Title: Theory / Unimportant / Math / Category / Mac Lane=
Tags: examplesSwitchExample1 toBinaryRelation fromMath hasSeenConjDef mathToSee closed duplicateToOtherBook TypToProperty defToFunctor markedtoBinaryRelation todo typToBinary
On every note of this deck, unless a note says otherwise:
Chapter: 1

Name: A metagraph
Definition: Objects, arrows, and two operations Domain and Codomain sending each arrow to an object
Page: 7

Name: A metacategory
Definition: Identity which sends an object \(O\) to an arrow \(O\to O\)
Page: 7
Definition2: Compositions on pairs of arrows \((f,g)\) such that \(\text{dom }g=\text{cod } f\), to an arrow \(\text{dom}g\to\text{cod }f\)
such that Composition is associative and the image of Identity is an identity for Composition
Conjdef: A metagraph with the additional operations

Name: The identity function
Definition: \(s\mapsto s\)
Page: 9
Variables: \(S\) a set
Notation: \(1_S\)

Name: The inclusion function
Definition: \(s\mapsto s\)
Page: 9
Variables: \(S\subseteq Y\) sets
Name2: The insertion function

Name: A category
Definition: An interpretation of the category axioms within set theory
Page: 10
Section: 2

Name: A directed graph
Definition: A set \(O\) of objects, a set \(A\) of arrows, and two functions
Page: 10
Section: 2
Name2: A diagram scheme

Name: The set of composable pairs
Definition: The set of elements \((g,f)\in A^2\) such that \(\text{dom }g=\text{cod }f\)
Page: 10
Section: 2
Variables: a directed graph
Notation: \(A\times_O A\)
Extra: c2?

Definition: The set of functions \(f\) in \(C\) such that \(\text{dom }f=b\) and \(\text{cod }f=c\)
Page: 10
Section: 2
Variables: \(C\) a category, \(b,c\in C\)
Notation: \(\text{hom}(b,c)\)

Objects: One object
Arrows: Its identity arrow
Page: 10
Chapter: 1
Section: 2
Compositions: \((1,1)\mapsto1\)

Objects: none
Page: 10
Chapter: 1
Section: 2
Notation: 0

Objects: Two objects \(a,b\)
Arrows: An arrow \(a\to b\)
Page: 10
Chapter: 1
Section: 2

Objects: Three objects
Page: 11
Chapter: 1
Section: 2
Notation: 3

Objects: Two objects \(a,b\)
Arrows: Two arrows from \(a\) to \(b\)
Page: 11
Chapter: 1
Section: 2
Compositions: Only the trivial ones

Name: Parallel arrows
Definition: Arrows with same domain and codomain
Page: 11
Section: 2
Variables: C a category

Objects: Arbitrary
Arrows: Only identities
Page: 11
Chapter: 1
Section: 2
Name: A discrete category
Compositions: Trivial

Objects: One object
Arrows: Arbitrary
Page: 11, 37
Chapter: 1, 3
Section: 2
Name: A (categorical) monoid
Extra: , for programmers

Objects: One object
Arrows: Each arrow have a two-sided inverse
Page: 11
Chapter: 1
Section: 2
Name: A group

Objects: Positive integers
Arrows: An arrow \(m\to n\) is a \(m\times n\) matrix
Page: 11
Chapter: 1
Section: 2
Notation: Matr_K
Variables: K a commutative ring
Compositions: usual matrix product

Objects: Elements of V
Arrows: Functions from X to Y
Page: 11
Chapter: 1
Section: 2
Notation: Ens_V
Variables: V a set

Objects: Arbitrary
Arrows: At most one between each pair of objects
Page: 11
Chapter: 1
Section: 2
Name: A preorder
Compositions: The composition of \(A\to B\) and \(B\to C\) is the only arrow \(A\to C\)

Objects: Finite ordinals
Arrows: Order preserving functions
Page: 12
Chapter: 1
Section: 2
Name: Simplicial category

Objects: Finite ordinals
Arrows: arbitrary functions
Page: 12
Chapter: 1
Section: 2
Notation: Finord

Name: A functor
Definition: A function sending objects to objects, arrows to arrows, identity to identity, and preserving compositions
Page: 13
Section: 3
Example: The powerset function, sending sets to their power set and functions \(f:X\to Y\) to \(A\mapsto \{f(a)\mid a\in A\}\)

Name: A forgetful functor
Definition: A functor which forgets a part of some structure of an algebraic object
Page: 14
Section: 3
Name2: An underlying functor

Name: An isomorphism of category
Definition: with a two-sided inverse.
Page: 14
Section: 3
Variables: \(\mathbf {C,D}\) categories
Definition2: which is a bijection on objects and arrows.
Conjdef: A functor, equivalently

Name: A full functor \(\mathbf{C\to D}\)
Definition: \(F:\mathbf{C\to D}\) such that, for each \(c,c'\in C\), for each \(g:F(c)\to F(c')\), there exists \(f:c\to c'\) such that \(g=Tf\)
Page: 14
Section: 3
Variables: \(\mathbf {C,D}\) categories
Definition2: surjective on hom-sets
Conjdef: A functor, equivalently

Name: A faithful functor \(\mathbf{C\to D}\)
Definition: \(F:\mathbf{C\to D}\) such that, for each \(c,c'\in C\), for each \(f,f':c\to c'\), \(Tf=Tf'\) implies \(f=f'\).
Page: 15
Section: 3
Variables: \(\mathbf {C,D}\) categories
Definition2: injective on hom-sets
Conjdef: A functor, equivalently
Name2: An embedding

Name: Faithful functors
Page: 15
Section: 3
Chapter: 1
Under: composition

Name: Full functors
Page: 14
Section: 3
Chapter: 1
Under: composition

Name: A subcategory of C
Definition: A category which is a collection of some of the objects and some of the arrows of C
Page: 15
Section: 3
Variables: C a category

Page: 15
Object: Identity
Domain: S
Section: 3
Chapter: 1
Codomain: C
Variables: C a category
S a subcategory
Name: The inclusion functor from S to C
Faithfull: Faithful

Name: A full subcategory
Definition: A subcategory whose inclusion functor is full
Page: 15
Section: 3
Variables: C a category
Example: Inclusion from finite sets to sets

Name: A natural transformation from \(S\) to \(C\)
Definition: every arrow \(f:c\to c'\) in \(C\) is such that \(\tau_c\circ Tf=Sf\circ\tau_{c'}\).
Page: 16
Section: 4
Variables: \(B,C\) categories
\(S,T:C\to B\) functors
Example: Let \(GL_n\) the morphism sending a field \(K\) to \(GL_n(K)\) and a function \(f\) to its application term by term.
Let \((\ )^*\) the morphism sending a field \(K\) to its set of units, restricting a morphism \(f:K\to K'\) to \(K^*\to K^{\prime *}\)
Determinant, as a transformation\(Gl_n\to (\ )^*\) is natural two functors CRng-> Grp .
Conjdef: A function assigning to each \(c\in C\) an arrow \(\tau_c:Sc\to Tc\) in \(B\), in such a way that
Name2: A morphism of functors
Example2: Todo:algebraic
Denoted by: \(\tau\)

Name: The arrow natural in \(c\)
Definition: \(\tau_c\)
Page: 16
Section: 4
Variables: \(B,C\) categories
\(S,T:C\to B\) functors
\(\tau:S\to T\) a natural transformation
Intuition: If we think of the functor S as giving a picture in B of (all the objects and arrows of) C, then a natural transformation r is a set of arrows mapping (or, translating) the picture S to the picture T, with all squares (and parallelograms 1) like that above commutative:

Name: The components of \(\tau\)
Definition: \(\tau_a\) for \(a\in C\)
Page: 16
Section: 4
Variables: \(B,C\) categories
\(S,T:C\to B\) functors
\(\tau:S\to T\) a natural transformation

Page: 16, 17
Definition: A natural transformation \(\tau:S\to T\) with every component \(\tau c\) invertible in \(B\)
Chapter: 1
Section: 4
Variables: \(B,C\) categories
\(S,T:C\to B\) functors
Notation: \(\cong\)
Name: A natural equivalence
Context: Categories
Example: Morphism from finite vector space to its double dual
Name2: Natural isomorphism
Example2: Let \(\#:\mathbf{Set}_f\to\mathbf{Ord}\) the functor sending a finite set to its ordinal. With \(\theta_X:X\to\#X\) an arbitrary bijection, and \(\theta (f:X\to Y)\) being \(\theta_Y f\theta^{-1}_X\)
Let \(S:\mathbf{Ord}\to\mathbf{Set}_f\) the inclusion functor
\(\#S\) and \(S\#\) are natural isomorphism

Definition: [$]\defFun{T}{S}{c}{(\tau c)^{-1}}[/$]
Page: 16
Section: 4
Variables: \(B,C\) categories
\(S,T:C\to B\) functors
\(\tau:S\to T\) a natural equivalence
Context: Categories
Notation: \(\tau^{-1}\)

Name: An equivalence between \(C\) and \(D\)
Definition: A pair of functors \(S:C\to D\) and \(T:D\to C\) together with natural isomorphism \(T\circ S\) and \(S\circ T\).
Page: 18
Section: 4
Variables: \(C,D\) categories
Example: Let \(\#:\mathbf{Set}_f\to\mathbf{Ord}\) the functor sending a finite set to its ordinal. With \(\theta_X:X\to\#X\) an arbitrary bijection, and \(\theta (f:X\to Y)\) being \(\theta_Y f\theta^{-1}_X\)
Let \(S:\mathbf{Ord}\to\mathbf{Set}_f\) the inclusion functor
\(\#S\) and \(S\#\) are natural isomorphism

Name: An invertible arrow
Definition: An arrow \(e:a\to b\) such that therre exists \(e':b\to a\) with \(e'e=1_a\) and \(ee'=1_b\)
Page: 19
Section: 5
Variables: \(\mathbf C\) a category
Name2: An isomorphism

Name: A left inverse of \(e\)
Definition: An arrow \(e':b\to a\) such that \(e'e=1_a\)
Page: 19
Section: 5
Variables: \(\mathbf C\) a category,
\(e:a\to b\)
Name2: A retraction for \(e\)
Cardinal: At most one

Name: An inverse of \(e\)
Definition: An arrow \(e':b\to a\) such that \(e'e=1_a\) and \(ee'=1_b\)
Page: 19
Section: 5
Variables: \(\mathbf C\) a category,
\(e:a\to b\)
Notation: \(e^{-1}\)
Cardinal: At most one

Name: A right inverse of \(e\)
Definition: An arrow \(e':b\to a\) such that \(ee'=1_b\)
Page: 19
Section: 5
Variables: \(\mathbf C\) a category,
\(e:a\to b\)
Name2: A section of \(e\)
Cardinal: At most one

Question: \((ee')^{-1}\)
Answer: equals \(e^{\prime -1}e^{-1}\)
Page: 19
Chapter: 1
Section: 5
Variables: \(\mathbf C\) a category,
\(e:a\to b,e':b\to c\)

Page: 19
Definition: Two objects \(a,b\) such that there exists an invertible arrow \(a\to b\)
Chapter: 1
Section: 5
Variables: \(\mathbf C\) a category
Notation: \(\cong\)
Name: Isomorphic objects in \(\mathbf C\)
Equivalence: Equivalence relation
Extra: For programmer. Ch 5. 4. P 61
Mac Lane. Ch 1. 5. P 19

Name: A monic arrow in \(C\)
Definition: An arrow \(a:y\to z\) such that, for any \(f_1,f_2:x\to y\), \(a\circ f_1=a\circ f_2\) implies \(f_1=f_2\).
Page: 19
Section: 5
Variables: \(\mathbf C\) a category
Example: Injonctions in Set and Grp
Name2: A left-cancellable arrow
Denoted by: \(m\)

Name: An epic arrow in \(C\)
Definition: An arrow \(a:x\to y\) such that, for any \(f_1,f_2:y\to z\), \(f_1\circ a= f_2\circ a\) implies \(f_1=f_2\).
Page: 19
Section: 5
Variables: \(\mathbf C\) a category
Example: surjections
Name2: Right-inversible function
Applied to: Set

Smaller0: \(e\) has a right inverse
Greater0: \(e\) is epi
Page: 19
Chapter: 1
Section: 5
Variables: C a category
\(e:a\to b\) in C
Counterexample: Grp

Smaller0: \(e\) has a left inverse
Greater0: \(e\) is monic
Page: 19
Chapter: 1
Section: 5
Variables: C a category
\(e:a\to b\) in C

Name: A split epi
Definition: an arrow with a right inverse
Page: 19
Section: 5
Variables: C a category

Name: A split mono
Definition: an arrow with a left inverse
Page: 19
Section: 5
Variables: C a category

Smaller0: \(gh=1_b\)
Greater0: idempotent
Page: 19
Chapter: 1
Section: 5
Variables: C a category
\(g:a\to b\), \(h:b\to a\)
Prefix Greater: \(hg\) is

Name: An idempotent in C
Definition: An arrow \(e:a\to a\) such that \(e=ee\)
Page: 20
Section: 5
Variables: C a category

Name: A terminal object in C
Definition: An object \(a\) such that, for each object \(b\) there is a unique \(f:b\to a\) in C
Page: 20
Section: 5
Variables: C a category
Example: Singleton in Set
Example2: trivial group in Grp
Cardinal: At most one, up to iso

Name: A initial object in C
Definition: An object \(a\) such that, for each object \(b\) there is a unique \(f:a\to b\) in C
Page: 20
Section: 5
Variables: C a category
Example: Empty set
Example2: trivial group
Applied to: Set
Applied to2: Grp
Cardinal: At most one, up to iso

Name: A null object in C
Definition: An initial and terminal object
Page: 20
Section: 5
Variables: C a category
Example: 0 in Ab and in R-Mod
Name2: Biterminator object
Example2: Singleton in Set_*
Name3: Zero object
Cardinal: At most one, up to iso

Name: The zero arrow from \(a\) to \(b\)
Definition: The composition of the unique arrows from \(a\) to the zero object, and from the zero object to \(b\)
Page: 20
Section: 5
Variables: C a category,\(a,b\in\mathbf C\)
Assuming: C has a zero object

Name: Zero arrows
Page: 20
Section: 5
Chapter: 1
Variables: C a category
Under: composition

Name: A groupoid
Definition: A category in which every arrow is invertible.
Page: 20
Section: 5
Example: Fundamental groupoid \(\pi(X)\) of a topological space \(X\)

Objects: Points of \(X\)
Arrows: an arrow \(x\to x'\) is an homotopy class of paths \(f:x\to x'\).
Page: 20
Chapter: 1
Section: 5
Notation: \(\pi(X)\)
Name: The fundamental groupoid of \(X\)
Variables: \(X\) a topological class
Compositions: The homotopy class obtained by composing two pathes

Name: A path \(x\to x'\)
Definition: with \(f(0)=x\), \(f(1)=x'\)
Page: 20
Section: 5
Variables: \(X\) a topological class,
\(x,x'\in X\)
Definition2: continuous
Conjdef: \(f:[0,1]\to X\)

Name: An homotopy class \(x\to x'\)
Definition: The equivalence class for homotopic relation of a path \(x\to x'\)
Page: 20
Section: 5
Variables: \(X\) a topological class,
\(x,x'\in X\)

Page: 20
Definition: there is a continuous function \(F[0,1]^2\to X\) with \(F(t,0)=f(t)\), \(F(t,1)=g(t)\)
Chapter: 1
Section: 5
Variables: \(X\) a topological class,
Name: Homotopic pathes
Definition2: \(F(0,s)=F(0,s')\) and \(F(1,s)=F(1,s')\)
Symmetry: The relation on pair of path \(f,g:[0,1]\to X\) such that

Name: The inverse of \(f\)
Definition: The path \(x\mapsto f(1-x)\)
Page: 20
Section: 5
Variables: \(X\) a topological class,
\(f\) a path
Notation: \(f^{-1}\)

Page: 20
Definition: The relation on pair of path \(f,g:[0,1]\to X\) such that \(\text{hom}_{\pi(X)}(x,x)\)
Chapter: 1
Section: 5
Variables: \(X\) a topological class,
Name: The fundamental group of \(X\)
Assuming: \(X\) is path-connected

Page: 20
Definition: The path \(x\mapsto f(x/2)\) for \(x\le \frac12\) and \(x\mapsto g((x-1)/2)\) otherwise
Chapter: 1
Section: 5
Variables: \(X\) a topological class,
\(f,g\) path, with \(f(1)=g(0)\)
Notation: \(fg\)
Name: The composite of \(f\) and \(g\)
Symmetry: The relation on pair of pathe \(f,g:[0,1]\to X\) such that

Smaller0: C is a groupoid
Greater0: \(\text{hom}_G(x,x)\)
Page: 20
Chapter: 1
Section: 5
Variables: C a category,
\(x\) an object
Greater1: a group
Connect Greater0: is

Smaller0: C is a groupoid
Greater0: \(\text{hom}_G(x,x)\) is isomorphic to \(\text{hom}_G(x',x')\)
Page: 20
Chapter: 1
Section: 5
Variables: C a category
\(x,x'\) objects
\(f:x\to x'\)
Construction: [$]\defFun{\text{hom}_G(x,x)}{\text{hom}_G(x',x')}{g}{fgf^{-1}}[/$]

Smaller0: a groupoid
Greater0: determined by its set of objects and a group \(\text{hom}_G(x,x)\)
Page: 20
Chapter: 1
Section: 5
Variables: C a category
Prefix Greater: C is
Smaller1: connected