Topos Theory and the Connections between Category and Set Theory
Matthew Graham
Outline Topos Theory and the Connections between Why Category Theory? Category and Set Theory Definition
Basic Examples Matthew Graham Relations to Set Theory
Topos Definition and Examples March 13, 2008 Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Category and 1 Why Category Theory? Set Theory
Matthew Graham 2 Definition
Outline
Why Category Theory? 3 Basic Examples
Definition Basic 4 Relations to Set Theory Examples
Relations to Set Theory 5 Topos Definition and Examples Topos Definition and Examples
Motivation 6 Motivation and History of Topos theory and History of Topos theory Acknowledgements 7 Acknowledgements Topos Theory and the Connections between Why Category Theory? Category and Set Theory
Matthew Graham • “The fundamental idea of representing a function by an Outline arrow first appeared in topology around 1940” (Maclane) Why Category Theory? • Allows the abstraction of common structure. For example, Definition Groups, sets, topological spaces all have associativity Basic Examples intrinsic to them. Relations to • Unlike Set theory specifies the codomain of a function. Set Theory This allows it to distinguish the difference between an Topos Definition and identity map and an inclusion. Examples
Motivation • Leads to a different foundation for mathematics via Topos and History of Topos theory Theory, which allows one to explicitly construct different
Acknowledgements types of logic. Topos Theory and the Connections between Category and • In particular there exist “Intuitionist” logics that can be Set Theory constructed that have all of the properties of classical logic Matthew except for the excluded middle (The statement α ∨ ¬α is Graham not necessarily true). Outline • Think about the quantum mechanical two slit experiment Why Category and try to evaluate the truth value of the statement, “ The Theory? electron went through the top slit.” Definition • In these logics, proofs by contradiction do not hold: one Basic Examples must prove everything constructively.
Relations to • Some physicists are using topos theory to construct a Set Theory language to formulate physical theories with the hope of Topos (among other things) identifying possible unification Definition and Examples theories.
Motivation • Apparently they can construct theories where the logical and History of Topos theory structure will demarcate which questions can be answered theoretically, which can be tested physically, and which are Acknowledgements unanswerable! Topos Theory and the Connections between Category and Set Theory
Matthew Graham If Category Theory generalizes set theory then all of the familiar Outline objects and entities in set theory must be contained in Category Why Category theory somewhere. Which leads to the following questions: Theory?
Definition 1 How are Cartesian product, disjoint union, equivalence
Basic relations, inverse images, subsets, power sets, kernels, Examples . . . represented in Category theory? Relations to Set Theory 2 How are these notions and structures generalized by Topos Definition and category theory? Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between A Category is comprised of Category and Set Theory 1 a collection of C-objects and C-arrows denoted C→ with Matthew fixed domain and codomain. Graham 2 Identity arrows for every C-object in C. That is, for any Outline f g Why Category three C-objects A, B, C s.t. , A / B / C implies Theory? f Definition A / B ?? @@ Basic ?? 1 @@ Examples ? B @ f ?? g @@ Relations to Set Theory B g / C
Topos g f ·g Definition and f → → Examples 3 A / B , B / C ∈ C =⇒ A / B ∈ C Motivation f g h → and History of 4 Associativity holds: A / B / C / D ∈ C Topos theory f Acknowledgements implies the commutativity of A / B @@ ~ @@~~ g ~~@@ ~~ h @ DCo Topos Theory and the Connections between 1 vs. {∅} = 0 Category and Set Theory
Matthew The category with only one object and one arrow. Graham
Outline
Why Category Truck Theory? Ö Definition Banana . Basic Examples No matter what we call the two objects it must have this Relations to Set Theory structure. So we label the object 0 and the arrow h0, 0i = 10, Topos which gives the number zero by the diagram Definition and Examples
Motivation and History of 1 Topos theory 0 Acknowledgements Õ 0 Topos Theory and the Connections between 2, 3, ... Category and Set Theory
Matthew Graham Õ Õ Similarly, we can define the categories 2 : 0 / 1 Outline Õ Õ Õ Why Category and 3 : Theory? 0 / 1 /3 2 Definition These are examples of partial orderings. Compare them against Basic Examples the set theoretic definitions of the natural numbers
Relations to 0 = ∅, 1 = {∅}, 2 = {0, 1} = {∅, {∅}}, 3 = {0, 1, 2},.... Set Theory Topos A preorder category is a category that between any two Definition and Examples objects a, b ∈ C there exists at most one map a → b. This Motivation allows one to define a reflexive and transitive relation. If this and History of Topos theory relation is also antisymmetric then it is called a partial Acknowledgements ordering and the symbol v will be used to denote the relation. Topos Theory and the Connections between Posets and Skeletal Categories Category and Set Theory
Matthew Graham
Outline
Why Category Theory? A Poset is a pair P = hP, vi where P is a set and v is a
Definition partial ordering on P.
Basic Examples Relations to A Skeletal category is one in which “isomorphic” actually Set Theory means equal (i.e. a =∼ b =⇒ a = b). Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Discrete Categories vs. Sets Category and Set Theory
Matthew Graham
Outline If you took any set and added an identity arrow for each Why Category Theory? element, you would have a discrete category. In this sense
Definition discrete categories are sets.
Basic Examples
Relations to Set Theory {board, flower, skateboard} = Topos Definition and Examples
Motivation Ö Ö Ö and History of Board flower skateboard Topos theory
Acknowledgements Topos Theory and the Connections A monoid M is a category with one object. A monoid is between → Category and determined by C (M), the identity arrow 1e and the Set Theory composition rule ◦. Matthew Graham Example: Consider the category N comprised of a single object N and an infinite collection of arrows labeled 0, 1, 2, 3,... Outline which are the natural numbers. Why Category Theory? 1 Define composition law to be n ◦ m = n + m, which means Definition that the bottom left diagram commutes by definition. Basic Examples 2 Associative law holds (by associativity of addition) Hence Relations to Set Theory middle diagram commutes. Topos 3 Identity arrow 1N is defined to be the number 0. Which Definition and Examples implies commutativity of right diagram. m m n Motivation N / N N / N N / N and History of @ @ @ @ Topos theory @@ @@ ~~ @@ @@ m @ n @~ n @ 0 @ Acknowledgements n+m @@ ~~@@ n @@ @@ @ ~~ @ @ @ N NNo p N m / N Topos Theory and the Connections between Category and Set Theory For any category C and any a ∈ Cob, the set Hom(a, a) is a Matthew monoid. Graham Hom(a, b) = C(a, b) = f : f ∈ C→ s.t. a f , where Outline / b
Why Category ob Theory? a, b ∈ C .
Definition
Basic C is a Subcategory of D ( C ⊆ D) if Examples 1 ob ob Relations to a ∈ C =⇒ a ∈ D . Set Theory 2 for any two a, b ∈ Cob =⇒ C(a, b) ⊆ D(a, b). Topos Definition and Examples
Motivation C is a Full Subcategory of D if C ⊆ D and for any two and History of ob Topos theory a, b ∈ C , C(a, b) = D(a, b).
Acknowledgements Topos Theory and the Connections between Monic vs. injectivity Category and Set Theory Monic arrows are generalized from the notions of an injective
Matthew function. We can get the proper definition in terms of arrows Graham by realizing that injective functions are left cancelable i.e. Outline Why Category f (x) = f (y) =⇒ x = y Theory? Definition or Basic Examples f ◦ g(x) = f ◦ h(x) =⇒ g(x) = h(x).
Relations to Set Theory In arrows f is monic if for any two parallel functions / Topos g, h : C / A the following diagram commutes Definition and Examples
Motivation g and History of C / A . Topos theory
Acknowledgements h f f A / B
That is f ◦ g = f ◦ h =⇒ g = h. Topos Theory and the Connections between Epic vs. Surjectivity Category and Set Theory
Matthew Graham Epic arrows are derived in the same way except these functions are right cancelable. Outline
Why Category Theory?
Definition In arrows it amounts to the statement that f is an epic arrow if / Basic for any two parallel arrows g, h : B / C the following Examples diagram commutes Relations to Set Theory f Topos / . Definition and C A Examples f g Motivation and History of h Topos theory A / B Acknowledgements That is g ◦ f = h ◦ f =⇒ g = h. Topos Theory and the Connections between Iso arrow vs. Isomorphisms Category and Set Theory
Matthew Graham
Outline The notion of an Iso arrow is generalized from isomorphic
Why Category functions. These simply are the functions f where ∃ a function Theory? g s.t. f ◦ g = 1 = g ◦ f . Definition
Basic Examples In arrows it is the same: for f ∈ C→ f : a → b is iso (or Relations to → Set Theory invertible), in C if there is a g ∈ C , g : b → a s.t. Topos g ◦ f = 1a and f ◦ g = 1b. Definition and Examples Motivation ob ∼ and History of a, b ∈ C are isomorphic in C (a = b) if there is a C-arrow Topos theory f : a → b that is iso in C. Acknowledgements Topos Theory and the Connections between Category and In the category Set and in any Topos a monic and epic arrow Set Theory is an iso arrow. However, this is not true in general!! Even Matthew Graham though in any category an iso arrow is monic and epic.
Outline
Why Category Theory? Example: Take the category IN, every arrow is epic and monic Definition since you can cancel on the left and right. And the only iso is Basic 0 : IN → IN, since if m has an inverse n then m ◦ n = 1N or Examples m + n = 0 =⇒ m = n = 0. Relations to Set Theory
Topos Definition and Examples Example: In a Poset category if f : p → q has an inverse −1 Motivation f : q → p then p v q and and History of Topos theory q v p =⇒ p = q =⇒ f = 1p. Thus every arrow in a
Acknowledgements Poset is monic and epic but the only iso’s are the identities. Topos Theory and the Connections between Initial objects vs. ∅ Category and Set Theory
Matthew Initial objects were abstracted from Set by asking what Graham properties characterize the null set ∅? It turns out that for any Outline set A there exists only one function ∅ → A. In Set the initial Why Category object is simply ∅ and it happens to be unique. Theory?
Definition
Basic Examples An object0 is initial in category C if for every f Relations to a ∈ Cob, ∃!f ∈ C→ s.t. 0 / a Set Theory
Topos Definition and Examples Any two initial C-objects must be isomorphic in C 0 Motivation Proof: Suppose ∃0, 0 that are initial in C. then and History of f Topos theory ∃!f , g ∈ C→ s.t. 0 o / 00 . This means that Acknowledgements g 0 0 10 = g ◦ f : 0 → 0 and 100 = f ◦ g : 0 → 0 . Hence f has an inverse arrow and 0 =∼ 00. Topos Theory and the Connections between Category and Set Theory Example: of a category that has two initial objects that are Matthew Graham isomorphic but not equal.
Outline
Why Category Theory? Definition Ô ' Ô · g · Basic Examples
Relations to Set Theory Topos Example: In Grp and Mon the initial objects are any one Definition and Examples element algebra M = {e}. Each of these categories has Motivation and History of infinitely many objects (think products). Topos theory
Acknowledgements Topos Theory and the Connections between Terminal objects vs. singletons Category and Set Theory
Matthew Graham
Outline Why Category By reversing the direction of the arrows in the definition of Theory? initial object we get the idea of a terminal object. In Set Definition terminal objects are the singleton sets (the sets with only one Basic Examples element). Relations to Set Theory
Topos An object1 is terminal in a category C if for every Definition and ob → f Examples a ∈ C ∃!f ∈ C s.t. a / 1 . Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Duality Category and Set Theory
Matthew Duality is the process “reverse all arrows” with the following Graham identifications. ∗ Outline Statement Σ Dual Statement Σ Why Category f : a → b f : b → a Theory?
Definition a = domf a = codf
Basic i = 1a i = 1a Examples Isomorphic Isomorphic Relations to Set Theory h = g ◦ f h = f ◦ g
Topos f is monic f is epi Definition and Examples u is a right inverse of h u is a left inverse of h
Motivation f is invertible f is invertible and History of Topos theory t is a terminal object t is an initial object
Acknowledgements S, T : C → B are functors S, T : C → B are functors T is full T is full T is faithful T is faithful Topos Theory and the Connections between Category and Set Theory The Dual category Cop is comprised of the same objects that Matthew Graham are in C with all of the C-arrows reversed.
Outline Example: For any C:(Cop)op = C. Why Category Theory? Example: If C is discrete then Cop = C Definition op Basic Example: If C is a pre-prder (P, R), with R ⊆ P × P, then C Examples is the pre-order (P, R−1), where pR−1q ⇐⇒ qRp. That is R−1 Relations to Set Theory is the inverse relation to R.
Topos Definition and Example: The dual statement of the theorem “any two initial Examples C objects are isomorphic” is the theorem “any two terminal C Motivation and History of objects are isomorphic” Topos theory
Acknowledgements Topos Theory and the Connections between Products vs. Cartesian Product Category and Set Theory
Matthew Graham Set Definition: A × B = {(x, y): x ∈ A, y ∈ B} Outline What structure of this definition should we use to form our Why Category definition in category theory (since we can’t use elements)? Theory?
Definition
Basic If we define the projection maps Examples
Relations to Set Theory pA : A × B → A pA(x, y) = x Topos pB : A × B → B pB (x, y) = y Definition and Examples
Motivation and we are given another set C with a pair of maps f : C → A and History of Topos theory and g : C → B define p : C → A × B by the rule
Acknowledgements p(x) = hf (x), g(x)i. Topos Theory and the Connections between Category and The diagram Set Theory
Matthew C Graham x FF f xx FF g xx p FF Outline xx FF |xx pA pB F" Why Category A o A × B / B Theory? Definition commutes. Furthermore, the map p is the only arrow that can Basic make the diagram commute. Examples
Relations to Proof: Suppose that p(x) = hh, ki. Since we know that the Set Theory diagram commutes pA ◦ p(x) = f (x) and Topos Definition and pA(p(x)) = pA(hh(x), k(x)i) = h(x) =⇒ h(x) = f (x). Examples Similarly for k. Motivation and History of Topos theory The uniqueness of the map p is what is generalized in category Acknowledgements theory. Topos Theory and the Connections between Category theory product Category and Set Theory
Matthew Graham
Outline A product in a category C of two objects a and b is an object ob Why Category a × b ∈ C together with a pair of arrows Theory? (pra : a × b → a, prb : a × b → b) s.t. for any two arrows of the Definition form f : c → a and g : c → b ∃! hf , gi : c → a × b makes the Basic Examples following commute. Relations to Set Theory c y EE Topos f yy EE g Definition and yy hf ,gi EE Examples yy EE |yy pra prb E" Motivation a o a × b / b and History of Topos theory
Acknowledgements Topos Theory and the Connections between Products are not unique but are Category and Set Theory isomorphic Matthew Suppose that d is also a product of a × b then consider Graham
Outline d z DD Why Category f zz DD g Theory? zz hf ,gi DD zz DD Definition zz pra prb D" a o} a × b / b Basic aDD z< Examples DD zz DDhpra,prbi zz Relations to f DD zz g Set Theory D zz d Topos Definition and Examples Hence Motivation and History of hpra, prbi ◦ hf , gi = 1d . Topos theory ∼ Acknowledgements To complete the isomorphism of d = a × b just interchange the roles of d and a × b to get
hf , gi ◦ hpra, prbi = 1a×b. Topos Theory and the Connections between Co-products vs. Disjoint Union Category and Set Theory
Matthew Graham
Outline A Co-product of a, b ∈ C is an object a + b ∈ C together with
Why Category a pair of C arrows (ia, ib) (where ia : a → a + b and Theory? ib : b → a + b) s.t. for any C arrows of the form f : a → c and Definition g : b → c there is a unique arrow [f , g]: a + b → c making the Basic Examples following commute
Relations to Set Theory ia ib a / a + b o b Topos EE y Definition and EE yy Examples EE [f ,g] yy f EE yy g Motivation E" |yy and History of c Topos theory
Acknowledgements Topos Theory and the Connections between Category and Set Theory In Set the co-product of A and B is there disjoint union Matthew Graham [ A + B := (A × {0}) (B × {1}) . Outline
Why Category Theory? The injections are given by Definition Basic iA(a) = (a, 0) iB (b) = (b, 1) Examples
Relations to Set Theory In this case you need to define the function
Topos Definition and [f , g](x, d) = f (x)(1 − d) + dg(x) Examples
Motivation and History of We get to use elements since we are in Set. Topos theory
Acknowledgements Topos Theory and the Connections between Equalisers vs. Equalisers Category and Set Theory f / Matthew Set Definition: Given a pair of parallel functions A g / B Graham define E = {x : x ∈ A, f (x) = g(x)}. E equalizes f and g by Outline defining the inclusion E / A we get f ◦ i = g ◦ i. Why Category Theory? It turns out that i is a canonical equaliser of f and g. That is if Definition h : C → A is any other equaliser of f and g (i.e. f ◦ h = g ◦ h) Basic Examples then there exists a unique k : C → E s.t. i ◦ k = h (h ‘factors Relations to uniquely through’ i). Set Theory Topos f Definition and / Examples E / A g / B [c?? ? Motivation ?? ???? and History of k ???? h Topos theory ?? Acknowledgements C Uniqueness: If i ◦ k = h then i(k(c)) = k(c) = h(c) =⇒ f (h(c)) = g(h(c)) =⇒ h(c) ∈ E. Topos Theory and the Connections between Category Definition: Category and Set Theory An arrow i : e → a in C is an equaliser (in C) of parallel Matthew C→ f , g : a → b if: Graham 1 f ◦ i = g ◦ i Outline
Why Category 2 If h : c → a satisfies f ◦ h = g ◦ h then Theory? ∃!k ∈ C→ k : c → e s.t. i ◦ k = h Definition f Basic e / a / b Examples ]eCC = g CCCC {{ Relations to CCCC {{ Set Theory k CCCC {{ h CC {{ Topos c Definition and Examples • Every equaliser is monic Motivation and History of • Topos theory In any category, an epic equaliser is iso. Acknowledgements • In Set (and any Topos) monics are equalisers Topos Theory and the Connections between Why are the statements of Category and Set Theory category theory so complicated? Matthew Graham
Outline
Why Category Theory?
Definition
Basic Examples They don’t have to be. Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Limits and Co-limits vs. Category and Set Theory A Diagram D is simply a Subcategory of some other category
Matthew (i.e. a collection of C objects and some C-arrows between these Graham objects). Outline A D-Cone (denoted {fi : c → di }) of a diagram D consists of ob Why Category a c ∈ C with a C arrow fi : c → di , ∀di ∈ D s.t. Theory?
Definition di g / dj aD f = Basic DD fi j zz Examples DD zz DD zz Relations to D zz Set Theory c Topos commutes whenever g is an arrow in the diagram D. Definition and Examples A D-limit of a diagram D is a D-cone {fi : c → di } with the Motivation 0 0 and History of property that for any other D-cone{fi : c → di }, ∃! arrow Topos theory 0 f : c → c s.t. di Acknowledgements f 0 = aC i zz CC fi zz CC zz C zz CC 0 +3 c c f commutes for every object di ∈ D. Topos Theory and the Connections between Category and Set Theory Matthew Example: Consider the arrowless diagram of two C objects a Graham and b.A D-cone is an object c with two arrows f and g that Outline f g form ao c / b . What is another name for a D-limit of Why Category Theory? f / Definition this diagram? Example: Let D be the diagram a g / b . We Basic Examples
Relations to Set Theory can write a D-cone in this case as the commutative f Topos h / Definition and c / a g / b . What is another name for the D-limit of Examples this diagram? Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Co-cones and Co-limits Category and Set Theory
Matthew Graham By duality a Co-cone {fi : di → c} of a diagram D consists of Outline an object c and arrows fi : di → c for each object di ∈ D.A Why Category Theory? co-limit is defined analogously to limits. That is, a limit
Definition co-cone is a co-cone with the co-universal property that for any 0 0 0 Basic other co-cone {f : di → c }, ∃! arrow f : c → c s.t. the Examples i following commutes for every di ∈ D. Relations to Set Theory
Topos di g / dj Definition and DD z Examples DD zz f DD zzf Motivation i DD zz j and History of ! }z Topos theory c
Acknowledgements c • What is the limit of the empty diagram?
c0 +3 c • What is another name for c?
The limit of an empty diagram is a terminal object.
Topos Theory and the Connections between Pop QUIZ !!!!! Category and Set Theory
Matthew Graham
Outline • What is a cone of an empty diagram? Why Category Theory?
Definition
Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements • What is the limit of the empty diagram?
c0 +3 c • What is another name for c?
The limit of an empty diagram is a terminal object.
Topos Theory and the Connections between Pop QUIZ !!!!! Category and Set Theory
Matthew Graham
Outline • What is a cone of an empty diagram? Why Category Theory? c Definition
Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements c0 +3 c • What is another name for c?
The limit of an empty diagram is a terminal object.
Topos Theory and the Connections between Pop QUIZ !!!!! Category and Set Theory
Matthew Graham
Outline • What is a cone of an empty diagram? Why Category Theory? c Definition Basic • What is the limit of the empty diagram? Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements • What is another name for c?
The limit of an empty diagram is a terminal object.
Topos Theory and the Connections between Pop QUIZ !!!!! Category and Set Theory
Matthew Graham
Outline • What is a cone of an empty diagram? Why Category Theory? c Definition Basic • What is the limit of the empty diagram? Examples
Relations to Set Theory c0 +3 c Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements The limit of an empty diagram is a terminal object.
Topos Theory and the Connections between Pop QUIZ !!!!! Category and Set Theory
Matthew Graham
Outline • What is a cone of an empty diagram? Why Category Theory? c Definition Basic • What is the limit of the empty diagram? Examples
Relations to Set Theory c0 +3 c Topos Definition and • Examples What is another name for c?
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Pop QUIZ !!!!! Category and Set Theory
Matthew Graham
Outline • What is a cone of an empty diagram? Why Category Theory? c Definition Basic • What is the limit of the empty diagram? Examples
Relations to Set Theory c0 +3 c Topos Definition and • Examples What is another name for c?
Motivation and History of Topos theory The limit of an empty diagram is a terminal object.
Acknowledgements a b a o a × b / b aDD z< cGG KS w; DD z GG ww D zz GG ww DD zz GG ww D zz G ww c c0 Dually: • What is the co-limit of an empty diagram? An initial object. • What diagram is the coproduct a limit of?
a b
Topos Theory and the Connections between • What is a cone (and limit) for the following diagram? Category and Set Theory Matthew a b Graham
Outline
Why Category Theory?
Definition
Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements a o a × b / b cGG KS w; GG ww GG ww GG ww G ww c0 Dually: • What is the co-limit of an empty diagram? An initial object. • What diagram is the coproduct a limit of?
a b
Topos Theory and the Connections between • What is a cone (and limit) for the following diagram? Category and Set Theory Matthew a b Graham
Outline Why Category a b Theory? aDD < DD zz Definition D zz DD zz Basic D zz Examples c
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Dually: • What is the co-limit of an empty diagram? An initial object. • What diagram is the coproduct a limit of?
a b
Topos Theory and the Connections between • What is a cone (and limit) for the following diagram? Category and Set Theory Matthew a b Graham
Outline Why Category a b a o a × b / b Theory? aDD z< cGG KS w; DD zz GG ww Definition D z GG ww DD zz GG ww Basic D zz G w 0 w Examples c c Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements An initial object. • What diagram is the coproduct a limit of?
a b
Topos Theory and the Connections between • What is a cone (and limit) for the following diagram? Category and Set Theory Matthew a b Graham
Outline Why Category a b a o a × b / b Theory? aDD z< cGG KS w; DD zz GG ww Definition D z GG ww DD zz GG ww Basic D zz G w 0 w Examples c c Relations to Set Theory Dually: Topos • Definition and What is the co-limit of an empty diagram? Examples
Motivation and History of Topos theory
Acknowledgements • What diagram is the coproduct a limit of?
a b
Topos Theory and the Connections between • What is a cone (and limit) for the following diagram? Category and Set Theory Matthew a b Graham
Outline Why Category a b a o a × b / b Theory? aDD z< cGG KS w; DD zz GG ww Definition D z GG ww DD zz GG ww Basic D zz G w 0 w Examples c c Relations to Set Theory Dually: Topos • Definition and What is the co-limit of an empty diagram? Examples An initial object. Motivation and History of Topos theory
Acknowledgements a b
Topos Theory and the Connections between • What is a cone (and limit) for the following diagram? Category and Set Theory Matthew a b Graham
Outline Why Category a b a o a × b / b Theory? aDD z< cGG KS w; DD zz GG ww Definition D z GG ww DD zz GG ww Basic D zz G w 0 w Examples c c Relations to Set Theory Dually: Topos • Definition and What is the co-limit of an empty diagram? Examples An initial object. Motivation and History of • What diagram is the coproduct a limit of? Topos theory
Acknowledgements Topos Theory and the Connections between • What is a cone (and limit) for the following diagram? Category and Set Theory Matthew a b Graham
Outline Why Category a b a o a × b / b Theory? aDD z< cGG KS w; DD zz GG ww Definition D z GG ww DD zz GG ww Basic D zz G w 0 w Examples c c Relations to Set Theory Dually: Topos • Definition and What is the co-limit of an empty diagram? Examples An initial object. Motivation and History of • What diagram is the coproduct a limit of? Topos theory
Acknowledgements a b Topos Theory and the Connections between Co-equalisers vs. Equiv. relations Category and Set Theory
Matthew Graham The co-equaliser of a pair of parallel arrows f , g C-arrows is a f Outline / co-limit for the diagram a g / b . Or equivalently, it can be Why Category Theory? defined as a C-arrow q : b → c s.t. Definition 1 q ◦ f = q ◦ g Basic Examples 2 If h : b → c satisfies h ◦ f = h ◦ g in C then ∃! arrow Relations to Set Theory k : e → c s.t.
Topos f Definition and / q Examples a / b / e g D Motivation DD and History of DD k Topos theory h DD D" Acknowledgements c Topos Theory and the Connections between Definition of k Category and Set Theory
Matthew Graham
Outline Suppose that we have a k s.t. k ◦ fR = h. Then Why Category Theory?
Definition k([a]) = k(fR (a)) = h(a)
Basic Examples So define k([a]) = h(a). This is well defined since if [a] = [b] Relations to then Set Theory f (a) = f (b) =⇒ h(a) = h(b). Topos R R Definition and Examples This is a natural map (i.e. the only way to define this map) and Motivation and History of hence unique. Topos theory
Acknowledgements Topos Theory and the Connections between Pullback vs. Inverse Image Category and Set Theory Matthew f g Graham A Pullback of a diagram a / c o b is a limit in C for
Outline the diagram
Why Category b Theory? g Definition f Basic a / c Examples Relations to A cone and universal cone for this diagram is Set Theory
Topos f 0 Definition and d / b Examples ? ?? h Motivation g 0 ?? g and History of ?? Topos theory a / c Acknowledgements f Topos Theory and the Connections between Category and Set Theory In Set, given a function f : A → B, the inverse image of a set Matthew C ⊂ B is simply Graham −1 Outline f (C) = {x : x ∈ A, f (x) ∈ C} . Why Category Theory? Categorically this is represented by the pullback diagram below Definition
Basic f ∗ Examples f −1(C) / C _ Relations to _ Set Theory Topos Definition and A / B Examples f Motivation and History of ∗ −1 Topos theory where f is the restriction of f to f (C). Acknowledgements Topos Theory and the Connections between Pushouts Category and Set Theory
Matthew Graham
Outline
Why Category Theory? Are constructed by dualizing Pullbacks. They are the co-limit Definition of this diagram. Basic a Examples O Relations to Set Theory
Topos b o c Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Completeness Category and Set Theory
Matthew Graham
Outline
Why Category Theory? • A category C is complete if every diagram in C has a limit Definition in C. Basic Examples • Dually a category C is co-complete if every diagram in C
Relations to has a co-limit in C Set Theory • A category C is bi-complete if it is complete and Topos Definition and co-complete. Examples
Motivation and History of Topos theory
Acknowledgements ev has a universal property among functions of the form g C × A / B . ∀g of the above form ∃!g ˆ : C → BA s.t. BA × A KS GG The only choice ofg ˆ : C → BA GGev GG GG making this commute is G# gˆ×idA B defined: w; ww gc (a) = g(c, a) ∀a ∈ A ww wwg Theng ˆ(a) = g (a) ∀c ∈ C ww c C × A
Topos Theory and the Connections between Exponentiation (all products exist) Category and Set Theory In Set theory: We are familiar with Matthew Graham f BA = f : A / B . Outline
Why Category Theory? This set is related to products by the evaluation function.
Definition ev : BA × A → B s.t. ev(< f , x >) = f (x) Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Exponentiation (all products exist) Category and Set Theory In Set theory: We are familiar with Matthew Graham f BA = f : A / B . Outline
Why Category Theory? This set is related to products by the evaluation function.
Definition ev : BA × A → B s.t. ev(< f , x >) = f (x) Basic Examples
Relations to ev has a universal property among functions of the form Set Theory g A Topos C × A / B . ∀g of the above form ∃!g ˆ : C → B s.t. Definition and Examples BA × A KS G A Motivation G ev The only choice ofg ˆ : C → B and History of GG GG Topos theory GG making this commute is G# Acknowledgements gˆ×idA B defined: w; ww gc (a) = g(c, a) ∀a ∈ A ww wwg Theng ˆ(a) = g (a) ∀c ∈ C ww c C × A 3 and ∀c ∈ Cob and ∀g : c × a =⇒ ∃!ˆg : c → ba making the diagram commute (i.e. ∃!ˆg s.t. ev ◦ (ˆg × 1a) = g).
ba × a KS EE EEev EE EE E" gˆ×ida b x< xx xx xx g xx c × a This assignment ofg ˆ to g establishes a iso C(c × a, b) =∼ C(c, ba).
Topos Theory and the Connections C has exponentiation if a, b ∈ Cob =⇒ between Category and ob Set Theory 1 a × b ∈ C a ob a Matthew 2 ∃b ∈ C and an evaluation arrow ev : b × a → b Graham
Outline
Why Category Theory?
Definition
Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements This assignment ofg ˆ to g establishes a iso C(c × a, b) =∼ C(c, ba).
Topos Theory and the Connections C has exponentiation if a, b ∈ Cob =⇒ between Category and ob Set Theory 1 a × b ∈ C a ob a Matthew 2 ∃b ∈ C and an evaluation arrow ev : b × a → b Graham 3 and ∀c ∈ Cob and ∀g : c × a =⇒ ∃!ˆg : c → ba Outline making the diagram commute (i.e. Why Category Theory? ∃!ˆg s.t. ev ◦ (ˆg × 1a) = g). Definition Basic ba × a Examples KS EE EEev Relations to EE Set Theory EE E" Topos gˆ×ida b Definition and < xx Examples xx xxg Motivation xx and History of x Topos theory c × a
Acknowledgements Topos Theory and the Connections C has exponentiation if a, b ∈ Cob =⇒ between Category and ob Set Theory 1 a × b ∈ C a ob a Matthew 2 ∃b ∈ C and an evaluation arrow ev : b × a → b Graham 3 and ∀c ∈ Cob and ∀g : c × a =⇒ ∃!ˆg : c → ba Outline making the diagram commute (i.e. Why Category Theory? ∃!ˆg s.t. ev ◦ (ˆg × 1a) = g). Definition Basic ba × a Examples KS EE EEev Relations to EE Set Theory EE E" Topos gˆ×ida b Definition and < xx Examples xx xxg Motivation xx and History of x Topos theory c × a Acknowledgements This assignment ofg ˆ to g establishes a iso C(c × a, b) =∼ C(c, ba). • The following logic begins to paint a neat picture: • If A ⊂ B then A / B is injective, hence monic. f • On the other hand any monic function C / / B determines a subset of B. • In fact it creates a bijection Im(f ) =∼ B. • Hence the domain (of a monic arrow), up to isomorphism, is a subset of the codomain. • Should a subobject of an arrow g be any monic function that has the same codomain as g?
Topos Theory and the Connections between Subobjects vs. Subsets Category and Set Theory • In category theory we don’t have access to the elements Matthew Graham directly since we only have objects and arrows.
Outline
Why Category A ⊂ B iff x ∈ A =⇒ x ∈ B Theory? Definition • How does one get at the notion of a subset without Basic referring to the elements contained within each set? Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Subobjects vs. Subsets Category and Set Theory • In category theory we don’t have access to the elements Matthew Graham directly since we only have objects and arrows.
Outline
Why Category A ⊂ B iff x ∈ A =⇒ x ∈ B Theory? Definition • How does one get at the notion of a subset without Basic referring to the elements contained within each set? Examples • The following logic begins to paint a neat picture: Relations to Set Theory • If A ⊂ B then A / B is injective, hence monic. Topos Definition and • f Examples On the other hand any monic function C / / B determines a subset of B. Motivation and History of • In fact it creates a bijection Im(f ) =∼ B. Topos theory • Hence the domain (of a monic arrow), up to isomorphism, Acknowledgements is a subset of the codomain. • Should a subobject of an arrow g be any monic function that has the same codomain as g? • Which pushes us in the direction of defining the equivalent notion of subset on the arrows. • OK. Then we need to know (for g ∈ C→) how to determine which f ∈ C→ satisfy f ⊇ g or f ⊆ g? • And check that ⊆ makes (Sub(D), ⊆) a poset. Some interesting structure. • The relation of set inclusion is a partial ordering on the power set P(D) of a set D. • Hence (P(D), ⊆) is a poset which is a category where there exists an arrow A → B iff A ⊆ B. • When A ⊆ B, ∃ a vertical arrow that makes the diagram commute
Topos Theory and the • How do we make this analogy with subset stronger? Connections between Category and • We need to define subobject, the category equivalent of Set Theory subset. But it might be the case that the subset of some Matthew Graham set A is not even an object in our category!@!
Outline
Why Category Theory?
Definition
Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Some interesting structure. • The relation of set inclusion is a partial ordering on the power set P(D) of a set D. • Hence (P(D), ⊆) is a poset which is a category where there exists an arrow A → B iff A ⊆ B. • When A ⊆ B, ∃ a vertical arrow that makes the diagram commute
Topos Theory and the • How do we make this analogy with subset stronger? Connections between Category and • We need to define subobject, the category equivalent of Set Theory subset. But it might be the case that the subset of some Matthew Graham set A is not even an object in our category!@! • Outline Which pushes us in the direction of defining the equivalent
Why Category notion of subset on the arrows. Theory? • OK. Then we need to know (for g ∈ C→) how to Definition determine which f ∈ C→ satisfy f ⊇ g or f ⊆ g? Basic Examples • And check that ⊆ makes (Sub(D), ⊆) a poset. Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the • How do we make this analogy with subset stronger? Connections between Category and • We need to define subobject, the category equivalent of Set Theory subset. But it might be the case that the subset of some Matthew Graham set A is not even an object in our category!@! • Outline Which pushes us in the direction of defining the equivalent
Why Category notion of subset on the arrows. Theory? • OK. Then we need to know (for g ∈ C→) how to Definition determine which f ∈ C→ satisfy f ⊇ g or f ⊆ g? Basic Examples • And check that ⊆ makes (Sub(D), ⊆) a poset. Relations to Set Theory Topos Some interesting structure. Definition and Examples • The relation of set inclusion is a partial ordering on the Motivation and History of power set P(D) of a set D. Topos theory • Hence (P(D), ⊆) is a poset which is a category where Acknowledgements there exists an arrow A → B iff A ⊆ B. • When A ⊆ B, ∃ a vertical arrow that makes the diagram commute Topos Theory and the Connections between Subobject Category and Set Theory
Matthew Graham • Suppose that we defined subobject of d ∈ Cob to be a Outline monic arrow with codomain d. Why Category Theory? • Then consider the following definition of ‘inclusion’ Definition relation between subobjects. Basic Examples f g • Given a / d , b / d . f ⊆ g iff ∃h ∈ C→ s.t. Relations to Set Theory Topos b Definition and O ? ?? g Examples h ?? ?? Motivation ? and History of a / / d Topos theory f Acknowledgements Topos Theory • and the We need to check that this definition is: Connections • Reflexive (f ⊆ f ) a between O @ @ Category and @@ f Set Theory 1a @@ @@ Matthew a / / d Graham f • Outline Transitive (f ⊆ g, g ⊆ k =⇒ f ⊆ k) Why Category c Theory? F O B BB k Definition l BB BB Basic g B Examples h◦l b / / d O > Relations to || Set Theory h || || f Topos >|| Definition and a Examples ∼ Motivation • Antisymmetric (f = g =⇒ f = g) and History of Topos theory b V ? Acknowledgements ?? g l h ?? ?? ? a / / d f A subobject of d is the equivalence class of all monic arrows with codomain d. • This definition makes (Sub(D), ⊆) a poset. • This definition corresponds to the usual subobjects (defined via elements) in the categories Rng, Grp, Ab and R − Mod but apparently not in T op.
Topos Theory and the Connections between Actual definition of Subobject Category and Set Theory
Matthew Graham • The antisymmetric diagram does not guarantee that f = g Outline when it commutes only that f =∼ g. Easy check: it might Why Category Theory? be the case that a 6= b. Definition • Modify the definition to make equality hold: Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Actual definition of Subobject Category and Set Theory
Matthew Graham • The antisymmetric diagram does not guarantee that f = g Outline when it commutes only that f =∼ g. Easy check: it might Why Category Theory? be the case that a 6= b. Definition • Modify the definition to make equality hold: Basic Examples A subobject of d is the equivalence class of all monic arrows Relations to Set Theory with codomain d.
Topos Definition and • This definition makes (Sub(D), ⊆) a poset. Examples • This definition corresponds to the usual subobjects Motivation and History of (defined via elements) in the categories Rng, Grp, Ab and Topos theory R − Mod but apparently not in T op. Acknowledgements Topos Theory and the Connections between Category and Set Theory
Matthew Graham
Outline
Why Category • To each subset there is an associated characteristic Theory? function. Definition • Is the same true for every subobject? Basic Examples • If so how do you define it categorially? Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements • The characteristic function tells us if a particular element x ∈ X is also an element in A. • The characteristic functions ‘see’ the elements. • Is there a universal property that would allow us to define a similar function for category theory?
Topos Theory and the Connections between Properties of Power Set Category and Set Theory
Matthew Graham • P(D) =∼ 2D . There is a bijective correspondence between
Outline the functions D → 2 and the subsets of D. Why Category • Theory? You can get this by using the characteristic function χA of
Definition a subset A. 1 x ∈ A Basic χA = Examples 0 x 6∈ A Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Properties of Power Set Category and Set Theory
Matthew Graham • P(D) =∼ 2D . There is a bijective correspondence between
Outline the functions D → 2 and the subsets of D. Why Category • Theory? You can get this by using the characteristic function χA of
Definition a subset A. 1 x ∈ A Basic χA = Examples 0 x 6∈ A Relations to Set Theory
Topos • The characteristic function tells us if a particular element Definition and Examples x ∈ X is also an element in A. Motivation • The characteristic functions ‘see’ the elements. and History of Topos theory • Is there a universal property that would allow us to define Acknowledgements a similar function for category theory? • For a set A ⊂ D, A / D
χA {1} / 2 is a pullback square.
• The universal property is that χA is the only function that can make this a pullback square for the set A.
Topos Theory and the Connections • It turns out that there is a universal property. between −1 Category and • Let Af = {x : x ∈ D, f (x) = 1} = f ({1}). Set Theory Then the set Af arises by the pullback square below. Matthew Graham Af / D Outline Why Category f Theory? Definition {1} / 2 Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections • It turns out that there is a universal property. between −1 Category and • Let Af = {x : x ∈ D, f (x) = 1} = f ({1}). Set Theory Then the set Af arises by the pullback square below. Matthew Graham Af / D Outline Why Category f Theory? Definition {1} / 2 Basic Examples • For a set A ⊂ D, Relations to Set Theory A / D Topos Definition and χA Examples Motivation {1} / 2 and History of Topos theory
Acknowledgements is a pullback square. • The universal property is that χA is the only function that can make this a pullback square for the set A. • For x ∈ A, g(x) = 1 =⇒ x ∈ Ag . Thus, A ⊆ Ag . • Since the outer square commutes and i and j are inclusions then so is k. Hence, Ag ⊆ A. • But then f is the characteristic function of A therefore f = χA.
Topos Theory and the Connections between Proof: Category and Set Theory • Suppose there was some other function g that would make Matthew Graham the previous diagram a pullback. Then we get the following diagram Outline
Why Category Theory? Ag t @OO Definition @@@OO @@@ OOOi @@@@ OO Basic k @@@ OOO Examples # OO j A '/ D Relations to Set Theory g Topos Definition and # true Examples 1 / 2
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Proof: Category and Set Theory • Suppose there was some other function g that would make Matthew Graham the previous diagram a pullback. Then we get the following diagram Outline
Why Category Theory? Ag t @OO Definition @@@OO @@@ OOOi @@@@ OO Basic k @@@ OOO Examples # OO j A '/ D Relations to Set Theory g Topos Definition and # true Examples 1 / 2
Motivation and History of • For x ∈ A, g(x) = 1 =⇒ x ∈ Ag . Thus, A ⊆ Ag . Topos theory • Since the outer square commutes and i and j are Acknowledgements inclusions then so is k. Hence, Ag ⊆ A. • But then f is the characteristic function of A therefore f = χA. Topos Theory and the Connections between Subobject Classifier Category and Set Theory
Matthew Graham Given a category C with initial object 1. A subobject classifier Outline for C is Ω ∈ Cob together with a C-arrow true : 1 → Ω that Why Category Theory? satisfies the Ω-axiom below.
Definition Ω-axiom: For each monic a f there is one and only one Basic / / d Examples χf : d → Ω s.t. the following is a pullback square. Relations to Set Theory f Topos a / / d Definition and Examples χf Motivation true and History of / Topos theory 1 Ω
Acknowledgements 1 ⇐⇒ E has a terminal object and pullbacks. 2 ⇐⇒ E has an initial object and pushouts. Also (1), (3), and (4) imply (2). So another way of defining a topos E is to say that E is cartesian closed category with a subobject classifier.
Topos Theory and the Connections between Topos Category and Set Theory
Matthew Graham An Elementary Topos is a category E s.t. Outline 1 E is finitely complete. Why Category Theory? 2 E is finitely co-complete. Definition 3 E has exponentiation. Basic Examples 4 E has a subobject classifier. Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Also (1), (3), and (4) imply (2). So another way of defining a topos E is to say that E is cartesian closed category with a subobject classifier.
Topos Theory and the Connections between Topos Category and Set Theory
Matthew Graham An Elementary Topos is a category E s.t. Outline 1 E is finitely complete. Why Category Theory? 2 E is finitely co-complete. Definition 3 E has exponentiation. Basic Examples 4 E has a subobject classifier. Relations to Set Theory 1 ⇐⇒ E has a terminal object and pullbacks. Topos Definition and 2 ⇐⇒ E has an initial object and pushouts. Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Topos Category and Set Theory
Matthew Graham An Elementary Topos is a category E s.t. Outline 1 E is finitely complete. Why Category Theory? 2 E is finitely co-complete. Definition 3 E has exponentiation. Basic Examples 4 E has a subobject classifier. Relations to Set Theory 1 ⇐⇒ E has a terminal object and pullbacks. Topos Definition and 2 ⇐⇒ E has an initial object and pushouts. Examples Motivation Also (1), (3), and (4) imply (2). So another way of defining a and History of Topos theory topos E is to say that E is cartesian closed category with a Acknowledgements subobject classifier. Topos Theory and the Connections between Topoi Examples Category and Set Theory
Matthew Graham
Outline
Why Category Theory? • Set - the prime example and the motivation for the
Definition concept in the first place Basic • Examples Finset is a topos with limits, exponentials and > : 1 → Ω
Relations to exactly as in Set. Set Theory • Set2 Topos Definition and → Examples • Set the category of functions.
Motivation and History of Topos theory
Acknowledgements • So Topos theory was invented from a merger of ideas from geometry and logic. • Topos is a category with some extra properties that make it look like Set. • Many people study a space by studying the sheaves on that space. Apparently the main utility of a topos arises in situations in math where topological intuition is very effective but the space does not allow a topology. It is sometimes possible to formalize the intuition via a topos.
Topos Theory and the Connections between Some true statements about Topoi Category and Set Theory • ∼ 1963 Lawvere figured out new foundations for Matthew Mathematics based on category theory. (i.e. what is so Graham great about sets?) Outline • Even earlier than this the concept of tracking structure of Why Category Theory? systems via category theory became important in algebraic Definition geometry (via work by Grothendieck on the Weil Basic conjectures). Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Some true statements about Topoi Category and Set Theory • ∼ 1963 Lawvere figured out new foundations for Matthew Mathematics based on category theory. (i.e. what is so Graham great about sets?) Outline • Even earlier than this the concept of tracking structure of Why Category Theory? systems via category theory became important in algebraic Definition geometry (via work by Grothendieck on the Weil Basic conjectures). Examples
Relations to • So Topos theory was invented from a merger of ideas from Set Theory geometry and logic. Topos Definition and • Topos is a category with some extra properties that make Examples it look like Set. Motivation and History of • Topos theory Many people study a space by studying the sheaves on
Acknowledgements that space. Apparently the main utility of a topos arises in situations in math where topological intuition is very effective but the space does not allow a topology. It is sometimes possible to formalize the intuition via a topos. It turns out that it does behave similarly. • 2A logically represents all the mappings from A to the two point set {0, 1}, which can be thought of true and false. What does Ωa represent? • 1 What if Ω = {0, 2 , 1}? At a very basic level this means that if we regard a as ‘containing’ statements of some sort then there are more options than just true and false. (i.e. the law of excluded middle does not hold!!)
Topos Theory and the Connections between Some hand wavy connections with Category and Set Theory logic Matthew Graham • The exponential Ωa is the topos analogue of 2A in Set. Outline
Why Category • Some questions: Theory? • 2A =∼ P(A) is the same true for Ωa? Is it the “power set” Definition of a? Basic Examples
Relations to Set Theory
Topos Definition and Examples
Motivation and History of Topos theory
Acknowledgements At a very basic level this means that if we regard a as ‘containing’ statements of some sort then there are more options than just true and false. (i.e. the law of excluded middle does not hold!!)
Topos Theory and the Connections between Some hand wavy connections with Category and Set Theory logic Matthew Graham • The exponential Ωa is the topos analogue of 2A in Set. Outline
Why Category • Some questions: Theory? • 2A =∼ P(A) is the same true for Ωa? Is it the “power set” Definition of a? Basic Examples It turns out that it does behave similarly. Relations to • 2A logically represents all the mappings from A to the two Set Theory point set {0, 1}, which can be thought of true and false. Topos a Definition and What does Ω represent? Examples • 1 What if Ω = {0, 2 , 1}? Motivation and History of Topos theory
Acknowledgements Topos Theory and the Connections between Some hand wavy connections with Category and Set Theory logic Matthew Graham • The exponential Ωa is the topos analogue of 2A in Set. Outline
Why Category • Some questions: Theory? • 2A =∼ P(A) is the same true for Ωa? Is it the “power set” Definition of a? Basic Examples It turns out that it does behave similarly. Relations to • 2A logically represents all the mappings from A to the two Set Theory point set {0, 1}, which can be thought of true and false. Topos a Definition and What does Ω represent? Examples • 1 What if Ω = {0, 2 , 1}? Motivation and History of Topos theory At a very basic level this means that if we regard a as
Acknowledgements ‘containing’ statements of some sort then there are more options than just true and false. (i.e. the law of excluded middle does not hold!!) Topos Theory and the Connections between Acknowledgements Category and Set Theory
Matthew Graham
Outline
Why Category Theory? I would like to thank: Definition • YOU ! Basic Examples • and the people that came to the last talk but found it too
Relations to boring to come to this one. Set Theory • and the fact that there exists a truck banana category, Topos Definition and without which I don’t think that I could sleep at night. Examples
Motivation and History of Topos theory
Acknowledgements