# Tag: Posets

# Badiou’s Vain Platonizing, or How the World is a Topos? Note Quote.

As regards the ‘logical completeness of the world’, we need to show that Badiou’s world of T-sets does indeed give rise to a topos.

Badiou’s world consisting of T-Sets – in other words pairs (A, Id) where Id : A × A → T satisfies the particular conditions in respect to the complete Heyting algebra structure of T—is ‘logically closed’, that is, it is an elementary topos. It thus encloses not only pull-backs but also the exponential functor. These make it possible for it to internalize a Badiou’s infinity arguments that operate on the power-functor and which can then be expressed from insde the situation despite its existential status.

We need to demonstrate that Badiou’s world is a topos. Rather than beginning from Badiou’s formalism of T -sets, we refer to the standard mathematical literature based on which T-sets can be regarded as sheaves over the particular Grothendieck-topology on the category T: there is a categorical equivalence between T-sets satisfying the ‘postulate of materialism’ and S hvs(T,J). The complications Badiou was caught up with while seeking to ‘Platonize’ the existence of a topos thus largely go in vain. We only need to show that Shvs(T,J) is a topos.

Consider the adjoint sheaf functor that always exists for the category of presheaves

Id_{α} : Sets^{Cop} → Shvs(C^{op},J)

, where J is the canonical topology. It then amounts to an equivalence of categories. Thus it suffices to replace this category by the one consisting of presheaves Sets^{Top}. This argument works for any category C rather than the specific category related to an external complete Heyting algebra T. In the category of Sets define Y^{X} as the set of functions X → Y. Then in the category of presheaves,

Sets^{Cop}Y^{X}(U) ≅ Hom(h_{U},Y^{X}) ≅ Hom(h_{U} × X,Y)

, where h_{U} is the representable sheaf h_{U}(V) = Hom(V,U). The adjunction on the right side needs to be shown to exist for all sheaves – not just the representable ones. The proof then follows by an argument based on categorically defined limits, which has an existence. It can also be verified directly that the presheaf Y^{X} is actually a sheaf. Finally, for the existence of the subobject-classifier Ω_{SetsCop}, it can be defined as

Ω_{SetsCop}(U) ≅ Hom(h_{U},Ω) ≅ {sub-presheaves of h_{U}} ≅ {sieves on U}, or alternatively, for the category of proper sheaves S_{hvs}(C,J), as

Ω_{Shvs}(C,J)(U) = {closed sieves on U}

Here it is worth reminding ourselves that the topology on T is defined by a basis K(p) = {Θ ⊂ T | ΣΘ = p}. Therefore, in the case of T-sets satisfying the strong ‘postulate of materialism’, Ω(p) consists of all sieves S (downward dense subsets) of T bounded by relation ΣS ≤ p. These sieves are further required to be closed. A sieve S with an envelope ΣS = s is closed if for any other r ≤ s, ie. for all r ≤ s, one has the implication

f^{∗}_{rs}(S) ∈ J(r) ⇒ f_{rs} ∈ S

, where f_{rs} : r → s is the unique arrow in the poset category. In particular, since ΣS = s for the topology whose basis consists of territories on s, we have the equation 1^{∗}_{s}(S) = f^{∗}_{ss}(S) = S ∈ J(s). Now the condition that the sieve is closed implies 1_{s} ∈ S. This is only possible when S is the maximal sieve on s—namely it consists of all arrows r → s for r ≤ s. In such a case S itself is closed. Therefore, in this particular case

Ω(p)={↓(s)|s ≤ p} = {hs | s ≤ p}

This is indeed a sheaf whose all amalgamations are ‘real’ in the sense of Badiou’s postulate of materialism. Thus it retains a suitable T-structure. Let us assume now that we are given an object A, which is basically a functor and thus a T-graded family of subsets A(p). For there to exist a sub-functor B ֒→ A comes down to stating that B(p) ⊂ A(p) for each p ∈ T. For each q ≤ p, we also have an injection B(q) ֒→ B(p) compatible (through the subset-representation with respect to A) with the injections A(q) ֒→ B(q). For any given x ∈ A(p), we can now consider the set

φp(x) = {q | q ≤ p and x q ∈ B(q)}

This is a sieve on p because of the compatibility condition for injections, and it is furthermore closed since the map x → Σφ_{p}(x) is in fact an atom and thus retains a real representative b ∈ B. Then it turns out that φ_{p}(x) =↓ (Eb). We now possess a transformation of functors φ : A → Ω which is natural (diagrammatically compatible). But in such a case we know that B ֒→ A is in turn the pull-back along φ of the arrow true, which is equivalent to the category of T-Sets.

# From Vector Spaces to Categories. Part 6.

We began by thinking of categories as “posets with extra arrows”. This analogy gives excellent intuition for the general facts about adjoint functors. However, our intuition from posets is insufficient to actually prove anything about adjoint functors.

To complete the proofs we will switch to a new analogy between categories and vector spaces. Let * V* be a vector space over a field

*and let*

**K***be the dual space consisting of*

**V ∗***. Now consider any*

**K-linear functions V → K***. We say that the function*

**K-bilinear function ⟨−,−⟩ ∶ V × V → K***is non-degenerate in both coordinates if we have*

**⟨−,−⟩****⟨u _{1},v⟩ = ⟨u_{2},v⟩ ∀ v ∈ V ⇒ u_{1} = u_{2}, ⟨u,v_{1}⟩ = ⟨u,v_{2}⟩ ∀ u ∈ V ⇒ v_{1} = v_{2}**

We say that two * K-linear operators L ∶ V ⇄ V ∶ R* define an adjunction with respect to

*if,*

**⟨−, −⟩***, we have*

**∀ vectors u,v ∈ V****⟨u, R(v)⟩ = ⟨L(u), v⟩**

Uniqueness of Adjoint Operators. Let * L ⊣ R* be an adjoint pair of operators with respect to a non-degenerate bilinear function

*. Then each of*

**⟨−, −⟩ ∶ V × V → K***and*

**L***determines*

**R**the other uniquely.

Proof: To show that * R* determines

*, suppose that*

**L***is another adjoint pair. Thus, ∀ vectors*

**L′ ⊣ R***we have*

**u,v ∈ V****⟨L(u), v⟩ = ⟨u, R(v)⟩ = ⟨L′(u), v⟩**

Now consider any vector * u ∈ V*. The non-degeneracy of

*tells us that*

**⟨−, −⟩****⟨L(u), v⟩ = ⟨L′(u), v⟩ ∀ v ∈ V ⇒ L(u) = L′(u)**

and since this is true * ∀ u ∈ V* we conclude that

**L = L′****RAPL for Operators:**

Suppose that the function * ⟨−, −⟩ ∶ V × V → K* is non-degenerate and continuous. Now let

*be any linear operator. If*

**T ∶ V → V***has a left or a right adjoint, then*

**T***is continuous.*

**T****Proof: **

Suppose that * T ∶ V → V* has a left adjoint

*, and suppose that the sequence of vectors*

**L ⊣ T***has a limit*

**v**_{i}∈ V*. Furthermore, suppose that the limit*

**lim**_{i}v_{i}∈ V*exists. Then for each*

**lim**_{i}T(v_{i}) ∈ V*, the continuity of*

**u ∈ V***in the second coordinate tells us that*

**⟨−, −⟩****⟨u, T (lim _{i}v_{i})⟩ = ⟨L(u), **

**lim**_{i}v_{i}⟩**= lim _{i}⟨L(u), **

**v**_{i}⟩**= lim _{i}⟨u,T(v_{i})⟩**

**= ⟨u, lim _{i}T (v_{i})⟩**

Since this is true for all * u ∈ V*, non-degeneracy gives

**T (lim _{i}v_{i}) = lim_{i}T (v_{i})**

The theorem can be made rigorous if we work with topological vector spaces. If * (V, ∥ − ∥)* is a normed (real or complex) vector space, then an operator

*is bounded if and only if it is continuous. Furthermore, if*

**T ∶ V → V***is a Hilbert space then an operator*

**(V,⟨−,−⟩)***having an adjoint is necessarily bounded, hence continuous. Many theorems of category have direct analogues in functional analysis. After all, Grothendieck began as a functional analyst.*

**T ∶ V → V**We can summarize these two results as follows. Let * ⟨−,−⟩ ∶ V ×V → K* be a

*. Then for each vector*

**K-bilinear function***we have two elements of the dual space*

**v ∈ V**

**H**^{v}, H_{v}∈ V^{∗}defined by

* H^{v} ∶= ⟨v,−⟩ ∶ V → K*,

**H _{v} ∶= ⟨−,v⟩ ∶ V → K**

The mappings * v ↦ H^{v}* and

*thus define two*

**v ↦ H**_{v}*from*

**K-linear functions***to*

**V***and*

**V**^{∗}: H(−) ∶V → V^{∗}

**H(−) ∶ V → V**^{∗}Furthermore, if the function is * ⟨−,−⟩* is non-degenerate and continuous then the functions

*are both injective and continuous.*

**H(−), H(−) ∶ V → V**^{∗}the hom bifunctor

* Hom_{C}(−,−) ∶ C^{op} × C → Set* behaves like a “non-degenerate and continuous bilinear function”……

# Functors. Part 5.

We have called * L ∶ P ⇄ Q ∶ R* an adjoint pair of functions, but of course they more than just functions. If

*is an adjunction,*

**L ⊣ R***says that*

**then property (2) of Galois connections***and*

**∀ p**_{1}, p_{2}∈ P*we have*

**q**_{1}, q_{2}∈ Q**p _{1} ≤ P p_{2} ⇒ L(p_{1}) ≤ Q L(p_{2}) and q_{1} ≤ Q q_{2} ⇒ R(q_{1}) ≤ P R(q_{2})**

That is, the functions * L ∶ P → Q* and

*are actually homomorphisms of posets.*

**R ∶ Q → P**Definition of Functor: Let * C* and

*be categories. A functor*

**D***consists a family of functions:*

**F ∶ C → D**• A function on objects * F ∶ Obj(C) → Obj(D)*,

• For each pair of objects * c_{1}, c_{2} ∈ C* a function on hom sets:

* F ∶ Hom_{C}(c_{1},c_{2}) → Hom_{D}(F(c_{1}),F(c_{2}))*. These functions must preserve the category structure:

(i) Identity: For all objects * c ∈ C* we have

*.*

**F (id**_{c}) = id_{F(c)}(ii) Composition: For all arrows * α, β ∈ C* such that

*is defined, we have*

**β ○ α****F (β ○ α) = F (β) ○ F (α)**

Functors compose in an associative way, and for each category * C* there is a distinguished identity functor

*. In other words, the collection of all categories with functors between them forms a (very big) category, which we denote by*

**id**_{C}∶ C → C*.*

**Cat**This definition is not surprising. It basically says that a functor * F ∶ C → D* sends commutative diagrams in

*to commutative diagrams in*

**C***. That is, for each diagram*

**D***in*

**D ∶ I → C***we have a diagram*

**C***in*

**F**^{I}(D) ∶ I → D*(defined by*

**D***), which is commutative if and only if*

**F**^{I}(D) ∶= F ○ D*is.*

**D**Now let’s try to guess the definition of an “adjunction of categories”. Let * C* and

*be categories and let*

**D***be any two functors. When*

**L ∶ C ⇄ D ∶ R***is a poset, recall that*

**C***we have*

**∀ x, y ∈ C***and we use the notations*

**∣Hom**_{C}(x, y)∣ ∈ {0, 1}**x ≤ y ⇐⇒ ∣Hom _{C}(x,y)∣ = 1**

**x ≤/ y ⇐⇒ ∣Hom _{C}(x,y)∣ = 0**

Thus if * C* and

*are posets, we can rephrase the definition of a poset adjunction*

**D***by stating that ∀ objects*

**L ∶ C ⇄ D ∶ R***and*

**c ∈ C***there exists a bijection of hom sets:*

**d ∈ D****Hom _{C} (c, R(d)) ←→ Hom_{D} (L(d), c)**

In this form the definition now applies to any pair of functors between categories.

However, if we want to preserve the important theorems (uniqueness of adjoints and RAPL) then we need to impose some “naturality” condition on the family of bijections between hom sets. This condition is automatic for posets, so we will have to look elsewhere for motivation.

**Adjoint Functors: **

Let * C* and

*be categories. We say that a pair of functors*

**D***is an adjunction if for all objects*

**L ∶ C ⇄ D ∶ R***and*

**c ∈ C***there exists a bijection of hom sets*

**d ∈ D****Hom _{C} (c, R(d)) ←→ Hom_{D} (L(d), c)**

Furthermore, we require that these bijections fit together in the following “natural” way. For each arrow * γ ∶ c_{2} → c_{1} in C* and each arrow

*we require that the following cube of functions commutes:*

**δ ∶ d**_{1}→ d_{2}in D**Natural Transformation: **

Let C and D be categories and consider two parallel functors * F_{1},F_{2} ∶ C → D*. A natural transformation

*consists of a family of arrows*

**Φ ∶ F ⇒ G***, one for each object*

**Φ**_{c}∶F(c) → R(c)*, such that for each arrow*

**c ∈ C***the following square commutes:*

**γ ∶ c**_{1}→ c_{2}in CThe figure below the square is called a “2-cell diagram”. It hints at the close relationship between category theory and topology.

Let * D^{C}* denote the collection of all functors from

*to*

**C***and natural transformations between them. One can check that this forms a category (called a functor category). Given*

**D***we say that*

**F**_{1}, F_{2}∈ D^{C}*and*

**F**_{1}*are naturally isomorphic if they are isomorphic in*

**F**_{2}*, i.e., if there exists a pair of natural transformations*

**D**^{C}*and*

**Φ ∶ F**_{1}⇒ F_{2}*such that*

**Ψ ∶ F**_{2}⇒ F_{1}*and*

**Ψ ○ Φ = id**_{F1}*are the identity natural transformations. In this case we will write*

**Ψ ○ Φ = id**_{F2}*and we will say that*

**F**_{1}≅ F_{2}*and*

**Φ***are natural isomorphisms.*

**Φ**^{−1}∶= ΨTo develop some intuition for this definition, let * I* be a small category and let

*be any category. We have previously referred to functors*

**C***as “diagrams of shape*

**D ∶ I → C***in*

**I***“. Now we can think of*

**C***as a category of diagrams. Given two such diagrams*

**C**^{I}*, we visualize a natural transformation*

**D**_{1}, D_{2}∈ C^{I}*as a “cylinder”:*

**Φ ∶ D**_{1}⇒ D_{2}The diagrams D1 and D2 need not be commutative, but if they are then the whole cylinder is commutative.

**Limit/Colimit: **

Consider a diagram * D ∶ I → C*. The limit of

*, if it exists, consists of an object*

**D***and a canonical natural transformation*

**lim**_{I}D ∈ C*such that for each object*

**Λ ∶ (lim**_{I}D)^{I}⇒ D*and natural transformation*

**c ∈ C***there exists a unique natural transformation*

**Φ ∶ c**^{I}⇒ D*making the following diagram in*

**υ**^{I}∶ c^{I}⇒ (lim_{I}D)^{I}*commute:*

**C**^{I}**Hom Functors:**

Let * C *be a category. For each object

*the mapping*

**c ∈ C***defines a functor from*

**d ↦ Hom**_{C}(c, d)*to the category of sets Set. We denote it by*

**C****H ^{c} ∶= Hom_{C}(c,−) ∶ C → Set**

To define the action of * H^{c}* on arrows, consider any

*. Then we must have a function*

**δ ∶ d**_{1}→ d_{2}in C*, i.e., a function*

**H**^{c}(δ) ∶ H^{c}(d_{1}) → H^{c}(d_{2})*. There is only one way to define this:*

**H**^{c}(δ) ∶ Hom_{C}(c,d_{1}) → Hom_{C}(c,d_{2})**H ^{c }(δ) (φ) ∶= δ ○ φ**

Similarly, for each arrow * δ ∶ c_{1} → c_{2}* we can define a function

*by*

**H**^{c}(δ) ∶ Hom_{C}(d_{2},c) → Hom_{C}(d_{1},c)*. This again defines a functor into sets, but this time it is from the opposite category*

**H**^{c }(δ) (φ) ∶= φ ○ δ*(which is defined by reversing all arrows in*

**C**^{op}*):*

**C****H _{c} ∶= Hom_{C}(−,c) ∶ C^{op} → Set**

Finally, we can put these two functors together to obtain the hom bifunctor

**Hom _{C}(−,−)∶ C^{op} ×C →Set **

which sends each pair of arrows (* γ ∶ c_{2} → c_{1}, δ ∶ d_{1} → d_{2}*) to the function

**Hom _{C}(γ,δ) ∶ Hom_{C}(c_{1},d_{1}) → Hom_{C}(c_{2},d_{2})**

defined by * φ ↦ δ ○ φ ○ γ*. The product category

**C**^{op}*is defined in the most obvious way.*

**× C****Adjoint Functors: **

Let * C, D* be categories and consider a pair of functors

*. By composing these with the hom bifunctors*

**L ∶ C ⇄ D ∶ R**

**Hom**_{C}*and*

**(−, −)**

**Hom**_{D}*we obtain two parallel bifunctors:*

**(−, −)*** Hom_{C}(−,R(−))∶ C^{op }× D → Set* and

**Hom**_{D}

**(L(−),−)∶ C**^{op }× D → SetWe say that * L ∶ C ⇄ D ∶ R* is an adjunction if the two bifunctors are naturally isomorphic:

* Hom_{C} (−, R(−))* ≅

**Hom**_{D}*……..*

**(L(−), −)**

# Marching Along Categories, Groups and Rings. Part 2

A category C consists of the following data:

A collection * Obj(C)* of objects. We will write “

*” to mean that “*

**x ∈ C***“*

**x ∈ Obj(C)**For each ordered pair * x, y ∈ C* there is a collection

*of arrows. We will write*

**Hom**_{C}(x, y)*to mean that*

**α∶x→y***. Each collection*

**α ∈ Hom**_{C}(x,y)*has a special element called the identity arrow*

**Hom**_{C}(x,x)*. We let*

**id**_{x}∶ x → x*denote the collection of all arrows in*

**Arr(C)***.*

**C**For each ordered triple of objects * x, y, z ∈ C* there is a function

* ○ ∶ Hom_{C} (x, y) × Hom_{C}(y, z) → Hom_{C} (x, z)*, which is called composition of arrows. If

*and*

**α ∶ x → y***then we denote the composite arrow by*

**β ∶ y → z***.*

**β ○ α ∶ x → z**If each collection of arrows * Hom_{C}(x,y)* is a set then we say that the category

*is locally small. If in addition the collection*

**C***is a set then we say that*

**Obj(C)***is small.*

**C**Identitiy: For each arrow * α ∶ x → y* the following diagram commutes:

Associative: For all arrows * α ∶ x → y, β ∶ y → z, γ ∶ z → w*, the following diagram commutes:

We say that * C′ ⊆ C* is a subcategory if

*and if*

**Obj(C′) ⊆ Obj(C)***we have*

**∀ x,y ∈ Obj(C′)***. We say that the subcategory is full if each inclusion of hom sets is an equality.*

**Hom**_{C′}(x,y) ⊆**Hom**_{C}**(x,y)**Let * C* be a category. A diagram

*is a collection of objects in*

**D ⊆ C***with some arrows between them. Repetition of objects and arrows is allowed. OR. Let I be any small category, which we think of as an “index category”. Then any functor*

**C***is called a diagram of shape*

**D ∶ I → C***in*

**I***. In either case, we say that the diagram*

**C***commutes if for all pairs of objects*

**D***in*

**x,y***, any two directed paths in*

**D***from*

**D***to*

**x***yield the same arrow under composition.*

**y**Identity arrows generalize the reflexive property of posets, and composition of arrows generalizes the transitive property of posets. But whatever happened to the antisymmetric property? Well, it’s the same issue we had before: we should really define equivalence of objects in terms of antisymmetry.

Isomorphism: Let * C* be a category. We say that two objects

*are isomorphic in*

**x,y ∈ C***if there exist arrows*

**C***and*

**α ∶ x → y***such that the following diagram commutes:*

**β ∶ y → x**

In this case we write * x ≅_{C} y*, or just

*if the category is understood.*

**x ≅ y**If * γ ∶ y → x* is any other arrow satisfying the same diagram as

*, then by the axioms of identity and associativity we must have*

**β****γ = γ ○ id _{y} = γ ○ (α ○ β) = (γ ○ α) ○ β = id_{x} ○ β = β**

This allows us to refer to * β* as the inverse of the arrow

*. We use the notations*

**α***and*

**β = α**^{−1}* β^{−1} = α*.

A category with one object is called a monoid. A monoid in which each arrow is invertible is called a group. A small category in which each arrow is invertible is called a groupoid.

Subcategories of Set are called concrete categories. Given a concrete category * C ⊆ Set* we can think of its objects as special kinds of sets and its arrows as special kinds of functions. Some famous examples of conrete categories are:

• Grp = groups & homomorphisms

• Ab = abelian groups & homomorphisms

• Rng = rings & homomorphisms

• CRng = commutative rings & homomorphisms

Note that Ab ⊆ Grp and CRng ⊆ Rng are both full subcategories. In general, the arrows of a concrete category are called morphisms or homomorphisms. This explains our notation of **Hom _{C}**

_{.}

* Homotopy*: The most famous example of a non-concrete category is the fundamental groupoid

*of a topological space*

**π**_{1}(X)*. Here the objects are points and the arrows are homotopy classes of continuous directed paths. The skeleton is the set*

**X***of path components (really a discrete category, i.e., in which the only arrows are the identities). Categories like this are the reason we prefer the name “arrow” instead of “morphism”.*

**π**_{0}(X)* Limit/Colimit*: Let

*be a diagram in a category*

**D ∶ I → C***(thus*

**C***is a functor and*

**D***is a small “index” category). A cone under*

**I***consists of*

**D**• an object * c ∈ C*,

• a collection of arrows * α_{i} ∶ x → D(i)*, one for each index

*,*

**i ∈ I**such that for each arrow * δ ∶ i → j* in

*we have*

**I**

**α**_{j}= D(δ) ○ α_{i }In visualizing this:

The cone * (c,(α_{i})_{i∈I})* is called a limit of the diagram

*if, for any cone*

**D***under*

**(z,(β**_{i})_{i∈I})*, the following picture holds:*

**D**[This picture means that there exists a unique arrow * υ ∶ z → c* such that, for each arrow

*in*

**δ ∶ i → j***(including the identity arrows), the following diagram commutes:*

**I**When * δ = id_{i}* this diagram just says that

*. We do not assume that*

**β**_{i}= α_{i}○ υ*itself is commutative. Dually, a cone over*

**D***consists of an object*

**D***and a set of arrows*

**c ∈ C***satisfying*

**α**_{i}∶ D(i) → c*for each arrow*

**α**_{i}= α_{j}○ D(δ)*in*

**δ ∶ i → j***. This cone is called a colimit of the diagram*

**I***if, for any cone*

**D***over*

**(z,(β**_{i})_{i∈I})*, the following picture holds:*

**D**When the (unique) limit or colimit of the diagram * D ∶ I → C* exists, we denote it by

*or*

**(lim**_{I}D, (φ_{i})_{i∈I})*, respectively. Sometimes we omit the canonical arrows*

**(colim**_{I}D, (φ_{i})_{i∈I})*from the notation and refer to the object*

**φ**_{i}*as “the limit of D”. However, we should not forget that the arrows are part of the structure, i.e., the limit is really a cone.*

**lim**_{I}D ∈ CPosets: Let P be a poset. We have already seen that the product/coproduct in P (if they exist) are the meet/join, respectively, and that the final/initial objects in P (if they exist) are the top/bottom elements, respectively. The only poset with a zero object is the one element poset.

Sets: The * empty set ∅ ∈ Set* is an initial object and the

**one point***is a final object. Note that two sets are isomorphic in Set precisely when there is a bijection between them, i.e., when they have the same cardinality. Since initial/final objects are unique up to isomorphism, we can identify the initial object with the cardinal number 0 and the final object with the cardinal number 1. There is no zero object in Set.*

**set ∗ ∈ Set**Products and coproducts exist in Set. The product of * S,T ∈ Set* consists of the Cartesian product

*together with the canonical projections*

**S × T***and*

**π**_{S}∶ S × T → S*. The coproduct of*

**π**_{T}∶ S × T → T*consists of the disjoint union*

**S, T ∈ Set***together with the canonical injections*

**S ∐ T***and*

**ι**_{S}∶ S → S ∐ T*. After passing to the skeleton, the product and coproduct of sets become the product and sum of cardinal numbers.*

**ι**_{T}∶ T → S ∐ T[Note: The “external disjoint union” * S ∐ T* is a formal concept. The familiar “internal disjoint union”

*is only defined when there exists a set*

**S ⊔ T***containing both*

**U***and*

**S***as subsets. Then the union*

**T***is the join operation in the Boolean lattice*

**S ∪ T***; we call the union “disjoint” when*

**2**^{U}*.]*

**S ∩ T = ∅**Groups: The trivial group * 1 ∈ Grp* is a zero object, and for any groups

*the zero homomorphism*

**G, H ∈ Grp***sends all elements of*

**1 ∶ G → H***to the identity element*

**G***. The product of groups*

**1**_{H}∈ H*is their direct product*

**G, H ∈ Grp***and the coproduct is their free product*

**G × H***, along with the usual canonical morphisms.*

**G ∗ H**Let * Ab ⊆ Grp* be the full subcategory of abelian groups. The zero object and product are inherited from

*, but we give them new names: we denote the zero object by*

**Grp***and for any*

**0 ∈ Ab***we denote the zero arrow by*

**A, B ∈ Ab***. We denote the Cartesian product by*

**0 ∶ A → B***and we rename it the direct sum. The big difference between*

**A ⊕ B***and*

**Grp***appears when we consider coproducts: it turns out that the product group*

**Ab***is also the coproduct group. We emphasize this fact by calling*

**A ⊕ B***the biproduct in*

**A ⊕ B***. It comes equipped with four canonical homomorphisms*

**Ab***satisfying the usual properties, as well as the following commutative diagram:*

**π**_{A}, π_{B}, ι_{A}, ι_{B}This diagram is the ultimate reason for matrix notation. The universal properties of product and coproduct tell us that each endomorphism * φ ∶ A ⊕ B → A ⊕ B* is uniquely determined by its four components

*for*

**φ**_{ij}∶= π_{i}○ φ ○ ι_{j}*,so we can represent it as a matrix:*

**i, j ∈ {A,B}**Then the composition of endomorphisms becomes matrix multiplication.

Rings. We let * Rng* denote the category of rings with unity, together with their homomorphisms. The initial object is the ring of integers

*and the final object is the zero ring*

**Z ∈ Rng***, i.e., the unique ring in which*

**0 ∈ Rng***. There is no zero object. The product of two rings*

**0**_{R}= 1_{R}*is the direct product*

**R, S ∈ Rng***with component wise addition and multiplication. Let*

**R × S ∈ Rng***be the full subcategory of commutative rings. The initial/final objects and product in*

**CRng ⊆ Rng***are inherited from*

**CRng***. The difference between*

**Rng***and*

**Rng***again appears when considering coproducts. The coproduct of*

**CRng***is denoted by*

**R,S ∈ CRng***and is called the tensor product over*

**R ⊗**_{Z}S*…..*

**Z**

# From Posets to Categories. Part 1

A poset (partially ordered set) is a pair * (P, ≤)*, where

* P* is a set,

* ≤* is a binary relation on P satisfying the three axioms of partial order:

(i) Reflexive: **∀x ∈ P, x ≤ x**

(ii) Antisymmetric: **∀x ,y ∈ P, x ≤ y & y ≤ x ⇒ x = y**

(iii) Transitive: * ∀x, y, z ∈ P, x ≤ y & y ≤ z ⇒ x ≤ z*.

And what does this have to do with category theory?

**“x ≤ y” ⇐⇒ “ x → y” “x = y” ⇐⇒ “ x ↔ y”**

Given * x, y ∈ P*,

we say that * u ∈ P* is a least upper bound of

*if we have*

**x, y ∈ P***&*

**x → u***, and for all*

**y → u***satisfying*

**z ∈ P***&*

**x → z***we must have*

**y → z***. It is more convenient to express this definition with a picture. We say that*

**u → z***is a least upper bound of*

**u ∈ P***,*

**x***if for all*

**y***the following picture holds:*

**z ∈ P**Dually, we say that * l ∈ P* is a greatest lower bound of

*if for all*

**x, y***the following picture holds:*

**z ∈ P**Now suppose that * u_{1}, u_{2} ∈ P* are two least upper bounds for

*. Applying the defininition in both directions gives*

**x, y*** u_{1} → u_{2}* and

*,*

**u**_{2}→ u_{1}and then from antisymmetry it follows that * u_{1}* =

*, which just means that*

**u**_{2}*and*

**u**_{1}*are indistinguishable within the structure of*

**u**_{2}*. For this reason we can speak of the least upper bound (or “join”) of*

**P***. If it exists, we denote it by*

**x, y****x ∨ y**

Dually, if it exists, we denote the greatest lower bound (or “meet”) by

**x ∧ y**

The definitions of meet and join are called “universal properties”. Whenever an object defined by a universal property exists, it is automatically unique in a certain canonical sense. However, since the object might not exist, maybe it is better to refer to a universal property as a “characterization,” or a “prescription,” rather than a “definition.”

Let * P* be a poset. We say that

*is a top element*

**t ∈ P**if for all * z ∈ P* the following picture holds:

**z —> t**

Dually, we say that * b ∈ P* is a bottom element if for all

*the following picture holds:*

**z ∈ P****b —> z**

For any subset of elements of a poset * S ⊆ P* we say that the element

*is its join if for all*

**⋁ S ∈ P***the following diagram is satisfied:*

**z ∈ P**Dually, we say that * ⋀ S ∈ P* is the meet of

*if for all*

**S***the following diagram is satisfied:*

**z ∈ P**If the objects * ⋁ S* and

*exist then they are uniquely characterized by their universal properties.*

**⋀ S**The universal properties in these diagrams will be called the “limit” and “colimit” properties when we move from posets to categories. Note that a limit/colimit diagram looks like a “cone over S”. This is one example of the link between category theory and topology.

Note that all definitions so far are included in this single (pair of) definition(s):

* ⋁ {x, y} = x∨ y *&

**⋀ {x, y} = x ∧ y*** ⋁∅ = 0* &

*.*

**⋀ ∅ = 1**