A logic for categories

arXiv:1003.5552v1 [math.CT] 29 Mar 2010 A LOGIC FOR CATEGORIES CLAUDIO PISANI ABSTRACT. We present a doctrinal approach to category theory, obtained by abstract- ing from the indexed inclusion (via discrete fibrations and opfibrations) of left and of right actions of X ∈Cat in categories over X. Namely, a “weak temporal doctrine” consists essentially of two indexed functors with the same codomain such that the in- duced functors have both left and right adjoints satisfying some exactness conditions, in the spirit of categorical logic. The derived logical rules include some adjunction-like laws involving the truth-values- enriched hom and tensor functors, which condense several basic categorical properties and display a nice symmetry. The symmetry becomes more apparent in the slightly stronger context of “temporal doctrines”, which we initially treat and which include as an instance the inclusion of lower and upper sets in the parts of a poset, as well as the inclusion of left and right actions of a graph in the graphs over it. Contents 1 Introduction 1 2 Enriching adjunctions 5 3 The logic of hyperdoctrines 7 4 Temporal doctrines 9 5 Basic properties 12 6 Functors valued in truth values 14 7 Limits, colimits and Yoneda properties 16 8 Exploiting comprehension 18 9 The sup and inf reflections 20 10 The logic of categories 21 1. Introduction Let X be a set endowed with an equivalence relation ∼, and let VX be the poset of closed parts, that is those subsets V of X such that x ∈V and x ∼y implies y ∈V . A part P ∈PX has both a “closure” ♦P and an “interior” ⊓⊔P, that is the inclusion i : VX →PX has both a left and a right adjoint: ♦⊣i ⊣⊓⊔: PX →VX 2000 Mathematics Subject Classification: 18A05, 18A30, 18A40, 18D99. Key words and phrases: Temporal doctrine, internal hom and tensor, Frobenius law, adjunction-like laws, internal limits and colimits, quantification formulas. c⃝Claudio Pisani, 2010. Permission to copy for private use granted. 1 2 Thus the (co)reflection maps (inclusions) εP : i ⊓⊔P →P and ηP : P →i♦P induce bijections (between 0-elements or 1-elements sets): PX(iV, i ⊓⊔P) PX(iV, P) ; PX(i♦P, iV ) PX(P, iV ) By taking ⊓⊔(P ⇒Q) as a VX-enrichment of PX(P, Q), it turns out that the above adjunctions are also enriched in VX giving isomorphisms: ⊓⊔(iV ⇒i ⊓⊔P) ⊓⊔(iV ⇒P) ; ⊓⊔(i♦P ⇒iV ) ⊓⊔(P ⇒iV ) We also have the related laws: i♦(iV × iW) iV × iW ; i ⊓⊔(iV ⇒iW) iV ⇒iW ; ♦(i♦P × iV ) ♦(P × iV ) the first two of them saying roughly that closed parts are closed with respect to product (intersection) and exponentiation (implication). Given a groupoid X, the same laws hold for the inclusion of the actions of X in the groupoids over X (via “covering groupoids”). The above situation will be placed in the proper general context in sections 2 and 3, where we develope some technical tools concerning enriched adjunctions and apply them to hyperdoctrines [Lawvere, 1970]. Now, let us drop the symmetry condition on ∼, that is suppose that X is a poset; then we have the poset of lower-closed parts DX and that of upper-closed parts UX. Again, the inclusions i : DX →PX and i′ : UX →PX have both a left and a right adjoint: ♦⊣i ⊣⊓⊔: PX →DX ; ♦′ ⊣i′ ⊣⊓⊔′ : PX →UX While some of the above laws still hold “on each 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 way: ⊓⊔′ (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 laws (1) through (5) hold also for the inclusion of the left and the right actions of a category X in categories over X (via discrete fibrations and opfibrations): i : SetXop →Cat/X ; i′ : SetX →Cat/X and, when they make sense, also for the inclusion of open and closed parts in the parts of a topological space (or, more generally, of local homeomorphisms and proper maps to a space X in spaces over X; see [Pisani, 2009]). Abstracting from these situations, we may define a “temporal algebra” as a cartesian closed category with two reflective and coreflective full subcategories satisfying the above laws (in fact, it is enough to assume either (3) or (4) or (5)). A “temporal doctrine” is then essentially an indexed temporal algebra ⟨iX : MX →PX ←M′X : i′ X ; X ∈C⟩ such that the inclusions i1 and i′ 1 over the terminal object 1 ∈C are isomorphic. Temporal doctrines and their basic properties are presented in sections 4 and 5. In Section 6 we show how the “truth-values” M1 ∼= M′1 serve as values for an enriching of PX, MX and M′X in which the adjunctions Σf ⊣f ∗⊣Πf : PX →PY ♦X ⊣iX ⊣⊓⊔X : PX →MX ; ♦′ X ⊣i′ X ⊣⊓⊔′ X : PX →M′X ∃f ⊣f· ⊣∀f : MX →MY ; ∃′ f ⊣f ′· ⊣∀′ f : M′X →M′Y (6) are also enriched (where, for f : X →Y in C, ∃fM ∼= ♦Y ΣfiXM). For example, the temporal doctrine of posets is two-valued while that of reflexive graphs is Set-valued, by identifying sets with discrete graphs. If C = Cat, the f

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