Logics of Worlds a mathematical interlude background image credit: Shafarevich's Conjecture London Tsai 1 / 40 Topics Pictures of Structure Categories Toposes Morphisms Functors Bundles Relations Natural Sheaves Quotients Transformations Elementary and Lattices Yoneda Lemma Grothendieck Toposes Topologies Limits Quotient spaces Adjunctions Covers Comma Categories Manifolds Algebras Logic 2 / 40 Morphisms A morphism is a type of mathematical object which has a domain and codomain. If the codomain of a morphism 푓 and the domain of the morphism 푔 are the same, we can form a new morphism 푔 ∘ 푓 . In set theory, a morphism is a function: Q: How many dierent morphisms exist between two sets? 3 / 40 Morphisms A morphism 푓 is mono (injective) i it is left-cancelable, i.e. when it satis|es 푓 ∘ 푔 = 푓 ∘ ℎ ⟹ 푔 = ℎ . For sets, this looks like: A morphism f is epi (surjective) iy it is right-cancelable, i.e. when it satis|es ℎ ∘ 푓 = 푔 ∘ 푓 ⟹ 푔 = ℎ . For sets, this looks like: 4 / 40 Morphisms An identity morphism 푖푑퐴 has the same domain and codomain and is both left and right cancelable for any morphisms it composes with. An isomorphism 푓 : 퐴 → 퐵 is a morphism which has an inverse, i.e. when −1 −1 푓 ∘ 푓 = 푖푑퐴 and 푓 ∘ 푓 = 푖푑퐵 . We can also say 푓 is invertible. Sometimes only one of the above conditions is true (then 푓 is said to have a left or right inverse only). Sets are isomorphic i they have the same cardinality. An endomorphism has the same domain and codomain. An automorphism is an invertible endomorphism. Q: How can permutations and combinations be thought of as morphisms? 5 / 40 Morphisms Generally, we care less about speci|c morphisms and more about equivalence classes of morphisms satisfying some condition. Finding a unique morphism that satis|es a condition is a very special and important case. The conditions or constraints we are allowed to put on morphisms can, in the most general sense, be called (categorical) structure. The structure of a mathematical object can be derived from its morphisms - it is just the structure of morphisms which take it as a codomain. 6 / 40 Morphisms Just as we can compose morphisms, we can also "decompose" them, or factor them into their constitutive parts. Commutative diagrams are important mainly because they let us "look inside" a morphism, meaning, we can look at the constraints of a morphism by looking at the constraints of its factors. 7 / 40 Relations Properties of relations: re}exivity: 푎푅푎 symmetry: 푎푅푏 ⟹ 푏푅푎 anti-symmetry: 푎푅푏 ∧ 푏푅푎 ⟹ 푎 = 푏 transitivity: 푎푅푏 ∧ 푏푅푐 ⟹ 푎푅푐 A relation which is re}exive and transitive is a pre-order. partial order relations are re}exive, transitive, anti-symmetric (typically written as ≤) total order relations are partial orders in which, for any two elements 푎 and 푏, either 푎푅푏 or 푏푅푎 (typically written as <) equivalence relations are re}exive, transitive, and symmetric (typically written as ≡) Equivalence relations produce equivalence classes of objects, essentially partitioning a space. 8 / 40 Relations An example of an equivalence relation depicted as a matrix of some set 푋 × 푋 : 9 / 40 Relations Here are some possible ways of conceiving a relation from a morphism viewpoint: 1. An injective function 푅 : 퐴 → 푋 × 푌 which "selects" ordered pairs of elements (푥, 푦). This can be extended to n-ary relations in 푋 × 푋×. 푋 . 2. The pre-image of any function establishes an equivalence relation on the domain. 3. A function 푅 : 푋 → 푌 where 푥푅푦 ⟺ 푅(푥) = 푦 for 푥 ∈ 푋, 푦 ∈ 푌 . This seems to be the standard in category theory literature since composition is a natural consequence of the transitivity condition of relations. So the properties of relations can be generalized as properties of morphisms. Except we want to adopt an "extrinsic" view which characterizes these properties in terms of composition. Option 3 in that sense gives a "natural" interpretation of composition. We can also talk about morphisms which "do not quite |t" a relation, that is, they can be viewed as a restriction of some proper relation morphism. 10 / 40 Posets A poset is a set with a partial order relation. Posets can be depicted in a Hasse diagram: Weisstein, Eric W. "Hasse Diagram." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HasseDiagram.html A function between posets that preserves order is called monotonic. Preserving order doesn't necessarily require injectivity (nor surjectivity), it only requires that 푎 ≤ 푏 → 푓(푎) ≤ 푓(푏) . 11 / 40 Posets - Adjoints Given a monotone function 퐹 : 퐶 → 퐷, a right adjoint of 퐹 is 푅 : 퐷 → 퐶 such that 퐹(푐) ≤ 푑 ⟺ 푐 ≤ 푅(푑) and a left adjoint 퐿 satis|es 푐 ≤ 퐿(푑) ⟺ 퐹(푐) ≤ 푑 for all 푐 ∈ 퐶 and 푑 ∈ 퐷 no adjoint adjoint 12 / 40 Posets - Adjoints - Galois connnection A Galois connection is formed by any pair of left and right adjoint functions. There are monotone and antitone (order-reversing, so the RHS becomes ≥) Galois connections. As a historical note, the original Galois connection is antitone and can be written as: 퐸 ↦ 퐺푎푙(퐿/퐸) ⟺ 퐺 ↦ 퐹푖푥(퐺) where 퐸 is a sub|eld of a |eld extension 퐿/퐾 퐺푎푙(퐿/퐸) are the automorphisms of 퐿 that preserve 퐸 퐺 is a subgroup of 퐺푎푙(퐿/퐾) 퐹푖푥(퐺) is a sub|eld of 퐿 consisting of elements that are preserved by automorphisms in 퐺 See: https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory 13 / 40 Quotients An equivalence relation can be viewed as a morphism, called a quotient, which takes an object to its equivalence classes. Quotients are always epimorphisms. Why? 14 / 40 Lattices A lattice is a poset in which any two elements have a unique join (least upper bound) and meet (greatest lower bound). A complete lattice P is a lattice where every subset of P has a unique join and meet. A semilattice has only meets or joins, but not both. not a lattice A lattice chain A lattice |lter An important example of a complete lattice is the set of subsets, or the power set of some set 퐴. In that case, joins are unions and meets are intersections, top is 퐴 and bottom empty set. 15 / 40 Lattices The distributive law doesn't necessarily hold in lattices. In other words, if joins are denoted with ∨ and meets are denoted with ∧, the following doesn't necessarily hold: 푎 ∧ (푏 ∨ 푐) = (푎 ∧ 푏) ∨ (푎 ∧ 푐) A frame is a lattice in which the above holds. A locale is a frame with the order reversed. A morphism between two lattices is called a lattice homomorphism if it preserves joins and meets. 16 / 40 Topologies An open set is a set of points without a boundary. This means for any point we choose in that set, we can |nd a sphere around it that is also entirely in the set. A topology is a bounded lattice of open sets closed under in|nite numbers of unions and |nite intersections. 17 / 40 Topologies A set along with a topology (of open sets in the set) is a topological space. Terminological note: For any property 휙 and some 푓 : 퐶 → 퐷 we say that 푓 preserves 휙 when 휙(퐶) ⟹ 휙(푖푚(푓)) and we say that 푓 re}ects 휙 when 휙(푖푚(푓)) ⟹ 휙(퐶). A function between two topological spaces that re}ects open sets is called continuous. A topological space 푆 is called a metric space if there exists a morphism 푑 : 푆 × 푆 → 푅 that takes any two points of S to a distance in 푅 and satis|es: 1. 푑(푥, 푦) = 0 ⟺ 푥 = 푦 2. 푑(푥, 푦) = 푑(푦, 푥) 3. 푑(푥, 푧) ≤ 푑(푥, 푦) + 푑(푦, 푧) 18 / 40 Quotient Spaces A quotient space is the result of a quotient morphism on a space. Also see the real projective plane: https://en.wikipedia.org/wiki/Real_projective_plane 19 / 40 Covers A cover of a subset Y of X is a collection of subsets of X which together contain Y. A re|nement V of a cover U is a cover consisting of 푉푖 such that for every 푈푖 ∈ 푈 , 푉푖 ⊂ 푈푖 . The re|nement relation is therefore a partial order on the set of covers of Y. In topology, covers can be used to characterize spaces. The following picture depicts the notion of "commuting sections" of covers. 20 / 40 Manifold One type of topological space is a manifold, which locally looks like Euclidean space but can globally be very messy. This local vs. global theme can be characterized as a morphism from a space called the tangent space to the manifold. Locality in this case is then described by the point-wise sections of this map. 21 / 40 Algebras An algebra is a complete lattice in which every element is the join of compact elements. Another de|nition, more common and closely related, is that an algebra is a set together with operators on its elements. Operators take 푛 elements of a set (where 푛 is called the arity) and produce another element in the set. This property that no matter what operators are applied, we always "remain in" the set, is called closure. 22 / 40 Logics A boolean algebra is often used in conjunction with a set of propositions (themselves a collection of symbols that can be combined together according to certain rules). The simplest boolean algebra can also be speci|ed in terms of a truth table. a b 푎 ∧ 푏 푎 ∨ 푏 ¬푎 푎 → 푏 F F F F T T T F F T F F F T F T T T T T T T F T However, we can have boolean algebras of more than two elements.