Title: Theory / Unimportant / Math / Category / Awodey=
Tags: fromMath mathToDef mathToSee examplesSwitchExample1 hasSeenConjDef complex toBinaryRelation defToFunctor CorrectType automaticImplies récupéré similarity TypToQuestion duplicateFromOtherBook exampleToQuestion onMultipleBook toBinaryFunction toCategory

Name: The identity arrow of \(A\)
Page: 004
Chapter: 1
Section: 1.3
Variables: \(\mathbf C\) a category,
\(A\in \mathbf C_0\).
Notation: \(1_A\)
Kind: Def
Property: A left and right identity for \(\circ\)

Name: A partially ordered set
Definition: A set \(A\) with a reflexive transitive antisymmetric binary relation.
Page: 003
Chapter: 1
Section: 1.4
Index: 2
Kind: Ex
Abbreviation: A poset

Page: 006
Definition: A relation \(R\subseteq A\times A\) such that \(a R a\) for all \(a\in A\).
Chapter: 1
Section: 1.4
Name: A reflexive relation
Kind: Ex
Index: 3

Page: 006
Definition: A relation \(\le\subseteq S^2\) such that, for all \(x,y\), either \(x\le y\) or \(y\le x\).
Chapter: 1
Section: 1.4
Variables: \(S\) a set
Name: A linear order on \(S\)
Kind: Ex
Index: 3

Name: A monoid category
Definition: A category which contains only one object.
Page: 011
Chapter: 1
Section: 1.4
Index: 13
Kind: Ex

Left: categories
Right: generalized monoids
Left2: functors
Right2: generalized homomorphism
Chapter: 1
Section: 1.4

Page: 012
Definition: An arrow \(f:A\to B\) such that there exists an arrow \(g\) such that \(f\circ g=1_b\) and \(g\circ f=1_A\).
Chapter: 1
Variables: \(\mathbf C\) a category
Name: An isomorphism
Kind: Def
Index: 3
Extra: Category theory for programmer, page 61, chap 51 section 4
Awodey chapter 1, page 12

Name: A group category
Definition: A category with a single object, each arrows are isomorphism.
Chapter: 1
Section: 1.5
Variables: \(\mathbf C\) a category.

Name: The projection functor
Definition: \(\pi_i:\mathbf C_1\times\mathbf C_2:\mathbf C_i\) associating to a pair the \(i\)-th element.
Page: 015
Chapter: 1
Section: 1.6
Variables: \(\mathbf C_1,\mathbf C_ 2\) two categories

Definition: \(\downarrow(p)\), the principal ideal of elements \(q\) such that \(q\le p\).
Page: 017
Chapter: 1
Section: 1.6
Variables: \(\mathbf P\) a poset category and
\(p\in\mathbf P\).
Notation: \(P/p\)
Index: 4

Name: Universal Mapping Property
Page: 019
Chapter: 1
Section: 1.7
Abbreviation: UMP

Name: The Universal Mapping Property of \(M(A)\)
Definition: There is a function \(i:A\to|M(A)|\), and given any monoid \(N\) and any function \(f:A\to|N|\) there is a unique monoid homomorphism \(\overline f:M(A)\to N\) such that \(|\overline f|\circ i=f\).
Page: 019
Chapter: 1
Section: 1.7
Variables: \(A\) a set.

Definition: The underlying oriented graphs of \(\mathbf C\).
Page: 021
Chapter: 1
Section: 1.7
Variables: \(\mathbf C\) a category.
Notation: \(U(\mathbf C)\)

Definition: The objects of \(\mathbf C\)
Page: 022
Chapter: 1
Section: 1.7
Variables: \(\mathbf C\) a category.
Notation: \(\mathbf C_0\)

Definition: The arrows of \(\mathbf C\).
Page: 022
Chapter: 1
Section: 1.7
Variables: \(\mathbf C\) a category
Notation: \(\mathbf C_1\)

Definition: The pairs \((f,g)\) of arrows of \(\mathbf C\) such that cod(\(f\))=dom(\(g\)).
Page: 022
Chapter: 1
Section: 1.7
Variables: \(\mathbf C\) a category.
Notation: \(\mathbf C_2\)

Name: The commutative diagram of a functor
Page: 022
Chapter: 1
Section: 1.7
Variables: \(\mathbf C\) a category

Name: The commutative diagram of functors
Page: 022
Chapter: 1
Section: 1.7
Variables: \(\mathbf {C,D}\) categories

Name: A small category
Definition: A category such that both the collection of objects and of arrows are sets.
Page: 024
Chapter: 1
Section: 1.8
Variables: \(\mathbf C\) a category.

Name: A locally small category \(\mathbf C\)
Definition: A category such that, for all objects \(X, Y\), the collection \(\text{Hom}_C(X,Y)=\{f\in \mathbf C_1\mid f:X\to Y\}\) is a set.
Page: 024
Chapter: 1
Section: 1.8

Name: A hom-set
Definition: The set of arrows from an object to another one.
Page: 025
Chapter: 1
Section: 1.8
Variables: \(\mathbf C\) a category.
Index: 2
Kind: Def

Name: A monomorphism
Definition: An arrow \(f:A\to B\) such that for all \(g,h:C\to A\) in \(\mathbf C\), \(fg=fh\) implies \(g=h\). That is, \(f\) cancels on the left.
Page: 029
Chapter: 2
Section: 2.1
Variables: \(\mathbf C\) a category.
Notation: \(f:A\rightarrowtail B\)
Index: 1
Kind: Def
Intuition: Generalization of injection

Name: An epimorphism
Definition: for all \(i,j:B\to D\), \(if=jf\) implies \(i=j\), . That is, if \(f\) cancels on the right.
Page: 029
Chapter: 2
Section: 2.1
Variables: \(\mathbf C\) a category
\(f:A\to B\).
Notation: \(f:A\twoheadrightarrow B\)
Index: 1
Kind: Def

Name: An epi arrow which is not a surjective function
Page: 031
Chapter: 2
Section: 2.1
Index: 5
Kind: Ex
Example: The monoid inclusion from \(\mathbf N\) to \(\mathbf Z\).

Name: A split mono
Definition: An arrow with a left inverse.
Page: 032
Chapter: 2
Section: 2.1.1
Variables: \(\mathbf C\) a category
Index: 7
Kind: Def

Name: A split epi
Definition: An arrow with a right inverse.
Page: 032
Chapter: 2
Section: 2.1.1
Variables: \(\mathbf C\) a category.
Index: 7
Kind: Definition

Smaller0: every epi splits
Greater0: the axiom of choice
Page: 033
Chapter: 2
Section: 2.1.1
Smaller to greater: is equivalent to
Suffix Greater: in set theory
Suffix Smaller: in category theory

Name: Point
Definition: An arrow from a final element.
Page: 036
Chapter: 2
Section: 2.2
Name2: Constant
Name3: Global element

Name: A generalized elements of \(A\)
Definition: An arbitrary arrow \(x:X\to A\) with arbitrary domain \(X\).
Page: 036
Chapter: 2
Section: 2.2
Variables: \(A\) a category
Name2: Variable element of \(A\)

Name: A category with enough point
Definition: For \(f,g:A\to B\), if for all \(x:1\to A\), \(fx=gx\), then \(f=g\).
Page: 036
Chapter: 2
Section: 2.2
Variables: \(\mathbf C\) a category.

Definition: [$]\defFun{A_1\times A_2}{B_1\times B_2}{}{\langle f_1\circ p_1,f_2\circ p_2 \rangle}[/$].
Page: 041
Chapter: 2
Section: 2.6
Variables: \(\mathbf C\) a category,
\(X,A_1,A_2\in\mathbf C_0\)
\(f_i: A_i\to B_i\)
Notation: \(f_1\times f_2\)

Name: Category with finite product
Definition: A category with terminal object and all binary product.
Page: 048
Chapter: 2
Section: 2.6
Index: 19
Kind: Def

Name: A category which have all products
Definition: A category such that every set of objects in \(\mathbf C\) has a product.
Page: 048
Chapter: 2
Section: 2.6
Variables: \(\mathbf C\) a category.
Index: 19
Kind: Def

Name: A functor which preserve binary products
Definition: \(F:\mathbf C\to\mathbf D\) which takes every product diagram \(A_{1}\xleftarrow{p_1} A_1\times A_2\xrightarrow{p_2} A_{2}\) to a product diagram \(FA_{1}\xleftarrow{Fp_1} F(A_1\times A_2)\xrightarrow{Fp_2} FA_{2}\).
Page: 049
Chapter: 2
Section: 7
Variables: \(\mathbf {C,D}\) categories
Index: 21
Kind: Def

Question: If category theory implies a sentence \(\Sigma\) it implies
Answer: \(\Sigma^*\).
Page: 054
Chapter: 3
Section: 1
Variables: \(\mathbf C\) a category,
\(A,B,C\in\mathbf C_0\),
\(g:C\to B\) and
\(f:B\to A\).
Kind: Prop
Name: Formal duality

Question: If a statement implies some diagram
then
Answer: its dual statements imply
Page: 054
Chapter: 3
Section: 1
Variables: \(\mathbf C\) a category, \(A,B,C\in\mathbf C_0\), \(g:C\to B\) and \(f:B\to A\).
Kind: Prop

Name: A coprojection
Definition: the arrow \(i_i:A_i\to A_1+A_2\).
Page: 055
Chapter: 3
Section: 3.2
Variables: \(\mathbf C\) a category,
\(i\in\{1,2\}\)
\(A_1,A_2\in\mathbf C_0\).
Kind: Def

Smaller0: \(O(X+Y)\)
Greater0: \(O(X)\times O(Y)\)
Page: 057
Chapter: 3
Section: 2
2
Variables: \(X,Y\) topologies.
Index: 6
Kind: Def
Smaller to greater: =
Hide middle: yes
Extra: O ?
Context: \(\mathbf{Top}\)

Name: A rooted poset
Definition: A poset with a distinguished minimal element 0.
Page: 058
Chapter: 3
Section: 2

Name: Functor which does not preserves coproduct
Page: 060
Chapter: 3
Section: 2
Example: The forgetful functors \(\mathbf{Mon}\to\mathbf{Set}\)

Name: Parallel arrows
Definition: Two arrows which have the same domains and codomains.
Page: 062
Chapter: 3

Definition: The \(\sim\)-equivalence class of \(x\).
Page: 065
Chapter: 3
Section: 3.4
Variables: -\(S\): a set with an equivalence relation \(\sim\)
-\(x\in X\).
Notation: \([x]\)

Name: The quotient of \(X\) by \(\sim\)
Definition: the set of equivalence-classes according to \(\sim\).
Page: 066
Chapter: 3
Section: 3.4
Variables: \(S\) a set
An equivalence relation \(\sim\).

Name: The coequalizer of \(\mathcal F\)
Definition: An arrow \(h:B\to Q\) universal such that \(h\circ f=h\circ f'\) for all \(f,f'\in\mathcal F\). That is, for all \(z:B\to Z\) such that \(z\circ f=z\circ f'\) for all \(f,f'\in \mathcal F\), there exists a unique \(u:Q\to Z\) such that \(u\circ h=z\).
Page: 066
Chapter: 3
Section: 3.4
Variables: \(\mathbf C\) a category,
\(A,B\in\mathbf C_0\),
\(\mathcal F\) a set of arrows \(A\to B\)
Index: 18, 19
Kind: Def, Prop
Example: \(H/im(f-g)\)
Example2: the monoid object \((\mathbb N,+)\)
Typ: epi
Applied to: In an universal algebra of \(f,g:G\to H\)
Applied to2: In Cat, the coequalizer of the two functors from 1 to 2

Smaller0: an arrow from \(1\) to a poset
Greater0: the coequalizer of \(1\xrightarrow[0_Q]{0_P} P+Q\)
Page: 068
Chapter: 3
Section: 3.4
Hide middle: yes
Prefix Greater: in Pos,
Greater1: \(P+Q/(0_P=0_Q)\), the coproduct of pointed set
Connect Greater0: is
Prefix Smaller: A pointed poset is

Smaller0: \((g_1\circ h_1)\star (g_2\circ h_2)=(g_1\star g_2)\circ (h_1\star h_2)\)
Greater0: \(1^ \star=1^ \circ\), \(\star=\circ\)
Page: 078
Chapter: 4
Section: 1
Variables: \(G\) a set with two binary operations
\(\circ, \star\)
\(g_1,g_2,h_1,h_2\in G\).
Hide middle: yes
Name: Eckmann-Hilton argument
Greater1: they are commutative.

Name: A strict monoidal category
Definition: A category equiped with an associative operation \(\otimes\) on objects and a distinguished object \(I\), which is an identity for \(\otimes\).
Page: 079
Chapter: 4
Section: 1
Variables: \(\mathbf C\) a category.
Index: 9
Kind: Def
Definition2: A monoidal category \(\left(\mathbf C,\otimes,I,(\lambda_A)_{A\in\mathbf C_0},(\rho_A)_{A\in\mathbf C_0},(\alpha_{ABC})_{A,B,C\in\mathbf C_0}\right)\) such that \(\alpha,\lambda, \rho\) are identities.
Example: In poset: The meet operation \(\land\) and 1
The join operation \(\lor\) and 0
Definition3: a monoid in the category \(\mathbf{Cat}\)

Name: The kernel of \(h\)
Definition: The equalizer of \(h\) and of \(u\).
Page: 080
Chapter: 4
Section: 2
Variables: \(G\) and \(H\) be groups in the category \(\mathbf{Sets}\),
\(u,h:G\to H\) be a group homomorphism,
with \(u\) sending all elements of \(G\) to the identity element of \(H\)

Q: \(\phi\) has [...] which is [...] with the property that, [...]
A: a kernel \(ker(h)=h^{-1}(1)\); a normal subgroup of \(G\); for any normal subgroup \(N\subseteq G\), \(N\subseteq ker(h)\) iff there is a (necessarilay unique) homomorphism \(\overline h: G/N\to H\) with \(\overline h\circ \pi=h\).
Page: 082
Chapter: 4
Section: 2
Variables: \(G,H\) groups and \(h:G\to H\) a group homomorphism.
Index: 10
Kind: Theorem

Question: \(h\) factors
Answer: as a quotient followed by an injective homomorphism.
Page: 082
Chapter: 4
Section: 2
Variables: \(G,H\) groups
\(h:G\to H\) a group homomorphism.
Index: 11
Kind: Corollary

Name: A linear representation of \(G\)
Definition: A function from \(G\) to a vector space.
Chapter: 4
Variables: \(G\) a group.

Name: A congruence \(\sim\) on \(\mathbf C\)
Definition: An equivalence relation such that \(f\sim g\) implies that \(f\) and \(g\) are parallel and \(bfa\sim bga\) for all arrows \(a,b\).
Page: 083
Chapter: 4
Section: 4.3
Variables: \(\mathbf C\) a category.
Index: 12
Kind: Def

Page: 084
Definition: The relation between parallel arrows sent to the same arrow.
Chapter: 4
Section: 4.3
Variables: \(\mathbf {C,D}\) categories
\(F:\mathbf C\to\mathbf D\) a functor
Notation: \(\sim_F\)

Name: The kernel category of \(F\)
Definition: \(\mathbf C^{\sim_F}\).
Page: 085
Chapter: 4
Section: 4.3
Variables: \(\mathbf {C,D}\) categories and \(F:\mathbf C\to\mathbf D\) a functor.
Notation: \(ker(F)\)
Index: 13
Kind: Theo

Smaller0: \(f\sim g\implies f\sim_F g\)
Greater0: there is a factorization \(\tilde F:\mathbf C/\sim\to \mathbf D\) such that \(\tilde F\circ \pi=F\)
Page: 085
Chapter: 4
Section: 4.3
Variables: \(\mathbf {C,D}\) categories and \(F:\mathbf C\to\mathbf D\) a functor, \(\sim\) a congruence on \(\mathbf C\) and \(\pi:\mathbf C\to\mathbf C/\sim\) the projection.
Index: 13
Kind: Theo
Smaller to greater: iff

Q: Every function \(F:\mathbf C\to\mathbf D\) factors as [...] where \(\pi\) is [property] and \(\tilde F\) is [property]
A: \(F=\tilde F\circ \pi\); bijective on objects and surjective on Hom-sets; is injective on Hom-sets
Page: 085
Chapter: 4
Section: 3
4.3
Index: 14
Kind: Corollary

Name: Finite presentation of a category
Definition: A directed graph \(G\) and a set of relations of the form \[(g_1\circ \dots\circ g_n)=(g'_1\circ\dots\circ g'_n)\] with \(g_i\) parallel to \(g_i'\)
Page: 085
Chapter: 4

Name: The category presented by this pair
Definition: the category whose objects are the vertices of \(G\) and whose arrows are the smallest congruence \(\sim\) such that for all relations, \((g_1\circ \dots\circ g_n)\sim(g'_1\circ\dots\circ g'_n)\)
Page: 086
Chapter: 4
Variables: \(G\) a directed graph,
\(\Sigma\) a set of relations of the form \[(g_1\circ \dots\circ g_n)=(g'_1\circ\dots\circ g'_n)\] with \(g_i\) parallel to \(g_i'\)
Notation: \((G,\Sigma)\)
Definition2: the coequalizer of \(\mathbf C(n\times 2)=\mathbf C(2)+\dots+\mathbf C(2)\) sending each arrow to one side f the relation. Where \(n\) is the number of relation
Applied to4: yes

Name: A subobject of \(X\)
Definition: A monomorphism \(m:M\to X\) of \(C\).
Page: 089
Chapter: 5
Section: 1
Kind: Def
Denoted by: \(M\) if the arrow is clear

Smaller0: the arrow \(e\) is an equalizer of \(f\pi_1\) and \(g\pi_2\)
Greater0: \(E,p_1,p_2\) is a pullback of \(f\) and \(g\)
Page: 093
Chapter: 5
Section: 2
Variables: In a category with products and equalizer, given a corner of arrows
Consider the diagram
\(p_i=\pi_ie\).
Kind: Proposition
Smaller to greater: iff

Q: [...] is [...].
A: The pullback of \(f\) and \(i\); \(f^{-1}(V)\)
Page: 097
Chapter: 5
Section: 2
Variables: In Sets, let \(f:A\to B\) and \(V\subseteq B\) with inclusion function \(i\).
Index: 7
Kind: Ex

\(f\) is monic implies \(g'\) is monic.
Chapter: 5
Section: 2
Variables: In any category, given \(f:A\to C\) and \(g:B\to C\), let \(f':A\times_ C B\to A\) \(g':A\times_ C B\to B\) be a pullback.

The two squares is/are pullback implies the rectangle is/are a pullback, thus
\[A\times_B(B\times_CD)\cong A\times_ CD.\]
Page: 095
Chapter: 5
Section: 3
Index: 8
Kind: Lem

Q: [...] and the outer rectangle are pullback implies [...] is a pullback.
A: the right square; the left square
Page: 095
Chapter: 5
Section: 3
Index: 8
Kind: Lem

Q: one can form the pullbacks [the three arrows on the left] as on the left end, [...] there exists a unique [...] as indicated, making the left end a commutative triangle, and the upper face a commutative rectangle, and indeed a pullback.
A: \(\alpha'\) and \(\beta'\); implies; \(\gamma'\)
Page: 096
Chapter: 5
Section: 3
Variables: \(\mathbf C\) a category,
\(A,B,C,A',B',C'\) objects
\(\alpha:A\to C\), \(\beta:B\to C\), \(\gamma:A\to B\) and \(h:C'\to C\).
Index: 9
Kind: Corollary

Q: there is the following diagram of categories and functors:

-there is [...]
A: a functor \(f^*:\mathbf C/C\to\mathbf C/C'\) defined by \((A\xrightarrow{\alpha}C)\mapsto(C'\times_CA\xrightarrow{\alpha'}C')\) where \(\alpha'\) is the pullback of \(\alpha\) along \(f\).
Page: 096, 097
Chapter: 5
Section: 3
Variables: \(\mathbf C\) be a category with pullbacks and \(f:A\to B\) in \(\mathbf C\).
Index: 10, 11
Kind: Prop, CorCor
Assuming: \(\mathbf C\) has pullbacks

Name: The coproduct of \(\alpha\) and of \(p\)
Definition: \(\coprod_{j\in J}A_{\alpha(j)}\).
Page: 099
Chapter: 5
Section: 3
Variables: \(I\) be an index.
\((A_i)_{i\in I}\) a family of set.
\(\alpha:J\to I\).
\(p_i:A_i\to I\).
\(p=[p_i]:\coprod_{i\in I}A_i\to I\).
Index: 13
Kind: Ex

Smaller0: finite products
Greater0: pullbacks
Page: 100
Chapter: 5
Section: 4
Variables: \(\mathbf C\) be a category.
Index: 14
Kind: Proposition
Smaller to greater: iff
Prefix Greater: \(\mathbf C\) has
Smaller1: equalizers
Greater1: a terminal object
Extra: Strong than cor 2.6 page 94

Name: Pullback
Definition: Produit Fibré
Counterexample: Syncer.remove: Remove from grave

Name: A diagram of type \(\mathbf J\) in \(\mathbf C\)
Definition: A functor \(D:\mathbf J\to\mathbf C\).
Page: 101
Chapter: 5
Section: 4
Variables: \(\mathbf{J,C}\) be categories.
Index: 15
Kind: Def

Name: A cone to \(D\)
Definition: An object \(C\in\mathbf C\) and a family of arrows in \(\mathbf C\), \(c_j:C\to D_j\), such that for each arrow \(\alpha:I\to j\in \mathbf J\), \(c_j=D_{\alpha}\circ c_i\).
Page: 101
Chapter: 5
Section: 4
Variables: \(\mathbf{J,C}\) be categories and \(D:\mathbf J\to\mathbf C\) a diagram of type \(\mathbf J\).
Index: 15
Kind: Definition
Extra: to est un indice

Name: A morphism of cone
Definition: \(\theta:(C,c_j)\to(C',c'_j)\) such that for each \(j\in\mathbf J\), \(c_j=c'_j\circ\theta\).
.
Page: 102
Chapter: 5
Section: 4
Variables: \(\mathbf{J,C}\) be categories and \(D:\mathbf J\to\mathbf C\) a diagram of type \(\mathbf J\). Let \((C,c_j),(C',c'_j)\) be cones to \(\mathbf D\).
Index: 15
Kind: Definition

Name: A finite limit for \(D\)
Definition: A terminal object in \(\mathbf{Cone}(D)\) such that \(\mathbf{J}\) is a finite index category.
Page: 102
Chapter: 5
Section: 4
Variables: \(\mathbf{J,C}\) be categories
\(D:\mathbf J\to\mathbf C\) a diagram of type \(\mathbf J\).
Index: 16
Kind: Definition

Q: Intuitively, a limit for a diagram \(D\) is [...] because [...].
A: the closest cone to the diagram \(D\); any other cone comes from the limit by composing with the unique function \(u\)
Page: 102
Chapter: 5
Section: 4
Variables: \(\mathbf{J,C}\) be categories
\(D:\mathbf J\to\mathbf C\) a diagram of type \(\mathbf J\).
Index: 16
Kind: Definition

Q: Let \(\mathbf{J}\) be [a precise category]. Let \(D:\mathbf{J}\to \mathbf{C}\) be [...].
A limit for \(D\) is [...].
A: \(\{1,2\}\) without nonidentity arrow; a pair of objects \((D_1,D_2)\); the product \(D_1\times D_2\)
Page: 103
Chapter: 5
Section: 4
Variables: \(\mathbf C\) a category,
\(D_1,D_2\in\mathbf C_0\).
Index: 17
Kind: Ex

Q: Let \(\mathbf{J}\) be [a precise category]. Let \(D:\mathbf{J}\to \mathbf{C}\) be [...].
A limit for \(D\) is [...].
A: \(\{1,2\}\) with two arrows \(\alpha,\beta:1\to2\); a pair of objects \((D_1,D_2)\) with arrows \(D_\alpha,D_\beta\); an equalizer for \(D_\alpha\) and \(D_\beta\)
Page: 103
Chapter: 5
Section: 4
Variables: \(\mathbf C\) a category,
\(D_1,D_2\in\mathbf C_0\),
\(D_\alpha, D_\beta:D_1\to D_2\).
Index: 18
Kind: ex

Q: Let \(\mathbf{J}\) be [a precise category].
Let \(D:\mathbf{J}\to\mathbf{C}\) be [...].
A limit for \(D\) is [...].
A: \(\{\}\); the only diagram to the empty set; a terminal object in \(C\)
Page: 103
Chapter: 5
Section: 4
Variables: \(\mathbf C\) a category.
Index: 19
Kind: Ex

Q: Let \(\mathbf{J}\) be [a precise category]. Let \(D:\mathbf{J}\to\mathbf{C}\) be .
A limit for \(D\) is [...].
A: \(\left(\{1,2,3\},\{f:1\to2,g:3\to 2\}\right)\); a pullback of \(f\) and \(g\): \(A\times_C B\)
Page: 103
Chapter: 5
Section: 4
Variables: \(\mathbf C\) a category, \(A,B,C\in\mathbf C_0\), \(f:A\to C\), \(g:B\to C\).
Index: 20
Kind: Ex

Name: A contravariant functor on \(\mathbf C\)
Definition: A functor of the form \(F:C^{op}\to D\).
Page: 107
Chapter: 5
Variables: \(\mathbf{C,D}\) be categories.
Index: 26
Kind: Definition
Example: Let \(\mathbf C\) be locally small and \(C\in\mathbf C\). The representable functors \(\text{Hom}(-,C)\)

Q: [...] map [...] to [related to (co)limit].
A: Contravariant representable functors; colimits; limits
Page: 107
Chapter: 5
Index: 27
Kind: Definition

Page: 48
Object: \(X\mapsto\text{Hom}(A,X)\) and
Domain: \(\mathbf C_1\to\mathbf{Set}\)
Arrow: \((f:X\to Y)\mapsto(g\mapsto (g\circ f))\)
Section: 7
Chapter: 2
Variables: \(\mathbf{C}\) be a category and
\(A\in\mathbf C\).
Notation: The (covariant) representable functor of \(A\)
Extra: The representable functor of \(A\)

Smaller0: the two sphere \(S^2\)
Greater0: the pushout of \(i\) and \(i\)
Page: 108
Chapter: 5
Section: 6
Variables: \(D^2\) be the two-dimensional disk
\(S^1\) be the one dimensional sphere(the circle).
\(i:S^1\to D^2\) the inclusion.
Index: 28
Kind: Ex
Smaller to greater: is equal to
Hide middle: yes

Name: A cocone
Definition: An object \(C\) (the vertex) and arrows \(c_j:D_j\to C\) for each \(j\in \mathbf J\), such that for all \(\alpha:i\to j\in\mathbf J\), \(c_j\circ D(\alpha)=c_i\).
Page: 109
Chapter: 5
Section: 6
Variables: \(\mathbf{J,C}\) be categories and
\(D:\mathbf{J\to C}\) be a diagram.

Name: A colimit
Definition: An initial object in the category cocone from the base \(D\).
Page: 109
Chapter: 5
Section: 6
Variables: \(\mathbf{J,C}\) be categories
\(D:\mathbf{J\to C}\) be a diagram.
Notation: [$]\varinjlim_{j\in\mathbf J}D_j[/$]
Name2: An universal co-cone
Extra: Other name by wp

Name: A functor which creates limits of type \(J\)
Definition: A functor \(F:\mathbf{C\to D}\) such that, for every diagram \(C:\mathbf{J\to C}\) and limit \((p_j:L\to FC_j)_j\) in \(\mathbf D\) there is a unique cone \((\overline{p_j}:\overline{L}\to C_j)_j\) in \(\mathbf C\) with \(F(\overline L)=L\) and \(F(\overline {p_j})=p_j\) which, furthermore, is a limit.
Page: 110
Chapter: 5
Section: 6
Variables: \(J\) a diagram.
Index: 30
Kind: Definition
Extra: every limit in \(\mathbf D\) is the image of a unique cone in \(\mathbf C\), which is a limit in \(\mathbf C\).

Name: Functor which creates all limits and \(\omega\)-colimit
Page: 111
Chapter: 5
Section: 6
Index: 31
Kind: Proposition
Example: The forgetful functor \(U:\mathbf{Groups\to Sets}\).

Name: A large category
Definition: A category which is not small.

Name: An \(\omega\)CPO
Definition: A poset that has all colimits of type \(\omega=(\mathbb N,\le)\).
Page: 112
Chapter: 5
Section: 6
Index: 33
Kind: Ex

Smaller0: \(P\) is an \(\omega\)CPO
Greater0: for all chain \(d_0\le d_1\le\dots\) in \(P\) there exists a minimal \(d_\omega\), such that \(d_i\le d_\omega\) for all \(i\in\mathbb N\)
Page: 112
Chapter: 5
Section: 6
Variables: \(P\) a poset.
Index: 33
Kind: Ex
Smaller to greater: iff

Name: Continuous function from \(D\) to \(E\)
Definition: A function \(f:D\to E\) such that [$]f(\varinjlim d_n)=\varinjlim f(d_n)[/$].
Page: 112
Chapter: 5
Section: 6
Variables: \(D,E\) be two \(\omega\)CPO.
Index: 33
Kind: Ex

Smaller0: \(f\) is continuous and \(D\) has an initial element
Greater0: \(f\) has a least fixed point
Page: 112
Chapter: 5
Section: 6
Variables: \(D\) be a \(\omega\)CPO and
\(f:D\to E\).
Index: 33
Kind: Ex

Name: Finite \(\omega\)CPO
Page: 114
Chapter: 5
Section: 6
Example: \(\omega_n=\{i\le n\mid i\in \omega\}\).

Q: In [...], [...] is [...].
A: \(\mathbf {POS}\); the colimit of \(\omega_0\to \omega_1\to\dots\); \(\omega\)
Page: 114
Chapter: 5
Section: 6

Q: In [...], [...] is [...].
A: \(\omega\mathbf {CPO}\); the colimit of \(\omega_0\to \omega_1\to\dots\); \(\omega+1\)
Page: 114
Chapter: 5
Section: 6

Q: In [...], the colimit of \(\omega_0\to \omega_1\to\dots\) [...]
A: \(\omega\); does not exists.
Page: 114
Chapter: 5
Section: 6

Name: Universal property which is not a (co)limit
Page: 119
Chapter: 6
Example: Free algebra.

Name: An exponential of \(B\) and \(C\)
Definition: A pair composed of an object \(C^B\) and an arrow \(\epsilon:C^B\times B\to C\) such that for any \(A\in\mathbf C\) and \(f:A\times B\to C\) there is a unique arrow \(\tilde f:A\to C^B\) such that \(\epsilon(\tilde f\times 1_B)=f\).
Page: 120
Chapter: 6
Section: 1
Variables: \(\mathbf C\) be a category with products. Let \(B,C\in\mathbf C\).
Kind: Definition

Name: Evaluation function
Definition: The function \(C^B\times B\to C\).
Page: 121
Chapter: 6
Section: 1
Variables: \(\mathbf C\) be a category with products.
\(B,C\in\mathbf C\).
Notation: \(\epsilon\)
Kind: Definition

Name: The exponential transpose of a function \(f\)
Definition: For \(f:A\times B\to C\), an arrow \(g:A\to C^B\) such that \(\epsilon\circ(g\times 1_B)=f\).
Page: 121
Chapter: 6
Section: 1
Variables: \(\mathbf C\) be a category with products. Let \(A,B,C\in\mathbf C_0\).
Notation: \(\tilde f\)
Kind: Definition

Name: The transpose of a function \(f\)
Definition: Given a function \(f:A\to C^B\), the function \(g:A\times B\to C\) such that \(f\circ(g\times 1_B)\).
Page: 121
Chapter: 6
Section: 1
Variables: \(\mathbf C\) a category
Notation: \(\overline f\)
Kind: Definition
Assuming: The category has products

Question: \(\tilde{\overline f}=\)
Answer: \(f\).
Page: 121
Chapter: 6
Section: 1
Variables: \(\mathbf C\) be a category with products.
\(A,B,C\in\mathbf C\).
\(f:A\to C^B\) .
Kind: Definition

Question: \(\overline{\tilde f}=\)
Answer: \(f\).
Page: 121
Chapter: 6
Section: 1
Variables: \(\mathbf C\) be a category with products.
\(A,B,C\in\mathbf C\).
\(f:A\to C^B\) .
Kind: Definition

Q: \([...]\simeq [...] \)
A: \text{Hom}_{\textbf{C} }(A\times B,C); \text{Hom}_{\textbf{C} }(A,C^B)
Page: 121
Chapter: 6
Section: 1
Variables: \(\mathbf C\) be a category with products.
\(A,B,C\in\mathbf C\).
Kind: Definition

Q: The [...] is [...].
A: exponential transpose of \(\epsilon:B^A\times A\to B\); \(1_{B^A}\)
Page: 125
Chapter: 6
Section: 2
Variables: \(\mathbf C\) a category and \(A,B\in\mathbf C\).

Name: The exponentiation by \(A\)
Definition: [$]\defFun{C_0}{\mathbf C_0}{B}{B^A}[/$].
Page: 126
Chapter: 6
Section: 2
Variables: \(\mathbf C\) a category
\(A\in\mathbf C_0\).
Index: 7
Kind: Proposition
Typ: functorial

Definition: \((\neg a\lor b) \).
Page: 129
Section: 3
Notation: \(a\rightarrow b\)
Notation2: \(b^a\)

Q: In \(\mathbf C\) [...] [...] [...].
A: product; is distributive over; coproducts
Page: 130
Chapter: 6
Section: 3
Variables: \(\mathbf C\) a category
Index: 9
Kind: Rem
Assuming: \(\mathbf C\) is a CCC

Name: Complete poset
Definition: A poset which has all set-indexed meets \(\bigwedge_{i\in I}a_i\).
Page: 130
Chapter: 6
Section: 3
Index: 10
Kind: Definition

Smaller0: \(L\) is complete
Greater0: \(L\) is co-complete
Page: 130
Chapter: 6
Section: 3
Variables: \((L,\land,\lor)\) a lattice.
Index: 10
Kind: Definition
Smaller to greater: iff

Smaller0: \(L\) is complete
Greater0: \(L\) is a Heyting algebra
Page: 130
Chapter: 6
Section: 3
Variables: \(L\) a lattice
Index: 11
Kind: Proposition
Greater1: \(L\) satisfies the infinite distributive law \(a\land \left(\bigvee _ib_i\right)=\bigvee_i(a\land b_i) \)
Connect Greater0: iff

-\(\vdash\) is reflexive and transitive
-\(p\vdash \top\)
-\(\bot\vdash p\)
-\(p\vdash q\) and \(p\vdash r\) iff \(p\vdash q\land r\)
-\(p\vdash r\) and \(q\vdash r\) iff \(p\lor q\vdash r\)
-\(p\land q\vdash r\) iff \(p\vdash q\implies r\)
Page: 131
Chapter: 6
Section: 4
Name: Intuitionistic propositional calculus

Name: The Lindenbaum-Tarski algebra of \(\mathcal L\)
Definition: Equivalence class with \([p]\le [q]\) iff \(p\vdash q\)
Page: 133
Chapter: 6
Section: 4
Variables: \(\mathcal L\) be propositional formulas, variables, rules of inference of the intuitionist calculus and axioms.

Smaller0: \(p\) is provable
Greater0: \([p]=[1]\)
Page: 133
Chapter: 6
Section: 4
Variables: \(\mathcal L\) be propositional formulas, variables, rules of inference of the intuitionist calculus and axioms.
\(p\) a formula.
Index: 13
Kind: Lemma
Smaller to greater: iff

Name: An interpretation of \(\mathcal L\) to \(H\)
Definition: A function which associates to each proposition \(p\) of \(\mathcal L\) [$]\llbracket p\rrbracket\in H[/$], such that [$]\llbracket p\land q\rrbracket=\llbracket p\rrbracket\land \llbracket q\rrbracket[/$], etc.
Page: 134
Chapter: 6
Section: 4
Variables: \(\mathcal L\) be propositional formulas, variables, rules of inference of the intuitionist calculus and axioms.
\(H\) a Heyting algebra.
Example: Taking [$]\llbracket p\rrbracket[/$] to \([p]\) in \(\text{HA}(\mathcal L)\).

Name: A model of \(\mathcal L\)
Definition: A function such that [$]\llbracket p\rrbracket=1[/$] iff \(\top\vdash p\) for all sentences \(p\).
Page: 134
Chapter: 6
Section: 4
Variables: \(\mathcal L\) be propositional formulas, variables, rules of inference of the intuitionist calculus and axioms.
\(H\) a Heyting algebra and \(M\) an interpretation of \(\mathcal L\) to \(M\).

Q: The intuitionistic propositional calculus is [...] with respect to [...].
A: complete; models in Heyting algebras
Page: 134
Chapter: 6
Section: 4
Index: 14
Kind: Proposition

Name: A subcategory
Definition: A mono in \(\mathbf{Cat}\).
Page: 147
Chapter: 7
Section: 1

Name: A subcategory of \(\mathbf C\)
Definition: A category with objects in \(\mathbf U_0\) and arrows in \(\mathbf U_1\) that is closed under the operations dom, cod, id and \(\circ\).
Page: 147
Chapter: 7
Section: 1
Variables: \(\mathbf C\) be a category.

Name: A functor injective/surjective on objects (respectively, arrows)
Definition: \(F:\mathbf C\to\mathbf D\) a functor such that the function \(F_0:\mathbf C_0\to\mathbf D_0\) (respectively, \(F_1:\mathbf C_1\to\mathbf D_1\)) is injective/surjective.
Page: 148
Chapter: 7
Section: 1
Kind: Definition

Definition: The function [$]\defFun{\text{Hom}_{\mathbf C}(A,B)}{\text{Hom}_{\mathbf D}(FA,FB)}{f}{F(f)}[/$].
Page: 148
Chapter: 7
Section: 1
Variables: \(F:\mathbf C\to\mathbf D\), \(A,B\in\mathbf C_0\).
Notation: \(F_{A,B}\)
Kind: Definition

Name: A full functor
Definition: A functor \(F:\mathbf C\to\mathbf D\) such that for all \(A,B\in\mathbf C_0\), the map \(F_{A,B}\) is surjective.
Page: 148
Chapter: 7
Section: 1
Variables: \(F:\mathbf C\to\mathbf D\) a functor.
Kind: Definition

Q: [...].
A: A functor \(F:\mathbf C\to\mathbf D\) such that, for all \(A,B\in\mathbf C_0\), the map \(F_{A,B}\) is injective
Page: 148
Chapter: 7
Section: 1
Kind: Definition
Name: A faithful functor

Name: Faithful functor which is not injective on arrows
Page: 148
Chapter: 7
Section: 1
Example: \(\nabla:C+C\to C \).

Name: Full subcategory of \(\mathbf C\)
Definition: A subcategory \(\mathbf U\) of \(\mathbf C\) such that for two objects \(A,B\in\mathbf U_0\), \(\text{Hom}_{\mathbf U}(A,B)=\text{Hom}_{\mathbf C}(A,B)\).
Page: 148
Chapter: 7
Section: 1
Variables: \(\mathbf C\) a category
Kind: Def

Page: 148
Chapter: 7
Section: 1
Index: 2
Kind: Ex
Name: Full and faithful subcategory
The inclusion functor \(\mathbf{Sets}_{fin}\rightarrowtail\mathbf{Sets}\),
the forgetful function \(\mathbf{Pos}\to\mathbf{Cat}\),
the discrete category \(\mathbf{Sets}\to\mathbf{Cat}\)
and
the forgetful functor \(\mathbf{Groups}\rightarrowtail\mathbf{Cat}\).

Name: Faithful but not full subcategory
Page: 148
Chapter: 7
Section: 1
Example: The forgetful functor \(\mathbf{Groups}\rightarrowtail\mathbf{Sets}\).

Q: \(f\) on objects is:[...] and
on arrow is: [...].
A: the constant function sending every object to the unique object of the group; sends each element to the unique element, and \(g_{p,q}\cdot g_{q,r}=g_{p,r}\)
Page: 149
Chapter: 7
Section: 1
Variables: \(P\) a poset,
\(G\) a group and
\(f:P\to G\) a functor.
Index: 2
Kind: Ex 2
Q: [...]
A: Substraction: \((\mathbb R,\le)\to(\mathbb R,+)\)

Name: A cocomplete poset
Definition: A poset which has all set-indexed joins \(\bigvee_{i\in I}a_i\).
Page: 130
Chapter: 6
Section: 3
Index: 10
Kind: Def

Smaller0: faithful
Greater0: for any objects \(X,Y\), distinct arrows \(f,g:X\to Y\), there is an arrow \(x:C\to Y\) such that \(fx\ne gx\)
Page: 149
Chapter: 7
Section: 2
Variables: \(\mathbf C\) a locally small category and
\(C\in \mathbf C_0\)
Smaller to greater: iff
Hide middle: yes
Prefix Smaller: \(\text{Hom}_{\mathbf C}(C,-):\mathbf{C}\to\mathbf{Sets} \) is

Name: A generator for \(\mathbf C\)
Definition: \(C\in \mathbf C_0\) such that \(\text{Hom}_{\mathbf C}(C,-):\mathbf{C}\to\mathbf{Sets} \) is faithful.
Page: 149, 150
Chapter: 7
Section: 2
Variables: \(\mathbf C\) a locally small category

Definition: The free group generated by \(X\).
Page: 150
Chapter: 7
Section: 2
Variables: \(X\) a set
Notation: \(F(X)\)

Smaller0: \(\text{Hom}(F(1),G)\)
Greater0: \(U(G)\), where \(U:\mathbf{Group\to Sets}\) is the forgetful functor
Page: 150
Chapter: 7
Section: 2
Variables: \(G\) a group.
Smaller to greater: is isomorphic to
Hide middle: yes

Page: 150
Chapter: 7
Section: 2
Variables: \(\mathbf A\) an algebra defined by functions and equality,
\(\mathbf C\) a category
\(A\in\mathbf A\) in \(\mathbf C\)
\(X\in C\).
Notation: \(\text{Hom}(X,A)\)
Index: 3
Kind: Ex
Typ: An element of \(\mathbf A\)

Q: [definition]
A: A bounded lattice \((L,\land,\lor,0,1)\) with a unary operator \(\neg\) such that:
-\(a\le \neg b\) iff \(a\land b=0\)
-\(\neg\neg a=a\)
Page: 34
Chapter: 2
Index: 4
Kind: Ex
Name: Boolean algebra

Name: Meet
Page: 34
Chapter: 2
Variables: \((B,\land,\lor,\neg,0,1)\) a Boolean algebra.
Notation: \(\land\)
Index: 4
Kind: Ex

Name: join
Page: 34
Chapter: 2
Variables: \((B,\land,\lor,\neg,0,1)\) a Boolean algebra.
Notation: \(\lor\)
Index: 4
Kind: Ex

Name: Complementation
Page: 34
Chapter: 2
Variables: \((B,\land,\lor,\neg,0,1)\) a Boolean algebra.
Context: Boolean algebra
Notation: \(\neg\)
Index: 4
Kind: Ex

Name: A filter
Definition: non-empty
Page: 35
Chapter: 2
Section: 3
Variables: \((B,\land,\lor,\neg,0,1)\) a Boolean algebra
Definition2: closed upward
Conjdef: A subset of \(B\)
Definition3: closed under meets.

Name: The component of \(\theta\) at \(C\)
Definition: The \(\pmb D\)-arrow \(\theta_C\)
Page: 156
Chapter: 7
Section: 4
Variables: \(\pmb{C,D}\) categories
\(F,G:\pmb C\to\pmb D\) functors
\(C\in\pmb C_0\)
\(\theta:F\to G\) a natural transformation
Index: 6
Kind: Def

Name: A principal filter
Definition: A set of the form \(\{U\in X\mid x\le U\}\) for some \(x\in X\).
Page: 153
Chapter: 7
Section: 2
Variables: \(X\) a BA and \(F\) a filter

Name: The set of ultrafilter of \(B\)
Page: 153
Chapter: 7
Section: 2
Variables: \(B\) a BA
Abbreviation: \(Ult(B)\)

Definition: The functor \(\mathbf{BA}^{op}\to\mathbf{Sets}\), sending a Boolean algebra \(B\) to \(\text{Hom}_{\mathbf{BA} }(B,2)\) and \(h:B\to B'\) in \(\mathbf{BA}_1\) to \(h^{-1}\).
Page: 107, 153
Chapter: 7
Section: 2
Notation: Ult functor

Definition: The functor \(\mathbf{S}ets^{op}\to\mathbf{BA}\), sending a set \(X\) the set of subsets of \(X\) and \(f:X\to Y\) in \(\mathbf{S}ets_1\) to \(f^{-1}\).
Page: 152
Chapter: 7
Section: 2
Context: Category Theory
Notation: \(\mathbf P\)

Smaller0: \(\text{Ult}(\mathcal P(X))\)
Greater0: \(\{U\subseteq \mathcal P(X)\mid U\text{ is an ultrafilter}\} \)
Page: 153
Chapter: 7
Section: 3
Variables: \(X\) a set
Smaller to greater: =

Name: Natural functor in Sets
Page: 154
Chapter: 7
Section: 3
Example: \(\text{Ult}\circ\mathcal P\)

Name: Natural functor in \(\mathbf{BA}\)
Page: 155
Chapter: 7
Section: 3
Example: \(\mathcal P\circ\text{Ult}\)

Name: The Stone representation of \(B\)
Definition: \(\mathcal P\circ\text{Ult}(B)\).
Page: 155
Chapter: 7
Section: 3
Variables: \(B\) a BA

Question: Every BA is isomorphic to
Answer: a BA consisting of susbets of some set \(X\) with set-theoretical Boolean operations.
Page: 155
Chapter: 7
Section: 3
Index: 5
Kind: Prop
Name: Stone duality theorem

Smaller0: \(\theta\) is a natural isomorphism
Greater0: each \(\theta_C\) is an isomorphism
Page: 159
Chapter: 7
Section: 5
Variables: \(\mathbf{C,D}\) 2 categories
\(\theta\in\text{Fun}(\mathbf{C,D})\)
Index: 11
Kind: Lem
Smaller to greater: iff

Definition: Objects: Sending \(X\) to \(\mathcal P(X)\cong \mathbf{Sets}(X,2)\), and \(f:X\to Y\) to \(f^*=f^{-1}\).
Page: 160
Chapter: 7
Section: 5
Variables: \(X\) an object in the category Sets
Notation: \(X^*\) in \(\mathbf {Sets}\)

Smaller0: \(F\) is a functor
Greater0: -\(F\) is functorial in each argument \(F(A^i,--):\mathbf{A^{i-1}\to C}\) are functors for all \(A^i\in \mathbf A_i\) and
-\(F\) satisfies the following "interchangeable law". Given \(\alpha^i:A^i\to A^{\prime i}\in\mathbf A^i\), \(F(A^{\prime0},\alpha^1)\circ F(\alpha^0,A^1)=F(\alpha^0,A^{\prime1})\circ F(A^0,\alpha^1)\)
Page: 161
Chapter: 7
Section: 5
Variables: \(\mathbf{A^0,A^1,C}\) categories and
\(F:\mathbf{A^0\times A^1}\to C\)
Index: 14
Kind: Lem
Smaller to greater: iff

Smaller0: \(F\) is a functor
Greater0: \(F\) is functorial in each argument \(F(A^i,--):\mathbf{A^{i-1}\to C}\) are functors for all \(A^i\in \mathbf A_i\)
Page: 161
Chapter: 7
Section: 5
Variables: \(\mathbf{A^0,A^1,C}\) categories and
\(F:\mathbf{A^0\times A^1}\to C\)
Index: 14
Kind: Lem
Smaller to greater: iff
Greater1: \(F\) satisfies the following "interchangeable law". Given \(\alpha^i:A^i\to A^{\prime i}\in\mathbf A^i\), \(F(A^{\prime0},\alpha^1)\circ F(\alpha^0,A^1)=F(\alpha^0,A^{\prime1})\circ F(A^0,\alpha^1)\).

Smaller0: \(\mathbf{C^1}\)
Greater0: \(\mathbf C\)
Page: 164
Chapter: 7
Variables: \(\mathbf C\) a category and
\(\mathbf 1\) the final category
Index: 15
Kind: Ex
Smaller to greater: =

Smaller0: \(\mathbf{C^2}\)
Greater0: \(\mathbf C^\to\)
Page: 164
Chapter: 7
Variables: \(\mathbf C\) a category and
\(\mathbf 2\) the category
Index: 15
Kind: Ex
Smaller to greater: =

Smaller0: \(\mathbf{C^I}\)
Greater0: \(\Pi_{i\in \mathbf I}\mathbf C\)
Page: 164
Chapter: 7
Variables: \(\mathbf C\) a category and
\(\mathbf I\) a discrete category (without non identity arrow)
Index: 15
Kind: Ex
Smaller to greater: =

Question: \(\mathbf{Sets^{A\rightrightarrows B}}=\)
Answer: \(\mathbf{Graph} \).
Page: 165
Chapter: 7
Variables: \(A\rightrightarrows B\) the category with two objects and two arrows from the first to the second one, and identity arrow
Index: 17
Kind: Ex

Name: Functor which preserves CCC
Page: 167, 168
Chapter: 7
Index: 20, 22
Kind: Prop
Example: Pos to Cat
Example2: Grpd to Cat

Name: Groupoid
Definition: A category in which each arrow is iso.
Page: 167
Chapter: 7
Index: 22
Kind: Prop

Name: A monoidal category
Definition: A 5-tuple \(\left(\mathbf C,\otimes,I,(\lambda_A)_{A\in\mathbf C_0},(\rho_A)_{A\in\mathbf C_0},(\alpha_{ABC})_{A,B,C\in\mathbf C_0}\right)\) with:
-\(\mathbf C\) a category,
-\(\otimes:\mathbf C\times \mathbf C\to \mathbf C\),
-\(I\in\mathbf C\),
-\(\alpha_{ABC}:A\otimes (B\otimes C)\tilde\longrightarrow(A\otimes B)\otimes C\),
-\(\lambda_A:I\otimes A\tilde\longrightarrow A\),
-\(\rho_A:A\otimes I\tilde\longrightarrow A\),
such that:
-\(\left(\alpha_{ABC}\otimes 1\right)\circ \alpha_{A(B\otimes C) D}\circ\left(1\otimes \alpha_{BCD}\right)=\alpha_{(A\otimes B)CD}\circ \alpha_{AB(C\otimes D)}\)
-\(1\otimes \lambda_A=\left(\rho_A\otimes 1\right)\circ \alpha_{AIA}\),
-\(\lambda_{I}=\rho_I\),
Page: 169
Chapter: 7
Index: 23

Name: \(\mathbf{C_1}\) and \(\mathbf{C_2}\) are equivalent
Definition: Two functors \((E_i)_{i\in\{0,1\} }:\mathbf{C_i\to C_{i-1} }\) and \(\alpha_0,\alpha_1\) two natural transformations \(\alpha_i:1_{C_i}\tilde\longrightarrow E_{1-i}\circ E_i\) in \(\mathbf{C_i^{C_i} }\)
Page: 172, 176, 177, 178
Chapter: 7
Variables: \(\mathbf{C_0},\mathbf{C_1}\) categories
Index: 25, 28, 29, 31
Kind: def, prop, Ex, Prop
Example: \(\mathbf{Par}\simeq\mathbf{Sets}_*\).
From \(\mathbf {Par}\) to \(\mathbf{Sets}_*\): sends \((A,a)\) to \((A\setminus \{a\})\), \(f:(A,a)\to (B,b)\) is sent to similar the function \(f\) with domain \(U_f=A\setminus f^{-1}(b)\).
From \(\mathbf{Sets}_*\) to \(\mathbf {Par}\): sends \(A\) to \((A\cup\{*\},*)\) and \(f:A\rightharpoondown B\) to

Name: A pseudo-inverse of \(E_0\)
Definition: \(E_1\).
Page: 172
Chapter: 7
Variables: \(\mathbf{C_0,C_1}\) categories, \(E_0,E_1, \alpha_0,\alpha_2\) an equivalence of categories
Index: 25

Smaller0: \(E\) is part of an equivalence of categories
Greater0: \(E\) is full and faitful and for every \(D\in\mathbf D\), there exists \(C\in\mathbf C\) such that \(FC\cong D\)
Page: 173
Chapter: 7
Variables: \(\mathbf{C,D}\) categories,
\(E:\mathbf{C\to D}\)
Index: 26
Kind: Prop
Smaller to greater: iff

Name: A meet-semilattice
Definition: A pair with a set \(L\) and a binary operation \(\land\) such that \(\land\) is idempotent, associative and commutative
Page: 129
Chapter: 6
Section: 3
Definition2: An ordered set \((L,\le)\) such that, for every \(a,b\in L\), there is a greatest lower bound of \(a\) and \(b\)
Extra: Not in the book

Name: A lattice
Definition: A set \(L\) and two binary operations such that \((L,\land)\) and \((L,\lor)\) are meet- and join- semilattices respectively, and \(a\lor(a\land b)=a\) and \(a\land(a\lor b)=a\) for each \(a,b\in L\).
Page: 129
Chapter: 6
Section: 3
Extra: Not in the book
French: Un treillis

Name: A bounded lattice
Definition: A lattice \((L,\land,\lor)\) and two constants 0 and 1 such that \(a\lor 0=a\) and \(a\land 1=a\) for each \(a\in L\).
Page: 129
Chapter: 6
Section: 3
Extra: Not in the book

Name: A join-semilattice
Definition: A pair with a set \(L\) and a binary operation \(\lor\) such that \(\lor\) is idempotent, associative and commutative
Page: 129
Chapter: 6
Section: 3
Definition2: An ordered set \((L,\le)\) such that, for every \(a,b\in L\), there is least upper bound of \(a\) and \(b\).
Extra: Not in the book

Name: Distributive lattice
Definition: A lattice such that \((a\lor b)\land c=(a\land c)\lor (b\land c)\).
Page: 129
Chapter: 6
Section: 3

Name: \(a\le b\)
Definition: \(a=a\land b\).
Chapter: 6
Section: 3
Variables: \((L,\land)\) a meet-semilattice
\(a,b\in A\)

Name: \(a\le b\)
Definition: \(b=a\lor b\).
Chapter: 6
Section: 3
Variables: \((L,\lor)\) a join-semilattice and
\(a,b\in A\)

Question: The number of initial element is
Answer: at most 1 up to iso.
Page: 033
Chapter: 2
Section: 2.2
Variables: \(\mathbf C\) a category
Extra: 2 dans 4

Name: The images of \(i,j,\dots\) in \(\mathbf C\)
Page: 101
Chapter: 5
Section: 4
Notation: \(D_i, D_j,\dots\)
Index: 15
Kind: Def

Page: 090
Definition: Isomorphic object
Chapter: 5
Section: 1
Variables: \(\mathbf C\) a category
\(X\in\mathbf C^0\)
Notation: \(\cong \)
Name: Equivalent objects
Context: \(Sub_{\mathbf C}(X)\)

Definition: \(f\)
Page: 090
Chapter: 5
Section: 1
Variables: \(\mathbf C\) a category,
\(X\in\mathbf C^0\),
\(f\) an arrow in \(Sub_{\mathbf C}(X)\)
Typ: monic

Definition: There is an arrow from \(m\) to \(m'\) in \(Sub_{\mathbf C}(X)\).
Page: 090
Chapter: 5
Section: 1
Variables: \(\mathbf C\) a category,
\(X\in\mathbf C^0\),
\(m,m'\) objects in \(Sub_{\mathbf C}(X)\)
Notation: \(m\subseteq m'\)

Question: The number of arrow from \(m\) to \(m'\) in \(Sub_{\mathbf C}(X)\) is
Answer: at most one.
Page: 090
Chapter: 5
Section: 1
Variables: \(\mathbf C\) a category,
\(X\in\mathbf C^0\),
\(m,m'\) objects in \(Sub_{\mathbf C}(X)\)

Definition: A function \(U\to B\) for some \(U\subseteq A\).
Page: 175
Chapter: 7
Section: 10
Variables: \(A,B\) sets
Notation: An element of \(A\rightharpoondown B\)
Index: 27
Kind: Ex

Definition: \(U_f\to B, x\mapsto f(u)\).
Page: 175
Chapter: 7
Section: 10
Variables: \(A,B\) sets,
\(f:A\rightharpoondown B\)
Notation: \(|f|\)
Index: 27
Kind: Ex

Definition: The domain of definition of \(f\).
Page: 175
Chapter: 7
Section: 10
Variables: \(A,B\) sets,
\(f:A\rightharpoondown B\)
Notation: \(U_f\)
Index: 27
Kind: Ex

Name: An atom of \(\mathcal B\)
Definition: An element \(a\in\mathcal B\) such that \(0<a\) and there exists not \(b\in\mathcal B\) such that \(0<b<a\).
Page: 179
Chapter: 7
Section: 10
Variables: \(\mathcal B\) a BA
Index: 31
Kind: prop

Name: The set of atoms of \(\mathcal B\)
Page: 179
Chapter: 7
Section: 10
Variables: \(\mathcal B\) a BA
Notation: \(A(\mathcal B)\)
Index: 31
Kind: Prop

Name: An atomic Boolean algebra
Definition: A BA \(\mathcal B\) such that for all \(0\ne b\in\mathcal B\), there exists an atom \(0<a\le b\).
Page: 180
Chapter: 7
Section: 10
Index: 33
Kind: Lem

Name: The category of complete atomic BA
Page: 180
Chapter: 7
Section: 10
Notation: \(\mathbf{caBA}\)
Index: 33
Kind: Lem

Name: A limit for \(D\)
Definition: A terminal object in \(\mathbf{Cone}(D)\).
Page: 102
Chapter: 5
Section: 4
Variables: \(\mathbf{J,C}\) be categories
\(D:\mathbf J\to\mathbf C\) a diagram of type \(\mathbf J\).
Notation: [$$]p_i:\varprojlim_j D_j\to D_i [/$$]
Index: 16
Kind: Definition
Name2: Universal cone
Extra: Other name by wp

Name: The hom-set between \(A\) and \(B\)
Definition: The set of arrows with domain \(A\) and codomain \(B\).
Page: 25
Chapter: 1
Section: 1.8
Variables: \(\mathbf C\) a category,
\(A,B\in\mathbf C_0\)
Notation: \(\text{Hom}(A,B)\)
Index: 2
Kind: Def
Extra: Wikipedia helped to have a definition

Page: 25
Object: \(g\mapsto f\circ g\).
Domain: \(\text{Hom}(A,X)\to\text{Hom}(A,Y)\)
Section: 1.8
Chapter: 1
Variables: \(\mathbf C\) a category,
\(A\in\mathbf C_0\),
\(f:X\to Y\in\mathbf C_1\)
Kind: Def
Index: 2

Name: A functor injective/surjective on arrows
Definition: \(F:\mathbf C\to\mathbf D\) a functor such that the function \(F_1:\mathbf C_1\to\mathbf D_1\) is injective/surjective.
Page: 148
Chapter: 7
Section: 1
Kind: Def
Extra: implies faith/full

Name: A diagram on \(C\)
Definition: An element of type \(\mathbf C\to \mathbf{Sets}\)
Page: 185
Chapter: 8
Section: 1
Variables: \(\mathbf C\) a locally small category

Name: The evaluation functor at \(C\)
Page: 186
Chapter: 8
Section: 1
Variables: \(\mathbf C\) a locally small category,
\(C\in\mathbf C_0\)
Notation: \(ev_C\)
Typ: \(\mathbf{Sets^C}\to\mathbf{Sets}\)

Question: \(\mathbf{Sets^\Gamma}\) equals
Answer: Graphs
Page: 186
Chapter: 8
Section: 1
Variables: \(\Gamma\) be the category

Page: 187
Object: \(C\mapsto Hom_\mathbf{C}(-,C)\)
Domain: \(\mathbf{C}\)
Arrow: \(f:(C\to D)\mapsto\left(Hom_{\mathbf C}(-,f)\right)\)
Section: 1
Chapter: 8
Codomain: \(\mathbf{Sets^{C^{op}}}\)
Variables: \(\mathbf C\) a locally small category
Notation: The Yoneda embedding
Kind: Def

Smaller0: \(x\le y\)
Greater0: \(m(x)\le m(y)\)
Page: 7
Chapter: 1
Section: 1.4
Variables: \(A,B\) posets,
\(m:A\to B\),
\(x,y\in A\)
Index: 3
Kind: Ex

Name: A section of \(x\)
Definition: An element \(x'\) such that \(xx'=1_{X'}\)
Page: 32
Chapter: 2
Section: 2.1.1
Variables: \(\mathbf C\) a category,
\(X,X'\in\mathbf C_0\),
\(x:X\to X'\),
Context: Category
Index: 7
Kind: Def
Name2: A splitting of \(x\)

Name: A retraction of \(x\)
Definition: An element \(x'\) such that \(x'x=1_X\)
Page: 32
Chapter: 2
Section: 2.1.1
Variables: \(\mathbf C\) a category,
\(X,X'\in\mathbf C_0\),
\(x:X\to X'\).
Index: 7
Kind: Def

Name: The retract
Definition: \(A\)
Page: 32
Chapter: 2
Section: 2.1.1
Variables: \(\mathbf C\) a category,
\(X,Y\in\mathbf C_0\),
\(x:X\to Y\),
\(y:Y\to X\).
Index: 7
Kind: Def
Assuming: \(xy=1_Y\)

Question: First approximation of category theory:
Answer: «algebras of functions»
Page: 001
Chapter: 1

Page: 48
Object: \(X\mapsto\text{Hom}(X,C)\)
Domain: \(\mathbf C^{op}\)
Arrow: \(f\mapsto f^*\).
Section: 7
Chapter: 2
Codomain: \(\mathbf{Sets}\)
Variables: \(\mathbf{C}\) be a category
\(C\in\mathbf C\).
Name: The contravariant representable functor of \(A\)

Definition: The functor \(\mathbf C/C \to\mathbf C/D\), \(f\mapsto g\circ f\).
Page: 017
Chapter: 1
Section: 1.6
Variables: \(\mathbf C\) a category, \(C,D\in\mathbf C_0\) and \(g:C\to D\).
Notation: \(g_*\)
Index: 4

Name: Projective object
Definition: for any epi \(e:E\twoheadrightarrow X\)
Page: 033
Chapter: 2
Section: 2.1.1
Variables: \(\mathbf C\) a category,
Definition2: and arrow \(f:P\to X\)
Conjdef: An object \(P\) such that
Definition3: there is some arrow \(\overline f:P \to E\) such that
Definition4: \(e\circ \overline f=f\)
Hide remaining def: yes

Name: The dual of \(\Sigma\)
Definition: The formula obtained from \(\Sigma\) by replacing
\(f\circ g\) by \(g\circ f\),
cod by dom,
dom by cod.
Page: 053
Chapter: 3
Section: 1
Variables: \(\Sigma\) a formula.
Notation: \(\Sigma^*\)

Name: The coproduct of \(A\) and \(B\)
Definition: A diagram of the form \(A\xrightarrow{q_1} Q\xleftarrow{q_2}B\), such that, for any \(Z\) and \(A\xrightarrow{z_1} Z\xleftarrow{q_2}B\) there is a unique \(u:Q\to Z\) with \(u\circ q_i=z_i\).
Page: 055
Chapter: 3
Section: 3.2
Variables: \(\mathbf C\) a category,
\(A,B\in\mathbf C_0\).
Notation: \(A+B\)
Kind: Def
Example: Disjoint union
Applied to: set theory

Definition: \[\left\{\begin{array}{ll} f(x)&\delta=1\\ g(x)&\delta=2\end{array}\right.\]
Page: 056
Chapter: 3
Section: 3.2
Variables: \(\mathbf C\) a category,
\(A,B,Z\in\mathbf C_0\),
\(f:A\to Z\) and \(g:B\to Z\).
Notation: \([f,g](x,\delta)\)

Question: \(\mathbf C\) contains
Answer: objects and arrows
Page: 001
Chapter: 1
Variables: \(\mathbf C\) a category.

Question: \(f\) contains
Answer: a domain dom\((f)\) and a codomain cod\((f)\)
Chapter: 1
Variables: \(f\) an arrow in a category.

Name: A transitive relation
Definition: A relation \(R\subseteq A\times A\) such that \(a Rb\) and \(b R c\) implies \(a R c\).
Page: 006
Chapter: 1
Section: 1.4
Index: 3
Kind: Ex

Question: The number of inverse of \(f\) is
Answer: at most one.
Chapter: 1
Section: 1.5
Variables: \(\mathbf C\) a category, \(A,B\in\mathbf C_0\) and \(f:A\to B\).

Question: The functors from \(\mathbf C_1\) are
Answer: Domain and codomain \(\mathbf C\leftarrow C^\to\rightarrow \mathbf C\).
Page: 016
Chapter: 1
Section: 1.6
Variables: \(\mathbf C\) a category.

Question: In the free monoid, \(i\) is
Answer: the function sending a letter to the words of length 1 containing this letter.
Page: 018
Chapter: 1
Section: 1.7

Name: Universal mapping property of \(\mathbf C(G)\)
Definition: There is a graph homomorphism \(i:G\to|\mathbf C(G)|\) , and given any category \(\mathbf D\) and any graph homomorphism \(h:G\to|\mathbf D|\), there is a unique functor \(\overline h:\mathbf C(G)\to \mathbf D\) with \(|\overline h|\circ i=h\).
Page: 023
Chapter: 1
Section: 1.7
Variables: \(G\) a graph.

Question: The free category on a graph with two vertices and one edge between them is
Answer: the finite category 2.
Page: 023
Chapter: 1
Section: 1.7

Question: The free category on the graph
\(f\xrightarrow e g\xrightarrow f f\) has, as arrows
Answer: e, f, ef, fe, efe, fef, ....
Page: 023
Chapter: 1
Section: 1.7

Question: Functors preserves
Answer: split epis and split monos
Page: 032
Chapter: 2
Section: 2.1.1

Question: In Mon, for \(h,j:M\to N\) and \(x:1\to M\) then
Answer: \(hx=jx\).
Page: 036
Chapter: 2
Section: 2.2
Index: 12 2
Kind: Ex

Question: \(\langle x_1,x_2 \rangle: X\to A_1\times A_2\) is such that
Answer: \(x_i=p_i\circ \langle x_1,x_2 \rangle\).
Page: 046
Chapter: 2
Variables: \(\mathbf C\) a category
\(X,A_1,A_2\in\mathbf C_0\)
\(x_1: X\to A_1\), \(x_2:X\to A_2\)

Question: \(f\) is equivalent to
Answer: a pair of arrows \(f_i:X\to A_i\).
Page: 039
Chapter: 2
Section: 2.6
Variables: \(\mathbf C\) a category,
\(X,A_1,A_2\in\mathbf C_0\)
\(f: X\to A_1\times A_2\)

Question: \(g\) is
Answer: a function of two variables.
Chapter: 2
Section: 2.6
Variables: \(\mathbf C\) a category
\(X,A,B\in\mathbf C_0\)
\(g:A\to B\to X\).

Question: An algebra can always be categorically characterized as
Answer: a coequalizer of a free algebra.
Page: 068
Chapter: 3
Section: 3.4
Index: 22
Kind: Ex

Name: A group in \(\mathbf C\)
Definition: An object \(G\) and three arrows
-\(i:G\to G\),
-\(u:1\to G\), and
-\(m:G\times G\to G\),
such that:
\(m\circ (1\times m)=m\circ (m\times 1):G\times G\times G\to G\) (associativity)
\(1_G=m\circ \langle 1_G,u\rangle=m\circ \langle u,1_G\rangle:G\to G\) (identity)
and
\(u=m\circ \langle 1_g,i\rangle \circ \Delta=m\circ \langle i,1_g\rangle \circ \Delta\) (inverse)
Page: 076
Chapter: 4
Section: 1
Variables: \(\mathbf C\) a category.
Kind: Def
Assuming: \(\mathbf C\) has finite product and terminal object

Name: A maximal filter
Definition: it is a \(\subseteq\)-maximal filter which is a strict subset of \(B\)
Page: 35
Chapter: 2
Section: 3
Variables: \((B,\land,\lor,\neg,0,1)\) a Boolean algebra
Definition2: for each \(x\in B\), it contains \(x\) xor \(\neg x\)
Conjdef: A filter \(F\) such that, equivalently
Name2: An ultrafilter

Name: A natural transformation from \(F\) to \(G\)
Definition: A family of arrows \((\theta_O:FO\to GO\in \pmb D_1)_{O\in\pmb C_0}\), such that for any \(a:O\to O'\) in \(\pmb C\), \(\theta_{O'}\circ{}F(a)=G(a)\circ\theta_O\).
Page: 156, 157, 160
Chapter: 7
Section: 4
Variables: \(\pmb{C,D}\) categories
\(F,G:\pmb C\to\pmb D\) functors
Notation: \(F\to G\)
Index: 6
Kind: Def
Example: \(X\to Ult(\mathcal P(X))\) for \(X\) a set
Example2: The function \(^{**}\) on sets
Extra: Ult et P does not seems to be natural transformation
flèche dans D ?
Denoted by: \(\theta\)
Example3: For \(X\) a set, \(X\to \text{UM}(X)\), the free monoid generated by \(X\) and \(f\to \text{UM}(f)\) the morphism of monoid generated by \(f\)
Example4: \(B\to \mathcal P(\text{Ult}(B))\) for \(B\) a BA

Question: The number of terminal object is
Answer: at most one, up to iso.
Page: 33
Chapter: 2
Section: 2.2
Variables: \(\mathbf C\) a category

Question: The number of product diagram of \(A\) and \(B\) is
Answer: at most one, up to iso.
Page: 38
Chapter: 2
Section: 2.4
Variables: \(\mathbf C\) a category and
\(A,B\in\mathbf C_0\).
Index: 13
Kind: Def

Page: 004
Section: 1.3
Notation: \(\circ\)
Chapter: 1
Name: Composition of functions
Definition: A map on pair of arrows \((f,g)\) such that \(\text{dom}(f)=\text{cod}(g)\).
Kind: Def
Associative: associative

Name: A concrete category
Definition: Awodey's definition: A category whose objects are sets (possibly equiped with some structures) and arrows are functions (structure-preserving).
Page: 007
Chapter: 1
Section: 1.4
Index: 4
Kind: Ex
Definition2: wp's definition: A pair \((\mathbf C,U)\) with \(\mathbf C\) a category and \(U:\mathbf C\to\mathbf{Set}\) a faithful functor

Smaller0: \(R\) is a poset
Greater0: \(R\) is antisymmetric
Page: 009
Chapter: 1
Section: 1.4
Variables: \(A\) a set,
\(R\) a preorder on \(A\)
Index: 8
Kind: Ex
Smaller to greater: iff

Smaller0: \(\mathbf C_1\) is a set
Greater0: isomorphic to a category in which the objects are sets and the arrows are functions
Page: 013
Chapter: 1
Section: 1.5
Variables: \(\mathbf C\) a category.
Index: 6
Kind: Theorem
Hide middle: yes
Prefix Greater: \(\mathbf C\) is

Smaller0: 1/Sets
Greater0: Sets\(_*\)
Page: 017
Chapter: 1
Section: 1.6
Index: 8
Kind: Ex
Smaller to greater: is isomorphic to

Smaller0: \(i,j\) have the UMP of the free monoid on \(A\)
Greater0: a (unique) monoid isomorphism \(h:M\to N\) such that \(|h|\circ i=j\) and \(|h^{-1}|\circ j=i\), where \(h:|M|\to|N|\) is the result of the forgetful fonctors on \(h\)
Page: 019
Chapter: 1
Section: 1.7
Variables: \(A, M, N\) a set
\(i:A\to |M|\) and \(j: A\to |N|\)
Hide middle: yes
Prefix Greater: there is

Smaller0: The free monoid of a singleton
Greater0: \(\mathbb N\)
Page: 020
Chapter: 1
Section: 1.7
Smaller to greater: is isomorphic to

Smaller0: \(f\) is iso
Greater0: \(f\) is mono and epi
Page: 031
Chapter: 2
Section: 2.1
Variables: \(\mathbf C\) a category
\(f\in\mathbf C_1\).
Index: 6
Kind: Prop
Extra: inverse

Name: A terminal object
Definition: An object \(X\) such that for any object \(C\) there is a unique morphism \(C\to X\).
Page: 033
Chapter: 2
Section: 2.2
Variables: \(\mathbf C\) a category
Notation: 1
Example: Singleton in Set
Name2: A final object
Example2: trivial group in Grp
Extra: 2nd name by wp
Cardinal: At most one, up to iso

Smaller0: \(A\) is an ultrafilter
Greater0: there is an arrow \(A\to \mathbf 2\)
Page: 035
Chapter: 2
Section: 2.3
Variables: \(A\) a Boolean Alegra.
Hide middle: yes
Prefix Greater: in \(\mathbf{BA}\)

Smaller0: \(\text{Hom}_{\text{Sets} }(1,X)\)
Greater0: \(X\)
Page: 036
Chapter: 2
Section: 2.2
Variables: \(X\) a set
Smaller to greater: is equivalent to

Smaller0: \(f=g\)
Greater0: \(x:1\to P\), \(fx=gx\)
Page: 036
Chapter: 2
Section: 2.2
Variables: In Pos, \(f,g:P\to Q\).
Index: 12 1
Kind: Ex
Smaller to greater: iff
Hide middle: yes
Prefix Greater: for all

Name: A product for the objects \(A\) and \(B\)
Definition: An object \(P\) and arrows \(A\xleftarrow{p_1} P\xrightarrow{p_2}B\) satisfying the following UMP:
Given any diagram of the form \(A\xleftarrow{x_1} X\xrightarrow{x_2}B\) there exists a unique \(u:X\to P\) such that \(x_i=p_i u\).
Page: 038
Chapter: 2
Section: 2.4
Variables: \(\mathbf C\) a category and
\(A,B\in\mathbf C_0\).
Index: 13
Kind: Def

Smaller0: \(C\) is a nullary product
Greater0: \(C\) is a terminal product
Page: 047
Chapter: 2
Section: 2.6
Variables: \(\mathbf C\) a category and
\(C\in\mathbf C_0\).
Hide middle: yes

Smaller0: \(A\) is a finite coproducts of 1
Greater0: \(A\) is a finite set
Page: 056
Chapter: 3
Section: 3.2
Variables: \(A\) a set.
Hide middle: yes

Name: Cayley's theorem
Definition: Each group is isomorphic to a permutation group..
Page: 013
Chapter: 1
Index: 5
Kind: Theorem
Intuition: Each abstract group is isomorphic to a concrete group

Question: \(\Sigma\) holds for all categories, iff
Answer: so does the dual statement \(\Sigma^*\).
Page: 054
Chapter: 3
Section: 1
Variables: \(\Sigma\) a formula.
Index: 2
Kind: Prop
Name: Conceptual duality

Name: Commutative diagram of functor
Page: 009
Chapter: 1
Section: 1.4
Index: 6
Kind: Ex

Smaller0: \(M(A+B)\)
Greater0: \(M(A)+M(B)\)
Page: 056
Chapter: 3
Section: 2
Variables: \(A,B\) sets.
\(M\) the free-monoid functor
Index: 5
Kind: Ex
Smaller to greater: =

Smaller0: \(A+_{Ab}B\)
Greater0: \(A\times_{Ab} B\)
Page: 060
Chapter: 3
Section: 2
Variables: \(A,B\) abelian groups.
Index: 11
Kind: Prop
Smaller to greater: =
Hide middle: yes

Name: An equalizer of \(\mathcal F\)
Definition: An universal arrow \(h:E\to A\) such that \(f\circ h=f'\circ h\) for each \(f,f'\in\mathcal F\). That is, given \(z:Z\to A\) with \(f\circ z=f'\circ z\) for each \(f,f'\in\mathcal F\) there is a unique \(u:Z\to E\) with \(h\circ u=z\).
Page: 062
Chapter: 3
Variables: \(\mathbf C\) a category,
\(A,B\in\mathbf C_0\),
\(\mathcal F\) a set of arrows \(A\to B\)
Definition2: A subobject \(h\) of \(A\) with the property that \(z\in_A h\) iff \(f(z)=f'(z)\) for each \(f,f'\in \mathcal F\).
Example: As in etymology
Abbreviation: \(\text{Eq}(\mathcal F)\)
Extra: Ex 5.3,
Generalization to arbitrary set and notation by wp
French: monic
Etymology: Over set: the subset of the domain where each function agrees

Name: Cartesian closed category
Definition: has all finite products
Page: 122
Chapter: 6, 7
Section: 2, 6
Index: 2, 4, 13
Kind: Definition, Ex, Prop
Definition2: has exponential
Example: \(\mathbf{Sets}\), \(\mathbf{Sets_{fin} }\)
Conjdef: which
Example2: \(\mathbf{Pos}\)
Abbreviation: CCC
Example3: Small categories and functors, with exponential \(\mathbf{D^C}=\text{Fun}(\mathbf{C,D})\).
DefType: A category
Example4: \(\omega\textbf{CPO}\)

Smaller0: coequalizer of the inclusion function \(i\) and of the function \(u(x)=1\)
Greater0: \(G/H\)
Page: 080
Chapter: 4
Section: 2
Variables: \(G\) a group and \(H\le G\).
Smaller to greater: equals
Hide middle: yes
Assuming: \(H\) is normal

Name: The pullback of \(f\) and \(g\)
Definition: An object P and a pair \(\langle p_1,p_2\rangle\) with \(p_1:P\to A, p_2:P\to C\) such that \(fp_1=gp_2\), universal with this property.
Page: 092
Chapter: 5
Section: 2
Variables: \(\mathbf C\) be a category,
\(A,B,C,D\) objects,
\(f:A\to B\) and \(g:C\to D\).
Notation: \(A\times _CB\)
Index: 4
Kind: Def
Name2: Fiber product
Extra: other name from wp
Assuming: \(B=D\)
French: Produit fibré
Name3: Fibre/fibered product
Name4: Cartesian square

Name: Local membership relation
Definition: There exists \(f:Z\to M\) such that \(z=mf\)
.
Page: 091
Chapter: 5
Section: 1
Variables: -\(X\) be an object of a category \(\mathbf C\).
-\(m:M\rightarrowtail X\) be a subobject of \(X\).
-\(z:Z\to X\).
Notation: \(z\in_X M\)

Page: 158, 157-159
Definition: An isomorphism in the category of functors.
Chapter: 7
Section: 5
Name: A natural isomorphism
Kind: Def, Ex
Index: 10, 8
Example: \(\text{Hom}_{\mathbf{Grp} }(F(1),G)\cong U(G)\)
Example2: For \(F\) a field, \(V\mapsto V^{**}\), \(f:V\to W:x\mapsto\left(\phi\mapsto \phi(x)\right)\)
Example3: \(\text{Hom}_{\mathbf{Sets} }(X,\mathbf 2)\cong \mathcal P(X)\), \(\text{Hom}_{\mathbf{BA} }(B,\mathbf 2)\cong \text{Ult}(B)\),
Example4: \(A\times (B\times C)\cong (A\times B)\times C\), \(A\times B\cong B\times A\),

Name: A function which preserve limits of type \(J\)
Definition: whenever \(p_j:L\to D_j\) is a limit for a diagram \(D:\mathbf{J\to C}\); the cone \(Fp_j:FL\to FD_j\) is then a limit for the diagram \(FD:\mathbf{J\to D}\).
Page: 106
Chapter: 5
Variables: \(\mathbf{J,C,D}\) be categories.
Index: 24
Kind: Definition
Definition2: [$$]F(\varprojlim D_j)\cong \varprojlim F(D_j).[/$$]
Example: Let \(\mathbf C\) be locally small and \(C\in\mathbf C\). The representable functors \(\text{Hom}(C,-)\)
Conjdef: A functor \(F:\mathbf{C\to D}\) such that, equivalently

Name: A Heyting algebra
Definition: A poset with:
--finite meet: \(1\) and \(p\land q\),
--finite join: \(0\) and \(p\lor q\),
--Exponentials: for each \(a,b\), an element \(a\implies b\) such that \(a\land b\le c\) iff \(a\le b\implies c\).
Page: 129
Chapter: 6
Section: 3
Index: 8
Kind: Definition
Definition2: A bounded lattice \((L,\land,\lor,0,1)\) such that, for each \(a,b\in L\), there exists a greatest \(x\in L\) such that \(a\land x\le b\).
Example: the powerset of any set
Example2: the lattice of open sets of a topological space
Example3: Boolean algebras