the same is true fo r ev aluatio n; na mely , ev a luation at x is “represen ted” by the left and right “slices”: X/x := ∃ x ⊤ 1 ; x \ X := ∃ ′ x ⊤ ′ 1 (35) Note that slices can b e obtained as the (left or righ t) closure of the “singletons” { x } = Σ x 1 1 = c X x (see Section 3): X/x = ♦ X { x } ; x \ X = ♦ ′ X { x } 7.2. Coro llar y. [Y o neda pro p erties] Given a p oint x : 1 → X in C , ther e ar e isomorphisms j − 1 1 ( x · M ) ∼ = nat X ( X/x, M ) ; j − 1 1 ( x · M ) ∼ = ten X ( x \ X, M ) natur al in P ∈ P X (and dual ly for right close d p arts). Proo f. nat X ( ∃ x ⊤ 1 , M ) ∼ = nat 1 ( ⊤ 1 , x · M ) ∼ = lim 1 ( x · M ) ∼ = j − 1 1 ∀ 1 ( x · M ) ∼ = j − 1 1 ( x · M ) ten X ( ∃ ′ x ⊤ ′ 1 , M ) ∼ = ten 1 ( ⊤ ′ 1 , x · M ) ∼ = colim 1 ( x · M ) ∼ = j − 1 1 ∃ 1 ( x · M ) ∼ = j − 1 1 ( x · M ) 18 7.3. Coro llar y. [form ulas for quan tifications, in terior and closure] Given a m ap f : X → Y an d a p oin t y : 1 → Y in C , ther e ar e isomo rp hisms j − 1 1 ( y · ∀ f M ) ∼ = nat X ( f · Y /y , M ) ; j − 1 1 ( y · ∃ f M ) ∼ = ten X ( f ′ · y \ Y , M ) natur al in M ∈ M X (and dual ly for right close d p arts). T h e r e ar e isomorph isms j − 1 1 ( x · ⊓ ⊔ X P ) ∼ = hom X ( i X X/x, P ) ; j − 1 1 ( x · ♦ X P ) ∼ = meets X ( i ′ X x \ X, P ) natur al in P ∈ P X (and dual ly for right closur e). Proo f. Using Corollary 7 .2 and (31), (33), (28) and (30) resp ectiv ely , w e g et: j − 1 1 ( y · ∀ f M ) ∼ = nat Y ( Y /y , ∀ f M ) ∼ = nat X ( f · Y /y , M ) j − 1 1 ( y · ∃ f M ) ∼ = ten Y ( y \ Y , ∃ f M ) ∼ = ten X ( f ′ · y \ Y , M ) j − 1 1 ( x · ⊓ ⊔ X P ) ∼ = nat X ( X/x, ⊓ ⊔ X P ) ∼ = hom X ( i X X/x, P ) j − 1 1 ( x · ♦ X P ) ∼ = ten X ( x \ X, ♦ X P ) ∼ = meets X ( i ′ X x \ X, P ) 8. Ex ploiti ng compr ehen si on In this section and in the next one w e presen t some consequence s of t he comprehension adjunction c X ⊣ k X : P X → C /X a nd of the assumption t ha t it is fully faithful. 8.1. The components functor. W e define t he “comp onen ts” functor π 0 : C → B 1 b y: π 0 X := coend X 1 X = colim X ⊤ X = colim ′ X ⊤ ′ X 8.2. Remarks. Note tha t X is connected, that is π 0 X ∼ = true , iff X → 1 is final (or initial). Note also that the comp onents f unctor π 0 X = coend X 1 X = j − 1 1 ♦ 1 Σ X 1 X = j − 1 1 ♦ 1 c 1 X is left adjoin t to the full inclus ion k 1 i 1 j 1 : B 1 → C . Coheren tly , w e sa y that a space X ∈ C is “discre te” if X ∼ = k 1 i 1 j 1 V , for a truth v alue V ∈ B 1, so that π 0 yields in fact t he reflection in discrete spaces. Con vers ely , the co end functor can b e reduced to comp onents o r t o a colimit by co end X P ∼ = π 0 X ! k X P ∼ = colim X ! k X P ⊤ X ! k X P (36) Indeed w e hav e: co end X P ∼ = j − 1 1 ♦ 1 Σ X P ∼ = j − 1 1 ♦ 1 Σ X c X k X P ∼ = j − 1 1 ♦ 1 Σ X Σ k X P 1 X ! k X P ∼ = ∼ = j − 1 1 ♦ 1 Σ X ! k X P 1 X ! k X P ∼ = co end X ! k X P 1 X ! k X P ∼ = π 0 X ! k X P 19 8.3. The limit and colimit f ormulas for na t and ten. Since co end X can b e reduced to a colimit b y (36), the same is true for the functors meets X and ten X . In fact w e can do b etter, b y reducing ten X ( N , M ) to a colimit ov er the space corresp onding t o i ′ X N (or to i X M ), rather than to i ′ X N × X i X M ; similarly , nat X can be reduced to a limit: 8.4. Proposition. [(co)limit form ulas for ten and nat] meets X ( P , Q ) ∼ = co end X ! k X Q ( k X Q ) ∗ P ∼ = co end X ! k X P ( k X P ) ∗ Q (37) ten X ( N , M ) ∼ = colim X ! k X i ′ X N ( k X i ′ X N ) · M ∼ = colim ′ X ! k X i X M ( k X i X M ) ′ · N (38) hom X ( P , Q ) ∼ = end X ! k X Q ( k X Q ) ∗ P ∼ = end X ! k X P ( k X P ) ∗ Q (39) nat X ( L, M ) ∼ = lim X ! k X i X L ( k X i X L ) · M ; na t ′ X ( N , O ) ∼ = lim ′ X ! k X i ′ X N ( k X i ′ X N ) · O (40) Proo f. F or (37), by applying (27) of Prop osition 6.2 we get meets X ( P , Q ) ∼ = meets X ( P , Σ k X Q 1 X ! k X Q ) ∼ = ∼ = meets X ! k X Q (( k X Q ) ∗ P , 1 X ! k X Q ) ∼ = co end X ! k X Q ( k X Q ) ∗ P F or (38), w e then hav e: ten X ( N , M ) ∼ = meets X ( i ′ X N , i X M ) ∼ = co end X ! k X i ′ X N ( k X i ′ X N ) ∗ i X M ∼ = ∼ = co end X ! k X i ′ X N i X ! k X i ′ X N ( k X i ′ X N ) · M ∼ = colim X ! k X i ′ X N ( k X i ′ X N ) · M Similarly for (3 9), by applying t he second of (26) instead w e get: hom X ( P , Q ) ∼ = hom X (Σ k X P 1 X ! k X P , Q ) ∼ = ∼ = hom X ! k X P (1 X ! k X P , ( k X P ) ∗ Q ) ∼ = end X ! k X P ( k X P ) ∗ Q Finally , for (40) we hav e: nat X ( L, M ) ∼ = hom X ( i X L, i X M ) ∼ = end X ! k X i X L ( k X i X L ) ∗ i X M ∼ = ∼ = end X ! k X i X L i X ! k X i X L ( k X i X L ) · M ∼ = lim X ! k X i X L ( k X i X L ) · M By applying (38) or (40) to the fo r m ulas for quan tifications along a map f : X → Y of Corollary 7.3, we obtain a colimit and a limit form ula for ev aluation a t y : 1 → Y of ∃ f M (or ∃ ′ f N ) and ∀ f M (or ∀ ′ f N ), resp ectiv ely . Sa y that a part P ∈ P X is “left dense” if its left closure is terminal: ♦ X P ∼ = ⊤ X ; righ t densit y is of course defined dually . 8.5. Proposition. A p art P ∈ P X is left den s e iff k X P is final. A map f : X → Y in C is final iff c Y f is left de nse in Y . A sp ac e X ∈ C is c onn e cte d iff Σ X 1 X is dense. Proo f. (Note that w e implicitly use the canonical bijection betw een the ob jects of C /X and maps in C with co domain X .) F or the first one we hav e: ♦ X P ∼ = ♦ X Σ k X P 1 X ! k X P ∼ = ♦ X Σ k X P i X ! k X P ⊤ X ! k X P ∼ = ∃ k X P ⊤ X ! k X P and for the second one: ∃ f ⊤ X ∼ = ♦ Y Σ f 1 X ∼ = ♦ Y c Y f F or the last one, see Remarks 8.2. 20 9. Th e sup and i nf reflecti o ns F or X ∈ C , let X (resp. X ′ ) b e the full sub category of M X (resp. M ′ X ) generated by the left (resp. right) slices (35 ), and denote the inclusion functors b y h X : X → M X ; h ′ X : X ′ → M ′ X The partially defined left adjoin ts to i X h X , i ′ X h ′ X , k X i X h X and k X i ′ X h ′ X are denoted resp ectiv ely b y: sup X : P X → X ; inf X : P X → X ′ ; Sup X : C /X → X ; Inf X : C /X → X ′ 9.1. Remark. Of course, Sup X f exis ts iff sup X c X f do es, and in that case they are t he same. Note also that t he sup (resp. inf ) of a pa rt dep ends only on its left (resp. right) closure: sup X P exists iff ♦ X P has a reflection in X . 9.2. Proposition. The fol lowing ar e e quivalent for a sp ac e X ∈ C : 1. X has a final p oint x : 1 → X ; 2. ther e is a left dens e p art of X with a sup; 3. ther e is a final map T → X with a Sup; 4. id X : X → X has a Sup. Proo f. By Prop osition 8.5 and Remark 9.1, the last three conditions are equiv alen t. Since any p oin t x : 1 → X has the slice X/x as its Sup, 1) implies 3). Supp ose con v ersely that a left dense par t P ∈ P X has a sup; then b y Remark 9.1 ♦ X P ∼ = ⊤ X has a reflection X/x in X . By general w ell-kno wn facts ab out reflections, it follo ws that also X/x ∼ = ⊤ X , that is the p oint x : 1 → X is final. 9.3. Proposition. [final maps preserv e sups] L et t : S → T b e a fi n al map in C ; then f : T → X has a Sup iff f t : S → X do es, and in that c ase they c oin cide. Proo f. By Remark 9.1, it is enough to show that ♦ X c X f t ∼ = ♦ X c X f : ♦ X c X f t ∼ = ∃ f t ⊤ S ∼ = ∃ f ∃ t ⊤ S ∼ = ∃ f ⊤ T ∼ = ♦ X c X f 21 Note that t he sup (resp. inf ) reflections giv e, for the w eak temporal doctrine of cat- egories, the colimit (resp. limit) of the corresp onding functor (see also [Pisani, 20 0 7] and [Pisani, 2008]). Th us the ab ov e prop osition can b e seen a s the “external” corr espec- tiv e of Corollary 7 .1. 10. The logi c o f ca tegori es No w we w eake n the axioms of temp oral do ctrine so to include the motiv ating instance h Set X op → Cat /X ← Set X ; X ∈ Cat i and sho w that most o f t he la ws in Prop osition 6.2 still hold. 10.1. Weak te mporal doctrines. W eak temp oral do ctrines are defined as temp oral do ctrines except that: 1. W e do not require the existence of the Π f , rig h t adjoint t o f ∗ ; instead w e do require, for an y map f : X → Y , the existence of ∀ f : M X → M Y ; ∀ ′ f : M ′ X → M ′ Y righ t adjoint to f · and f ′ · resp ectiv ely . 2. W e do not assume that the catego r ies o f parts P X are cartesian closed, but only that they are cartesian and that the left or rig h t closed parts are exp onen tiable therein, that is i X M ⇒ P ; i ′ X N ⇒ P alw ays exist in P X . In a weak temp oral do ctrine w e still ha ve the co end X and end X functors a s in Section 6, left and right adjoint t o: i X X · j 1 = X ∗ i 1 j 1 = X ∗ i ′ 1 j ′ 1 = i ′ X X ′ · j ′ 1 with the difference that, among the factorizatons of end X in (25), only j − 1 1 ∀ X ⊓ ⊔ X ∼ = j ′ 1 − 1 ∀ ′ X ⊓ ⊔ ′ X : P X → B 1 (41) are alw a ys a v a ilable. W e also define meets X : P X × P X → B 1 ; hom X : ( P X ) op × P X → B 1 ten X : M ′ X × M X → B 1 nat X : ( M X ) op × M X → B 1 ; na t ′ X : ( M ′ X ) op × M ′ X → B 1 as in Section 6, but now the hom X ma y b e only partially defined, dep ending on whether the exp onen tial P ⇒ Q exists or do es not exist. On the con trary , the ab ov e a xiom on exp o nen tials assures that the na t X are alw ays defined. 22 10.2. Proposition. The functors nat X and nat ′ X enrich M X and M ′ X r esp e ctively in B 1 and, for any sp ac e X ∈ C or map f : X → Y , ther e ar e natur al isom orphisms: meets X ( f ∗ Q, P ) ∼ = meets Y ( Q, Σ f P ) ; meets X ( P , f ∗ Q ) ∼ = meets Y (Σ f P , Q ) (42) ten X ( N , ♦ X P ) ∼ = meets X ( i ′ X N , P ) ; ten X ( ♦ ′ X P , M ) ∼ = meets X ( P , i X M ) (43) nat X ( f · M , L ) ∼ = nat Y ( M , ∀ f L ) ; nat ′ X ( f ′ · O , N ) ∼ = nat ′ Y ( O , ∀ ′ f N ) (44) nat X ( L, f · M ) ∼ = nat Y ( ∃ f L, M ) ; nat ′ X ( N , f ′ · O ) ∼ = nat ′ Y ( ∃ ′ f N , O ) (45) ten X ( f ′ · N , L ) ∼ = ten Y ( N , ∃ f L ) ; ten X ( N , f · M ) ∼ = ten Y ( ∃ ′ f N , M ) (46) Proo f. Of course, the pro ofs of (42), (43) and (46) are as in Prop osition 6.2. F or (44) , b y a pplying Prop osition 2.2 to f · ⊣ ∀ f and recalling that expo nen tials in M X are giv en b y ⊓ ⊔ X ( i X L ⇒ i X M ), w e ha v e nat X ( f · M , L ) ∼ = end X ( i X f · M ⇒ i X L ) ∼ = j − 1 1 ∀ X ⊓ ⊔ X ( i X f · M ⇒ i X L ) ∼ = ∼ = j − 1 1 ∀ Y ∀ f ⊓ ⊔ X ( i X f · M ⇒ i X L ) ∼ = j − 1 1 ∀ Y ⊓ ⊔ Y ( i Y M ⇒ i Y ∀ f L ) ∼ = nat Y ( M , ∀ f L ) F or (45), since we ha v e adjunctions (with parameter L ∈ M X ) ∃ f ( L ∧ X X · j 1 V ) ⊣ nat X ( L, f · M ) ; ∃ f L ∧ Y Y · j 1 V ⊣ nat Y ( ∃ f L, M ) and since, by the restricted F r ob enius law of Prop osition 5 .6, ∃ f ( L ∧ X X · j 1 V ) ∼ = ∃ f ( L ∧ X f · Y · j 1 V ) ∼ = ∃ f L ∧ Y Y · j 1 V the result fo llows fro m Lemma 2 .1. 10.3. Proposition. Whenever they ar e define d, ther e ar e natur al isom orphisms: hom X ( f ∗ Q, P ) ∼ = hom Y ( Q, Π f P ) ; hom X ( P , f ∗ Q ) ∼ = hom Y (Σ f P , Q ) (47) nat X ( M , ⊓ ⊔ X P ) ∼ = hom X ( i X M , P ) ; nat ′ X ( N , ⊓ ⊔ ′ X P ) ∼ = hom X ( i ′ X N , P ) (48) nat X ( ♦ X P , M ) ∼ = hom X ( P , i X M ) ; nat ′ X ( ♦ ′ X P , N ) ∼ = hom X ( P , i ′ X N ) (49) Proo f. All of them can b e prov ed as (4 5) b y sho wing that their left adjoints are isomorphic; this r equires the F r o benius la w for the second of (47) and the restricted F rob enius la w for (49). W e lea ve them as an exercise to the reader. 23 Since all w e hav e pro ved in section 7, 8 and 9 (exc ept for (39) o f Prop osition 8.4, wherein hom X is only partially defined) depends o nly on the la ws in Prop osition 10.2 and on the comprehe nsion axiom, those results hold in an y weak temp oral do ctrine. In particular w e g et, for “generalized categories”, the Y oneda pro perties, the form ulas for quan tificatio ns (or “Kan extensions”), the form ulas for the (co)reflection in (left or rig ht) “closed part s” (or “actions”), and t he prop erties of final o r initial maps with respect to (co)limits (b oth “externally” and “in ternally”). Th us w e man ta in that the logic of w eak temp oral do ctrines we ll deserv es to b e called “a logic for categor ies”, in fact 1. b eing summarized b y a few adjunction-lik e laws , it lends itself to effectiv e and transparen t calculations; furthermore, a long with the ob vious “left-right” symmetry , there is a fa r more in teresting sort of duality: in man y cases laws and pro ofs on the “hom-side” corresp ond exactly to tho se on the “tensor-side”; 2. this calculus allows one to easily derive some ba sic non-trivial categorical facts; 3. it is “ autonomous”, providing its ow n truth v alues; 4. suitable natural strenghtenin gs or w eak enings can b e considered, so to o bta in more refined prop erties or a wider range of applications; some o f them will b e considered in a forthcoming work. References S. Ghilardi, G. Meloni (199 1), R elational and T op olo gic al Semantics for T emp or al and Mo dal Pr e dic ative L o gic s , A t t i del Congresso: Nuo vi problemi della Logica e della Filosofia della Scienza (Via r egg io), CLUEB Bologna, 59- 7 7. P .T. Johnstone (2002), Sk etches of an Elephan t: A T op os Theory Comp endium, Oxford Science Publications. D.M. Kan (195 8 ), A djoint F unctors , T rans. Amer. Math. So c. 87 , 294 - 329. F.W. La wv ere (1970), Equality in Hyp er do ctrines and the Compr ehensio n S cheme as an A djoint F unctor , Pro ceedings of the AMS Symp osium on Pure Mathematics, XVI I, 1-14. F.W. La wv ere (1989), Qualitative Dis tinctions b etwe en some T op ose s of Gener aliz e d Gr aphs , Pro ceedings o f the AMS Symp osium on Categories in Computer Science and Logic, Con temp orary Mathematics, v ol. 92, 261 - 299. C. Pisani (2007), Comp onen ts, Complemen ts and the Reflection F orm ula, The ory and Appl. Cat. 19 , 19- 40. C. Pisani (20 0 8), Balanced Category Theory , The ory and Appl. Cat. 20 , 85 -115. 24 C. Pisani (20 0 9), Balanced Category Theory I I, preprin t, arXiv:math.CT/0904.1 7 90v3. via Giob erti 86, 10128 T orino, Italy. Email: pisclau@ya hoo.it