Logics of Worlds a mathematical interlude
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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 di erent 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) i 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. In fact, the power set of any set ordered by inclusion is a boolean algebra (where ∩ corresponds to ∧, ∪ corresponds to ∨, and 푎푐 is ¬푎).
23 / 40 Logics
Intuitionistic (or constructive) logic is a generalization of classical logic, where law of excluded middle 퐴 ∨ ¬퐴 does not apply. Intuitionistic proofs also di er from classical ones in that they can always be viewed as the "realization" of an object, rather than simply asserting the existence.
The corresponding algebra to intuitionistic logic is the Heyting algebra. This algebra can take truth values besides True and False. It also allows for pseudocomplements, i.e. negating the complement of an element doesn't return the original element.
24 / 40 Categories
A category is a collection of objects such that:
For any two objects 퐴 and 퐵 there are a collection of morphisms 퐻표푚(퐴, 퐵).
Consider a third object 퐶: we can nd all the morphisms from 퐴 to 퐶 that "factor through" 퐵. Such morphisms are the ones formed by the composition operator:
∘ : 퐻표푚(퐴, 퐵) × 퐻표푚(퐵, 퐶) → 퐻표푚(퐴, 퐶) . where composition obeys associativity: 푓 ∘ 푔 ∘ ℎ = (푓 ∘ 푔) ∘ ℎ = 푓 ∘ (푔 ∘ ℎ)
Finally, for any object 퐴 in the category, 퐻표푚(퐴, 퐴) contains at least the identity morphism.
25 / 40 Categories - Functors
A functor between two categories maps objects to objects and arrows to arrows such that they respect composition and identity. That is:
1. 퐹(푔 ∘ ℎ) = 퐹(푔) ∘ 퐹(ℎ) for any commuting arrows 푔 and ℎ
2. 퐹(푖푑퐴) = 푖푑퐹 (퐴)
Functors can also be composed, and the above two conditions will hold through composition.
An example is the one that takes all objects of some category C to a privileged object 퐴 of some (not necessarily other) category D, and all morphisms to 푖푑퐴. This is called the constant functor.
When a category is locally small, 퐻표푚 itself yields functors to the category of sets, two for each object 퐴 in the category: 퐻표푚(퐴, −) and 퐻표푚(−, 퐴). A representable functor is any functor which is isomorphic to either of these two, again for some xed object 퐴. 퐻표푚(−, −) itself is an example of a bi-functor. An endofunctor takes a category to a sub-category.
26 / 40 Categories - Duality
A category 퐶 with its arrows ipped is denoted 퐶 표푝 . Any true statement we make in 퐶 yields a true statement in 퐶 표푝 - this is referred to as duality.
A functor 퐹 : 퐶 표푝 → 퐷 is considered contravariant with respect to 퐶. If 퐷 is the category of sets, then 퐹 is considered a presheaf. Dually, a functor 퐹 : 퐶 → Set is a co-presheaf.
A presheaf which is isomorphic to 퐻표푚(−, 퐴) for some xed 퐴 is a representable presheaf. Dually, a representable co-presheaf is isomorphic to 퐻표푚(퐴, −) for some 퐴.
27 / 40 Categories - Natural Transformations
A morphism between functors is called a natural transformation. Given any object 퐴 in a category 퐶, and two functors 퐹 : 퐶 → 퐷 and 퐺 : 퐶 → 퐷, we de ne some 훼퐴 which takes 퐹(퐴) to 퐺(퐷) such that composition is respected. This means for any 푓 : 퐴 → 퐵 we get a corresponding 훼퐵. This can be pictured as a "naturality square":
Natural transformations, just like functors and morphisms, can also compose. Likewise, we can form a category, called a 2-category, where functors are 1- morphisms with natural transformations as 2-morphisms. Closely related is the functor category, usually written as [퐶, 퐷] consisting of functors from 퐶 to 퐷 as objects.
28 / 40 Categories - Yoneda Lemma
Recall that for any category 퐶, there exists a category of presheaves [퐶 표푝, 푆푒푡].
There exists a special functor 푦 : 퐶 → [퐶 표푝, 푆푒푡] called the Yoneda embedding. For any object 푐 in 퐶, 푐 ↦ 퐻표푚퐶 (−, 푐).
The Yoneda Lemma says the set of natural transformations from the Yoneda embedding of an object 푐 to any other Set-valued presheaf is isomorphic to the value of 푐 under that presheaf.
표푝 퐻표푚[퐶 표푝,푆푒푡](푦(푐), 퐹) ≅ 퐹(푐) for some 퐹 : 퐶 → 푆푒푡
29 / 40 Categories - Limits
Let us consider a special category called a diagram category D. The objects are points and the morphisms are just arrows. A functor from D to a category C will pick objects and morphisms in C to "incarnate" the points and arrows of the diagram.
For any such "incarnation" functor, a cone is a xed object 퐴 and a collection of morphisms for each incarnated object 푓푖 : 퐴 → 퐷푖 for 퐷푖 in D. A co-cone is the dual of a cone.
30 / 40 Categories - Limits
A limit for a cone is a special cone, such that all other cones "factor uniquely" through it. Dually, a co-limit is a special co-cone, such that all other co-cones are a "factor of" it.
A picture of the uniqueness condition of a universal limit:
Examples of limits and colimits:
Example of a "universal mapping property" that is not a limit:
31 / 40 Categories - Limits
Limits (along with exponentials) can be generalized as universal mapping properties, which are themselves part of the theory of adjoint functors. They are characterized by the triangle:
where (퐹(퐶), 푢) is the universal element of 퐹
"Russell’s paradox drove set theory out of Frege’s Paradise. As set theory was reconstructed to escape the paradoxes, the set-universal for a property was always non-self-participating. Thus set theory became not 'the theory of universals' but the theory of non-self-participating universals. The reformulation of set theory cleared the ground for a separate theory of always-self-participating universals. That idea was realized in category theory by the objects having universal mapping properties. The self-participating universal for a property (if it exists) is the paradigmatic or archetypical example of the property. All instances of the property are determined to have the property by a morphism 'participating' in that paradigmatic instance (where the universal 'participates' in itself by the identity morphism)."
Ellerman, Theory of Adjoint Functors
32 / 40 Categories - Adjunctions
An adjunction is a pair of functors 퐹 : 퐶 → 퐷 and 퐺 : 퐷 → 퐶 together with two natural transformations:
1. The unit of the adjunction: 휂 : 푖푑퐶 → 퐺퐹 2. The co-unit of the adjunction: 휖 : 퐹퐺 → 푖푑퐷 which obey the following commutative conditions:
We say 퐹 is left adjoint to 퐺 or 퐺 is right adjoint to 퐹 , or simply 퐹 ⊣ 퐺.
An alternate (algebraic) formulation using 퐻표푚 is to say there is a natural isomorphism:
퐻표푚퐶 (퐹(푐), 푑) ≅ 퐻표푚퐷(푐, 퐺(푑))
33 / 40 Categories - Adjunctions - Examples
In Boolean algebra, considered as a category, take one object 푎 as xed and de ne conjunction functor as 퐹(푥) = (푎 ∧ 푥) , then a right adjunction follows 퐺(푦) = (푦 ∨ ¬푎) = (푎 ⟹ 푦) .
If we are in a category with products, we can de ne the exponential 푌 푋 as the right adjoint of a functor (−) × 푋 . This is called the product-hom adjunction because the exponential looks exactly like 퐻표푚(푋, 푌 ) but is de ned internally instead of as a set. Compare with the above example.
34 / 40 Comma Categories
For three categories 퐀, 퐁 and 퐂 and functors 푆 : 퐀 → 퐂 and 푇 : 퐁 → 퐂 we produce a comma category consisting of objects as the triples (퐴, 퐵, ℎ) where 퐴 is an object in 퐀, 퐵 in 퐁, and ℎ : 푆(퐴) → 푇(퐵) is in 퐂. Morphisms are pairs of arrows (푓, 푔) where 푓 comes from 퐀 and 푔 comes from 퐁.
We can imagine this category as two pictures 푆(퐀) and 푇(퐁) embedded in 퐂 connected pointwise with arrows.
35 / 40 Slice Categories
A special case of the comma category is when 퐀 is 퐂, 퐁 = 1 (the category of one object) and 푆 = 1퐶 (the identity functor on 퐂). This is a category where every object is determined by a morphism 휋 in 퐶 which has a xed codomain but whose domain varies through objects of 퐶. Therefore, any morphism in the slice category is just a morphism that respects composition with 휋:
36 / 40 Bundles
A bundle arises for any morphism as the collection of "germs over stalks", and equivalently, sections of subsets of the codomain. The total space comprised of all sections (when the morphism is locally a homeomorphism) is called etale.
We can formulate bundles as slice categories of Top, the category of topological spaces.
37 / 40 Sheaves
Sheaves are presheaves that satisfy uniqueness and gluing properties. In other words, the functor taking a topological space (or category with a topology) to the category of sets should be contravariant and restrict in a consistent way.
Any bundle automatically de nes sheaf, namely, the sheaf of germs over a space. 38 / 40 Elementary and Grothendieck Toposes
Grothendieck invented a way to treat objects of any category like the open sets of a space. A category with a topology given by families of covering morphisms is called a site. The category of sheaves over a site is known as a Grothendieck topos.
Lawvere and Tierney invented a simpli cation of this, which they named elementary topos. The idea is to generalize the notion of site as morphisms into a subobject classi er, and to allow all nite limits and exponents such that we can formulate set theory internally.
Every Grothendieck topos is also an elementary topos.
A morphism between toposes is called a geometric morphism, and automatically come with two adjoint functors, called the direct and inverse images. These adjoints generalize morphisms between locales and frames.
39 / 40 Resources
morphism, ncatlab, https://ncatlab.org/nlab/show/morphism functor, ncatlab, https://ncatlab.org/nlab/show/functor natural transformation, ncatlab, https://ncatlab.org/nlab/show/natural+transformation over category, ncatlab, https://ncatlab.org/nlab/show/over+category bundle, ncatlab, https://ncatlab.org/nlab/show/bundle sheaf, ncatlab, https://ncatlab.org/nlab/show/sheaf localic topos, ncatlab, https://ncatlab.org/nlab/show/localic+topos Mac Lane, Saunders and Moerdijk, Ieke (1992) Sheaves in Geometry and Logic – A rst introduction to topos theory, Springer Verlag Bradley, Tai-Danae, Math3ma blog, https://www.math3ma.com/categories
The last one is an amazing expositor of mathematics in general. All links were accessed July 2020.
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