Topos Theory Olivia Caramello Introduction Interpreting logic in categories First-order logic First-order languages First-order theories Categorical semantics Topos Theory Classes of ‘logical’ categories The interpretation of logic in categories The interpretation of formulae Examples Soundness and completeness Olivia Caramello Toposes as mathematical universes The internal language Kripke-Joyal semantics For further reading Topos Theory Interpreting first-order logic in categories Olivia Caramello Introduction • In Logic, first-order languages are a wide class of formal Interpreting logic in categories languages used for talking about mathematical structures of First-order logic any kind (where the restriction ‘first-order’ means that First-order languages quantification is allowed only over individuals rather than over First-order theories collections of individuals or higher-order constructions on Categorical semantics them). Classes of ‘logical’ categories • A first-order language contains sorts, which are meant to The interpretation of formulae represent different kinds of individuals, terms, which denote Examples individuals, and formulae, which make assertions about the Soundness and completeness individuals. Compound terms and formulae are formed by Toposes as mathematical using various logical operators. universes • It is well-known that first-order languages can always be The internal language interpreted in the context of (a given model of) set theory. In Kripke-Joyal semantics this lecture, we will show that these languages can also be For further reading meaningfully interpreted in a category, provided that the latter possesses enough categorical structure to allow the interpretation of the given fragment of logic. In fact, sorts will be interpreted as objects, terms as arrows and formulae as subobjects, in a way that respects the logical structure of compound expressions. 2 / 40 Topos Theory Signatures Olivia Caramello Introduction Interpreting logic in categories Definition First-order logic First-order A first-order signature Σ consists of the following data. languages First-order theories a) A set Σ-Sort of sorts. Categorical semantics b) A set Σ-Fun of function symbols, together with a map Classes of ‘logical’ categories assigning to each f 2 Σ-Fun its type, which consists of a finite The interpretation of formulae non-empty list of sorts: we write Examples Soundness and completeness f : A1 ···An ! B Toposes as mathematical universes to indicate that f has type A1;:::;An;B (if n = 0, f is called a The internal constant of sort B). language Kripke-Joyal semantics c) A set Σ-Rel of relation symbols, together with a map assigning For further to each Σ-Rel its type, which consists of a finite list of sorts: reading we write R A1 ···An to indicate that R has type A1;:::An. 3 / 40 Topos Theory Terms Olivia Caramello Introduction Interpreting logic in categories First-order logic First-order For each sort A of a signature Σ we assume given a supply of languages First-order theories variables of sort A, used to denote individuals of kind A. Categorical Starting from variables, terms are built-up by repeated semantics Classes of ‘logical’ ‘applications’ of function symbols to them, as follows. categories The interpretation of formulae Definition Examples Let Σ be a signature. The collection of terms over Σ is defined Soundness and completeness recursively by the clauses below; simultaneously, we define the Toposes as sort of each term and write t : A to denote that t is a term of sort mathematical universes A. The internal language Kripke-Joyal a) x : A, if x is a variable of sort A. semantics For further b) f (t1;:::;tn): B if f : A1 ···An ! B is a function symbol and reading t1 : A1;:::;tn : An. 4 / 40 Topos Theory Formation rules for formulaeI Olivia Caramello Introduction Consider the following formation rules for recursively building Interpreting logic in categories classes of formulae F over Σ, together with, for each formula f, First-order logic the (finite) set FV (f) of free variables of f. First-order languages First-order theories (i) Relations: R(t1;:::;tn) is in F, if R A1 ···An is a relation Categorical symbol and t1 : A1;:::;tn : An are terms; the free variables of semantics Classes of ‘logical’ this formula are all the variables occurring in some ti . categories The interpretation of formulae (ii) Equality: (s = t) is in F if s and t are terms of the same sort; Examples FV (s = t) is the set of variables occurring in s or t (or both). Soundness and completeness (iii) Truth: > is in F; FV (>) = /0. Toposes as mathematical (iv) Binary conjunction: (f ^ y) is in F, if f and y are in F; universes The internal FV (f ^ y) = FV (f) [ FV (y). language Kripke-Joyal semantics (v) Falsity: ? is in F; FV (?) = /0. For further (vi) Binary disjunction: (f _ y) is in F, if f and y are in F; reading FV (f _ y) = FV (f) [ FV (y). (vii) Implication: (f )y) is in F, if f and y are in F; FV (f )y) = FV (f) [ FV (y). (viii) Negation: :f is in F, if f is in F; FV (:f) = FV (f). 5 / 40 Topos Theory Formation rules for formulaeII Olivia Caramello Introduction Interpreting logic in (ix) Existential quantification: (9x)f is in F, if f is in F and x is a categories First-order logic variable; FV ((9x)f) = FV (f) n fxg. First-order languages (x) Universal quantification: (8x)f is in F, if f is in F and x is a First-order theories variable; FV ((8x)f) = FV (f) n fxg. Categorical semantics Classes of ‘logical’ (xi) Infinitary disjunction: _fi is in F, if I is a set, fi is in F for categories i2I The interpretation of formulae Examples each i 2 I and FV (_fi ) := [FV (fi ) is finite. i2I i2I Soundness and completeness (xii) Infinitary conjunction: ^fi is in F, if I is a set, fi is in F for Toposes as i2I mathematical universes each i 2 I and FV (^fi ) := [FV (fi ) is finite. The internal language i2I i2I Kripke-Joyal semantics A context is a finite list ~x = x1;:::;xn of distinct variables (the For further empty context, for n = 0 is allowed and indicated by []). reading Notation: We will often consider formulae-in-context, that is formulae f equipped with a context ~x such that all the free variables of f occur among ~x; we will write either f(~x) or f~x : fg. 6 / 40 Topos Theory Classes of formulae Olivia Caramello Introduction Definition Interpreting logic in categories In relation to the above-mentioned forming rules: First-order logic • The set of atomic formulae over Σ is the smallest set closed First-order languages under Relations and Equality). First-order theories • The set of Horn formulae over Σ is the smallest set containing Categorical semantics the class of atomic formulae and closed under Truth and Binary Classes of ‘logical’ categories conjunction. The interpretation of formulae • The set of regular formulae over Σ is the smallest set containing Examples the class of atomic formulae and closed under Truth, Binary Soundness and completeness conjunction and Existential quantification. Toposes as • The set of coherent formulae over Σ is the smallest set mathematical universes containing the set of regular formulae and closed under False The internal language and Binary disjunction. Kripke-Joyal semantics • The set of first-order formulae over Σ is the smallest set closed For further under all the forming rules except for the infinitary ones. reading • The class of geometric formulae over Σ is the smallest class containing the class of coherent formulae and closed under Infinitary disjunction. • The class of infinitary first-order formulae over Σ is the smallest class closed under all the above-mentioned forming rules. 7 / 40 Topos Theory Sequents Olivia Caramello Introduction Interpreting logic in categories First-order logic Definition First-order languages First-order theories • By a sequent over a signature Σ we mean a formal Categorical expression of the form( f `~ y), where f and y are formulae semantics x Classes of ‘logical’ over Σ and ~x is a context suitable for both of them. The categories The interpretation of intended interpretation of this expression is that y is a logical formulae Examples consequence of f in the context ~x, i.e. that any assignment Soundness and of individual values to the variables in ~x which makes f true completeness Toposes as will also make y true. mathematical universes • We say a sequent (f `~x y) is Horn (resp. regular, coherent, The internal language ...) if both f and y are Horn (resp. regular, coherent, ...) Kripke-Joyal semantics formulae. For further reading Notice that, in full first-order logic, the general notion of sequent is not really needed, since the sequent (f `~x y) expresses the same idea as (> `[] (8~x)(f )y)). 8 / 40 Topos Theory First-order theories Olivia Caramello Introduction Interpreting logic in categories First-order logic First-order languages First-order theories Definition Categorical semantics • By a theory over a signature Σ, we mean a set T of sequents Classes of ‘logical’ categories over Σ, whose elements are called the (non-logical) axioms The interpretation of formulae of T. Examples Soundness and • We say that T is an algebraic theory if its signature Σ has a completeness single sort and no relation symbols (apart from equality) and Toposes as mathematical its axioms are all of the form > `~x f where f is an atomic universes ~ The internal formula (s = t) and x its canonical context. language Kripke-Joyal • We say is a Horn (resp. regular, coherent, ...) theory if all semantics T For further the sequents in T are Horn (resp.