Title: Theory / Unimportant / Math / Category / Examples=
Tags: functor

Objects: Finite sets
Arrows: Functions
Page: 005
Chapter: 1
Section: 1.4
Notation: Sets_fin
Name: Category of finite sets
Extra: merge
Kind: Ex
Ccc: Yes

Objects: Posets
Arrows: Monotone functions
Page: 007, 35, 35, 122, 36, 58
Chapter: 1,2, 2, 6, 2, 3
Section: 1.4, 2.2, 2.3, 2, 2.2, 2
Notation: Pos
Name: Category of Posets
Extra: , initial/terminal, arrow to initial, Exponential, arrow from terminal, coproduct
Index: 3, 5, 4, ,7
Kind: Ex, Ex, Ex, ,Ex
Initial: Empty set
Terminal: Singleton
Coequalizer: For \(f,g:X\to Y\), the quotient of \(Y\) by the equivalence generated by \(\{x\in X\mid f(x)\sim g(x)\}\)
Mono: each arrow
Ccc: Yes
Arrow from initial: Empty function
Arrow to initial: Identity
Arrow from terminal: An element of \(A\)
Exponential: \(A^B\) is the set of increasing function from \(B\) to \(A\)

Objects: Sets
Arrows: Binary relations
Page: 007
Chapter: 1
Section: 1.4
Notation: Rel
Extra: merge
Index: 4
Kind: Ex

Objects: A single object "*"
Arrows: The identity arrow
Page: 007, 95
Chapter: 1, 6 (for programmer)
Section: 1.4, ,
Notation: 1
Name: singleton category
Extra: merge
Index: 5
Kind: Ex

Objects: Two objects \(*\rightarrow \star\)
Arrows: The identity
Page: 007
Chapter: 1
Section: 1.4
Notation: 2
Extra: merge
Index: 5
Kind: Ex

Objects: None
Page: 008
Chapter: 1
Section: 1.4
Notation: 0
Extra: merge
Index: 5
Kind: Ex

Objects: 0, 1 and 2
Arrows: an unique arrow from 0 to 1 and 2 and from 1 to 2, plus the identity
Page: 008
Chapter: 1
Section: 1.4
Notation: 3
Extra: merge
Index: 5
Kind: Ex

Objects: The elements of \(A\)
Arrows: Single arrow from \(a\) to \(b\) such that \(a\le b\)
Page: 009
Chapter: 1
Section: 1.4
Name: The category of a preorder \(A\)
Extra: poset
merge
Index: 7
Kind: Ex
Variables: \(A\) a poset
Initial: Least element if it exists
Terminal: Greatest element if it exists
Coequalizer: The join/least upper bound
Arrow from initial: One by object
Arrow to initial: Identity
Arrow from terminal: The identity

Objects: The statements
Arrows: Proofs of \(B\) assuming \(A\)
Page: 010
Chapter: 1
Section: 1.4
Name: Category of proof
Extra: merge
Index: 10
Kind: ex

Objects: Data structure
Arrows: Pure functions
Page: 010
Chapter: 1
Section: 1.4
Name: The program category
Extra: merge
Index: 11
Kind: Ex

Objects: The elements of \(X\)
Arrows: Only identity
Page: 011
Chapter: 1
Section: 1.4
Notation: \(\text{Dis}(X)\)
Name: Discrete \(X\)
Index: 12
Kind: Ex
Variables: For \(X\) a set.
Initial: If \(X\) is a singleton, then its unique element, otherwise none
Subcategory: They are discrete

Objects: \((C,D)\) for \(C\in\mathbf C\) and \(D\in\mathbf D\)
Arrows: For \(A=(C_1,D_1)\) and \(B=(C_2,D_2)\), the pairs \((f_1,f_2)\) with \(f_i:C_i\to D_i\)
Page: 014
Chapter: 1
Section: 1.6
Notation: \(\mathbf C_1\times\mathbf C_2\)
Name: Product of the categories \((\mathbf C_1, \mathbf C_2)\)
Extra: merge
Variables: \(\mathbf C_1,\mathbf C_ 2\) two categories

Objects: \(C^*\) for \(C\in\mathbf C\)
Arrows: \(f^*:A^*\to B^*\) for \(f:B\to A\)
Page: 015
Chapter: 1
Section: 1.6
Notation: \(\mathbf C^{\text{op}}\)
Name: The opposite of \(\mathbf C\)
Extra: merge
Index: 2
Variables: \(\mathbf C\) a category.
Name2: The dual of \(\mathbf C\)

Objects: The arrows of \(C\).
Arrows: For \(A=f\) and \(B=f'\), a pair \((g_1,g_2)\) such that \(g_2\circ f=f'\circ g_1\).
Page: 015
Chapter: 1
Section: 1.6
Notation: \(C^\to\)
Name: The arrow category of \(C\)
Extra: merge
Index: 3
Variables: \(\mathbf C\) a category.

Objects: The arrows \(f\) such that cod\((f)=C\)
Arrows: An arrow from \(f:A\to C\) to \(f':B\to C\) is an arrow \(a:A\to B\) in \(\mathbf C\) such that \(f'\circ a=f\), as indicated in
Page: 016, 35
Chapter: 1, 2
Section: 1.6, 2.2
Notation: \(\mathbf C/C\)
Name: The slice category
Index: 4, 6
Kind: , ex
Variables: \(\mathbf C\) a category and
\(C\in\mathbf C_0\)
Initial: \(0\to X\) for \(0\) initial in \(\mathbf C\)
Terminal: \(1_X\)

Objects: Arrows of \(\mathbf C\) whose domain is \(C\)
Arrows: Arrows from \((f:C\to X)\) to \((f':C\to X')\) is an arrow \(h:X\to X'\) such that \(h\circ f=f'\)
Page: 017, 35
Chapter: 1,2
Section: 1.6, 2.2
Notation: \(C/\mathbf C\)
Name: The coslice category of \(\mathbf C\) by \(C\)
Extra: merge,
initial
Index: 4, 5
Kind: ,Ex
Variables: \(\mathbf C\) a category
\(C\in\mathbf C_0\)
Initial: \(1_X\)
Terminal: \(X\to1\) for \(1\) terminal in \(\mathbf C\)

Objects: Pairs \((A,a)\) with \(A\) a set and \(a\in A\)
Arrows: \(f:(A,a)\to(B,b)\) are functions from \(A\) to \(B\) sending \(a\) to \(b\)
Page: 017
Chapter: 1
Section: 1.6
Notation: Sets\(_*\)
Name: Category of Pointed sets
Index: 8
Kind: Ex
Pushout: E.g. For \(Z\) the one point space, \(X+_ZY\) is \(X\cup Y\) with their point unified

Objects: The vertices of \(G\)
Arrows: \(f:A\to B\) is a finite path from \(A\) to \(B\) in \(G\)
Page: 020, 30
Chapter: 1, 3
Section: 1.7,
Notation: \(\mathbf C(G)\)
Name: The (free) category of \(G\)
Extra: merge, for programmer
Variables: \(G\) an oriented graph.

Objects: Boolean algebras
Arrows: Morphism from \(A\) to \(B\)
Page: ,35
Chapter: 2
Section: 2.2, 2
Notation: \(\mathbf{BA}\)
Name: The category of Boolean Algebra
Extra: , initial/final
Index: 4
Kind: , Ex
Initial: \(\{0,1\}\) with usual truth value, equivalently, \(\mathcal P(\{0\})\)
Terminal: \(\mathcal P(\emptyset)\)
Symmetric monoidal category: with the tensor product of rings \(\otimes _{\mathbf Z}\)

Objects: Rooted posets
Arrows: Monotone functions sending roots to roots
Page: 058
Chapter: 3
Section: 2
Notation: \(\text{Pos}_0\)
Name: Category of rooted posets
Extra: merge
Coproduct: Sends \((A,a),(B,b)\) to \((A\oplus B\setminus \{b\},a)\), sending \(b\) to \(a\)

Objects: Finite ordinal
Arrows: Functions
Chapter: 4
Notation: \(\text{Ord}_{\text{fin} }\)
Extra: merge

Objects: The objects of \(C\)
Arrows: An arrow \(A\to B\) is a pair \(\langle f,g\rangle\) such that \(f\sim g:A\to B\)
Page: 084
Chapter: 4
Section: 3
Notation: \(\mathbf C^\sim\)
Extra: merge
Kind: The congruence category
Variables: \(\mathbf C\) a category
\(\sim\) a congruence on \(\mathbf C\).
Compositions: component-wise

Objects: The objects of \(C\)
Arrows: An arrow \(A\to B\) is a \(\sim\)-equivalence classes of arrows \(A\to B\) of \(\mathbf C\)
Page: 084
Chapter: 4
Section: 4.3
Notation: \(\mathbf C/\sim\)
Name: The quotient category
Variables: \(\mathbf C\) a category
\(\sim\) a congruence on \(\mathbf C\).

Objects: The subobjects of \(X\)
Arrows: \(f:m\to m'\in \mathbf C_1\) if \(m=f\circ m'\)
Page: 090
Chapter: 5
Section: 1
Notation: \(Sub_{\mathbf C}(X)\)
Name: The category of subobjects of \(X\)
Variables: \(\mathbf C\) a category,
\(X\in\mathbf C^0\)

Objects: Cones to \(D\)
Arrows: Morphism of cones
Page: 102
Chapter: 5
Section: 4
Notation: \(\mathbf{Cone}(D)\)
Extra: merge
Index: 15
Kind: Definition
Variables: \(\mathbf{J,C}\) be categories,
\(D:\mathbf J\to\mathbf C\) a diagram of type \(\mathbf J\).

Objects: \(\omega\text{CPO}\)s
Arrows: \(\omega\)-continuous functions
Page: 123
Notation: \(\omega\textbf{CPO}\)
Ccc: yes

Objects: Functors \(F:\mathbf{C}\to \mathbf D\)
Arrows: natural transformations \(\theta:F\to G\)
Page: 158
Chapter: 7
Section: 5
Notation: \(\text{Fun}(\mathbf{C,D})\), \([C,D]\) or \(D^C\)
Name: The functor category from \(C\) to \(D\)
Extra: merge
Index: 9
Kind: Def
Variables: \(\mathbf{C,D}\) 2 categories

Objects: Functors \(\mathbf{C\to D}\)
Arrows: functors \(\mathbf{C\times 2\to D}\), or equivalently, \(\mathbf{C\to D^2}\)
Page: 164
Chapter: 7
Notation: \(\mathbf{D^C}\)
Extra: merge, finctor category ?
Index: 16
Kind: Ex
Variables: \(\mathbf{C,D}\) categories.
Assuming: This category exists

Objects: Sets
Arrows: The partial functions \(f:A\rightharpoondown B\)
Page: 175
Chapter: 7
Section: 10
Notation: \(\mathbf{Par}\)
Extra: merge
Index: , 27
Kind: , Ex
Compositions: Sending \(f:A\rightharpoondown B\) and \(g:B\rightharpoondown C\) to the partial function, with domain \(f^{-1}(U_g)\), sending \(x\) to \(g(f(x))\)

Objects: \((A_i)_{i\in I}\), with \(A_i\) some sets
Arrows: \((f_i:_i\to B_i)_{i\in I}:(A_i)_{i\in I}\to (B_i)_{i\in I}\)
Page: 177
Chapter: 7
Section: 10
Notation: \(\mathbf{Sets^I}\)
Name: The category of \(I\)-indexed sets
Extra: merge
Kind: 29
Variables: \(I\) a sets
Compositions: pointwise composotion

Objects: The sets
Arrows: Fonctions
Page: 5, 30, 32, 32, 34, 35, 56, 63, 67, 93, 33, 122, 150, 108, 36
Chapter: 1, 2, 2, 2, 2, 2, 3, 3, 3, 5, 2, 6, 7, 5, 2
Section: 1.4, 2.1, 2.1, 2.1.1, 2.2, 2.3, 3.2, 3, 3.4, 2, 2.1.1, 2, 2, 6, 2.2
Notation: Sets
Name: Category of sets
Extra: def, mono and epi, iso, split mono, initial and final, arrow to initial, coproduct, equalizer, coequalizer,arrow from terminal
Index: 1, 2, ,8, 1,,, 15, ,,,3 , ,,
Kind: , Prop, ,Ex, Ex,,, ex, ,, Ex, , ,
Initial: Empty set
Terminal: Singletons
Coproduct: Disjoint union, \(\bigcup \{(a,i)\mid a\in A_i, i\in I\}\)
Coequalizer: Given \(A,B\) sets, \(f,g:A\to B\): the quotient of \(B\) by the equivalence relation generated by \((f(a),g(a))\) for \(a\in A\)
Mono: Injective maps
Pushout: Given \(f:A\to B\) and \(g:A\to C\): \((B+C)/\sim\) where \(\sim\) is the equivalence relation generated by \(f(a)\sim g(a)\) for all \(a\in A\)
Biproduct: No
Ccc: Yes
Arrow from initial: Empty function
Arrow to initial: exactly the only arrows from other initial elements
Iso: Bijective maps, mono and epi
Product: Cartesian product
Arrow from terminal: A singleton
Epi: Surjective maps
Equalizer: Given \(A,B\) sets, \(f,g:A\to B\): the identity function from \(\{x\in A\mid f(x)=g(x)\}\) to \(A\)
Exponential: Functions from \(B\) to \(A\)
Generator: 1
Pullback: Given \(f:A\to C \) and \(g:B\to C\): \(\{\langle a,b\rangle\in A\times B\mid fa=gb\}\).
Arrow to terminal: Constant function
Classifier: Two-element set
Power: power set
Projective: Exactly the sets whose cardinal admits choice
Split mono: If domain is empty then codomain is empty

Objects: Rings
Arrows: Ring morphisms
Page: 34
Chapter: 2
Section: 2.2
Name: Category of rings
Extra: initial and final, wp reste
Index: 3
Initial: \(\mathbb Z\)
Terminal: The singleton rings
Coproduct: Free product of associative algebras
Coequalizer: Given \(f,g:A\to B\): \(B/\{f(r)-g(r)\mid r\in R\}\)
Mono: Injective morphism
Cocomplete: yes
Iso: Bijective ring homomorphism
Product: Direct product of rings
Has zero morphisms: no
Connected: yes, thanks to initial object
Epi: contains strictly the surjective. Also inclusion \(\mathbb Z\to\mathbb Q\)
Equalizer: Set theoretic equalizer, it is a subring
Kernel: No
Bimorphism: e.g. \(\mathbb{Z\to Q}\) which is not iso
Extremal epi: Surjective homomorphisms

Objects: Groups
Arrows: Morphism of group
Page: 34, 79, 150
Chapter: 2, 4, 7, 3
Section: 2.2, 1, 2, 3
Notation: Grp
Name: Category of group
Extra: initial and final, remaining wp, equalizer
Index: , 6
Kind: , Cor
Initial: Singleton group
Coproduct: Free product
Coequalizer: For \(f,g:G\to H\), quotient of \(H\) by the normal closure of \(\{f(x)g(x)^{-1}\mid x\in X\}\).
Mono: Injective morphisms
Pushout: Free product with amalgamation
Biproduct: no
Cocomplete: yes
Iso: Bijective morphisms
Product: Direct product
Zero morphism: Group sending everything to the identity
Zero object: The trivial group
Epi: Surjective morphisms
Equalizer: sending \(f,g\) to the kernel of \(f-g\)
Abelian: No
Cokernel: of \(f:G\to H\) is the quotient of \(H\) by the normal closure of \(im(f)\)
Generator: \(F(1)\) e.g.
Group: Abelian groups

Objects: Triples \((A,B,h)\) with \(A\in\mathbf A_1\), \(B\in\mathbf B_1\) and \(h:S(A)\to T(B)\in\mathbf C_1\)
Arrows: \((A,B,h)\to (A',B',h')\): all pairs \((f:A\to A'\in\mathbf A_1,g:B\to B'\in\mathbf B_1)\) such that the following diagram commutes:
Page: wp
Notation: \((S\downarrow T)\)
Name: The comma category of \(S\) and \(T\)
Variables: \(\mathbf{A,B,C}\) categories,
\(S:\mathbf{A\to C}\),
\(T:\mathbf{B\to C}\),
Compositions: Component-wise combination
Identity morphism: For the object \((A,B,h)\): \((1_A,1_B)\)

Page: wp
Object: Set \(X\) to the discrete category over \(X\)
Domain: Set
Arrow: identity
Codomain: Cat
Left adjoint to: functor sending a small category to its set of objects

Page: wp
Object: Identity
Domain: Ring
Codomain: Ab
Right adjoint to: Functor which assigns to every abelian group \(X\) (thought of as a \(Z\)-module) the tensor ring \(T(X)\)

Page: wp
Object: Identity
Domain: Ring
Codomain: Mon
Right adjoint to: Functor which assigns to every monoid \(X\) the integral monoid ring \(Z[M]\).

Objects: Commutative rings
Arrows: Morphism of rings
Page: wp
Notation: CRing
Coproduct: Tensor product of rings
Pushout: Tensor product of rings with the morphisms \(g': A\to A\otimes C_B\) and \(f':B\to A\otimes _CB\) that satisfies \(f'\circ g=g'\circ f\).
Free: Given a base \(X\): \(\mathbf Z[X]\)
Pullback: Fibered product: \(\{(a,b)\in A\times B\mid \alpha(a)=\beta(b)\}\) and projections
Opposite: equivalent to the category of affine schemes

Objects: Fields
Arrows: Morphisms of field
Page: wp
Name: Field
Mono: every morphism
Cocomplete: not even finitely
Free: none
Connected: connected components are the set of fields with the same characteristic

Objects: Rings (potentially without identity)
Arrows: Morphisms of rings
Page: wp
Notation: Rng
Initial: Zero ring
Free: for a base \(X\), the sets of polynomials in \(X\) without constant terms
Has zero morphisms: yes
Preadditive: no

Objects: Fields of characteristic \(p\)
Arrows: Morphisms of field
Page: wp
Name: The category of fields of characteristic \(p\)
Variables: \(p\) prime or 0
Initial: The prime field
Mono: every morphism
Cocomplete: not even finitely
Regular: yes
Free: none
Connected: Yes

Objects: Abelian groups
Arrows: Morphism of groups
Page: wp
Notation: Ab
Name: Category of Abelian groups
Coequalizer: of \(f,g:G\to H\) is \(H/im(f-g)\)
Pushout: For \(f:Z\to X\), \(g:Z\to Y\), their pushout is \(X+Y/\{f(z)-g(z)\mid z\in Z\}\).
Biproduct: direct sum
Zero morphism: the morphism sending everything to identity
Has zero morphisms: yes
Zero object: Trivial group
Cokernel: of \(f:G\to H\) is \(H/im(f)\)

Objects: Topological space
Arrows: Continuous functions
Page: wp
Notation: Top
Name: Category of topological space
Pushout: E.g.:
For \(Z\subseteq Y\) with inclusion \(g\), and \(f:Z\to X\), the pushout is \(X\cup_fY\).
All adjunction spaces
Regular: No

Objects: Small categories
Arrows: Functors
Page: wp, 32, 34
Chapter: 2, , 2
Section: 2.2, , 2.2
Notation: Cat
Name: The category of small categories
Extra: , terminal, initial
Kind: , , Example
Initial: 0
Terminal: 1
Ccc: yes
Regular: No

Objects: Monoids
Arrows: Morphisms
Page: wp, 59, 36, 70
Chapter: , 3, 2, 4
Section: , 2, 2.2, 1
Notation: Mon
Name: Category of monoids
Extra: , coproduct, from terminal
Index: ,9, ,7
Kind: , Ex, , Rem
Coproduct: Sending \(A_1\) and \(A_2\) \(M(A_1+A_2)/\sim\) where \(\sim\) is the equivalence generated by addition/deletion of the identity element of the monoid and replacing an \(A_i\) subword of length 2 \(aa'\) by the product in \(A_i\) of \(aa'\).
Arrow from initial: To each element, the functions from identity to identity
Arrow to initial: From any monoid, the constant function to the identity
Iso: Bijection
Arrow from terminal: one arrow to each monoid, to the identity element.
Kernel: exists, don't have a lot of information
Monoid: are commutative

Objects: left \(R\)-modules
Arrows: morphisms
Page: wp
Name: The category of \(R\)-modules
Variables: \(R\) a Ring
Biproduct: direct product
Zero morphism: maps all of the module to the identity element
Has zero morphisms: yes
Zero object: Trivial module

Objects: A singleton \(\{e\}\)
Arrows: For each \(r\in R\), \(r:e\to e\)
Page: wp
Name: Its enriched category over Ab
Variables: \((R,0,+,1,\times)\) a rings
Compositions: \((x,y)\mapsto x+y\)
Monoidal operations: \((x,y)\mapsto x\times y\)

Objects: \(K\)-vector spaces
Arrows: Linear functions
Page: wp,
Chapter: 2
Section: 2.2
Name: The category of \(K\)-vector spaces
Variables: \(K\) a field
Initial: Vector space of dimension 0
Biproduct: direct sum
Zero object: 0-dim space