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Topos Theory Olivia Caramello Introduction The categorical point of view The language of Category Theory Categories Functors Natural transformations Topos Theory Properties of functors Lectures 2-3-4: Categorical preliminaries Full and faithful functors Equivalence of categories Basic categorical Olivia Caramello constructions Universal properties Limits and colimits Adjoint functors The Yoneda Lemma Elementary toposes For further reading Topos Theory The categorical point of view Olivia Caramello Introduction • Category Theory, introduced by Samuel Eilenberg and Saunders The categorical point of view Mac Lane in the years 1942-45 in the context of Algebraic The language of Topology, is a branch of Mathematics which provides an abstract Category Theory Categories language for expressing mathematical concepts and reasoning Functors Natural about them. In fact, the concepts of Category Theory are transformations unifying notions whose instances can be found in essentially Properties of functors every field of Mathematics. Full and faithful functors • Equivalence of The underlying philosophy of Category Theory is to replace the categories primitive notions of set and belonging relationship between sets, Basic categorical constructions which constitute the foundations of Set Theory, with abstractions Universal of the notions of set and function, namely the concepts of object properties Limits and colimits and arrow. Adjoint functors • Since it was introduced, this approach has entailed a deep The Yoneda Lemma paradigmatic shift in the way Mathematicians could look at their Elementary subject, and has paved the way to important discoveries which toposes would have hardly been possible before. One of the great For further reading achievements of Category Theory is Topos Theory, a subject entirely written in categorical language. 2 / 37 Topos Theory The definition of category (1/2) Olivia Caramello Introduction The categorical point of view Definition The language of Category Theory A (small) category C consists of Categories Functors (i) a set Ob(C), Natural transformations (ii) for any a; b 2 Ob(C), a set HomC(a; b), Properties of functors (iii) for any a; b; c 2 Ob(C), a map: Full and faithful functors Equivalence of categories ◦C : HomC(a; b) × HomC(b; c) ! HomC(a; c) Basic categorical constructions called the composition and denoted by (f ; g) ! g ◦ f , Universal properties these data satisfying Limits and colimits Adjoint functors a) the composition ◦ is associative, i.e., for The Yoneda f 2 HomC(a; b); g 2 HomC(b; c) and h 2 HomC(c; d), we have Lemma (h ◦ g) ◦ f = h ◦ (g ◦ f ), Elementary toposes b) for each a 2 Ob(C), there exists ida 2 HomC(a; a) such that For further reading f ◦ ida = f for all f 2 HomC(a; b) and ida ◦ g = g for all g 2 HomC(b; a). 3 / 37 Topos Theory The definition of category (2/2) Olivia Caramello Introduction The categorical point of view • An element of Ob(C) is called an object of C. The language of Category Theory • Categories For a; b 2 ob(C), an element f of HomC(a; b) is called an Functors arrow (from a to b) in C; we say that a is the domain of f , b is Natural transformations the codomain of f , and we write f : a ! b, a = dom(f ) and Properties of functors b = cod(f ). Full and faithful functors • The arrow ida is called the identity arrow on a. Equivalence of categories Basic categorical Remark constructions Universal The concept of category has a first-order axiomatization, in a properties language having two sorts O and A (respectively for objects and Limits and colimits Adjoint functors arrows), two unary function symbols (for domain and codomain) The Yoneda A ! O, one unary function symbol O ! A (formalizing the Lemma concept of identity arrow) and a ternary predicate of type A Elementary toposes (formalizing the notion of composition of arrows). For further reading We will also consider large categories, that is categories with a proper class (rather than a set) of objects or arrows. 4 / 37 Topos Theory The dual category Olivia Caramello Introduction The categorical point of view The language of Category Theory The concept of category is self-dual i.e. the axioms in the Categories Functors definition of category continue to hold if we formally reverse the Natural transformations direction of arrows while mantaining the same objects. Properties of functors Definition Full and faithful op functors Given a category C, the dual category C is defined by setting Equivalence of categories op op Basic categorical Ob(C ) = Ob(C), HomC (a; b) = HomC(b; a), constructions op Universal and defining the composition g ◦Cop f of f 2 HomC (a; b) and properties op Limits and colimits f 2 HomC (b; c) by Adjoint functors The Yoneda g ◦Cop f = f ◦C g. Lemma Elementary opop toposes Note that C = C for any category C. For further reading 5 / 37 Topos Theory The duality principle Olivia Caramello Introduction The categorical point of view Every statement formulated in the language of Category Theory The language of has a dual, obtained by formally reversing the arrows and the Category Theory Categories order of composition of them. Functors Natural • A statement is true in a category C if and only if the dual transformations statement is true in the dual category Cop. Hence Properties of functors a statement is valid in all categories if and only if its dual is. Full and faithful functors • Equivalence of Anyway, two dual statements in the language of Category categories Theory, when interpreted in a given ‘concrete’ category, may Basic categorical constructions specialize to two very different-looking (and even Universal inequivalent!) mathematical statements. properties Limits and colimits • Sometimes, it is possible to lift ordinary mathematical Adjoint functors statements to the level of categories (or at least to classes of The Yoneda Lemma categories closed under duality) and obtain abstract proofs of Elementary them in the language of Category Theory; if this is the case, toposes For further one can then invoke the duality principle to derive dual reading versions of them which can be specialized to the original context. 6 / 37 Topos Theory Properties of arrows Olivia Caramello Introduction We can consider various properties of arrows in a category, The categorical point of view expressed in categorical language. An arrow f : a ! b is: The language of Category Theory • a monomorphism (or monic) if f ◦ g1 = f ◦ g2 implies g1 = g2 Categories Functors for all arrows g1; g2 : x ! a. Natural transformations • an epimorphism (or epic) if g1 ◦ f = g2 ◦ f implies g1 = g2 for Properties of functors all arrows g1; g2 : b ! x. Full and faithful functors • an isomorphism if there exists an arrow g : b ! a with Equivalence of categories f ◦ g = 1b and g ◦ f = 1a. Basic categorical constructions Notice that monomorphisms are dual to epimorphisms i.e. an Universal arrow f of a category C is a monomorphism in C if and only if it is properties op op Limits and colimits an epimorphism in C (regarded as an arrow in C ). Adjoint functors The Yoneda Example Lemma In the category Set, an arrow is: Elementary toposes • a monomorphism if and only if it is an injective function. For further reading • an epimorphism if and only if it is a surjective function. • an isomorphism if and only if it is a bijection. 7 / 37 Topos Theory Categories of mathematical objects Olivia Caramello Introduction The categorical point of view The language of Category Theory Important mathematical objects can be organized into categories. Categories Functors Examples Natural transformations Properties of • The category Set of sets and functions between them. functors Full and faithful • The category Top of topological spaces and continuous functors Equivalence of maps between them. categories Basic categorical • The category of Gr of groups and group homomorphisms, constructions the category Rng of rings and ring homomorphisms, the Universal properties category VectK of vector spaces over a field K and K -linear Limits and colimits maps between them, etc. Adjoint functors The Yoneda In fact, given a first-order theory T, we have a category Lemma T-mod(Set) having as objects the (set-based) models of T Elementary toposes and as arrows the structure-preserving maps between them. For further reading 8 / 37 Topos Theory Mathematical objects as categories Olivia Caramello Introduction The categorical point of view The language of Category Theory Categories Functors On the other hand, important mathematical objects arise as Natural transformations particular kinds of categories: Properties of • functors A set can be seen as a discrete category i.e. a category Full and faithful functors whose only arrows are the identity arrows. Equivalence of categories • A preorder can be seen as a preorder category i.e. a Basic categorical category having at most one arrow from one object to constructions Universal another. properties Limits and colimits • A monoid can be seen as a category with just one object. Adjoint functors • A groupoid is a category whose arrows are all isomorphisms; The Yoneda Lemma in particular, a group is a groupoid with just one object. Elementary toposes For further reading 9 / 37 Topos Theory Functors Olivia Caramello Introduction The categorical point of view Functors are the natural structure-preserving maps between The language of Category Theory categories. Categories Functors Natural Definition transformations Let C and C0 be two categories. A functor F : C!C0 consists of a Properties of 0 functors map F : Ob(C) ! Ob(C ) and of maps Full and faithful functors F : HomC(a; b) ! HomC0 (F(a); F(b)) for all a; b 2 C, such that Equivalence of categories • F(ida) = idF(a) for all a 2 C, Basic categorical constructions • F(g ◦ f ) = F(g) ◦ F(f ) for all f : a ! b; g : b ! c.
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