By John Baez
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Additional resources for Category Theory Course [Lecture notes]
Q Q φ A χi 1 you can also check that χi : X → 2 is the unique morphism from X to 2 that makes the square a pullback. 2. 3. A (elementary) topos is a cartesian closed category with finite limits (limits of finite sized diagrams) and a subobject classifier. 48 Grothendieck in the 1960’s introduced a concept of topos, now Grothendieck topos, which is a special case of alementary topos, as part of proving the Weil hypothesis in number theory. Later in the late 60’s and early 70’s Lawrence and Trerney simplified the concept of topos to define an "elementary topos".
1. Suppose C is a category with terminal object 1 ∈ C. Then there’s a functor elt : C → Set with elt( X ) = Hom(1, X ), ∀ X ∈ C and given any morphism g : X → YinC, elt( g) : elt( X ) → elt(Y ) is defined as follows: 1 f X g elt( g) f = g◦ f Y 43 g Proof: elt preserve composition: given X Y h Z we need elt(h ◦ g) = elt(h) ◦ elt( g) f 1 X g Y h Z Given f ∈ elt( X ) we have elt(h ◦ g) f = = = = (h ◦ g) ◦ f h ◦ (g ◦ f ) h ◦ (elt( g) f ) elt(h)(elt( g)( f )) Similarly elt(1x ) f = 1x ◦ f = f , for all f ∈ elt( X ).
4. Examples of elementary topos 1. Set: category of sets and functions. 2. FinSet: category of finite sets and functions, this doesn’t have all limits only finite limits, so topos theory includes finitest mathematics. 3. Set : category of sets and functions as defined using ZF=Zermelo-Fraenkel axioms without axiom of choice. If this if true we say the epimorphism splits. In a general topos, not every epimorphisms splits so the axiom of choice need not hold. 4. Graphs: The category of graphs: s E V t 5.
Category Theory Course [Lecture notes] by John Baez