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DIAGONAL LEMMA

  • Diagonal lemma
  • Statement in mathematical logic

    In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence

    Diagonal lemma

    Diagonal_lemma

  • Cantor's diagonal argument
  • Proof in set theory

    over Cantor's theory Diagonal lemma the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's

    Cantor's diagonal argument

    Cantor's diagonal argument

    Cantor's_diagonal_argument

  • Tarski's undefinability theorem
  • Theorem that arithmetical truth cannot be defined in arithmetic

    wholly elementary except for the diagonalization that the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for

    Tarski's undefinability theorem

    Tarski's undefinability theorem

    Tarski's_undefinability_theorem

  • Diagonalization
  • Topics referred to by the same term

    techniques, including: Cantor's diagonal argument, used to prove that the set of real numbers is not countable Diagonal lemma, used to create self-referential

    Diagonalization

    Diagonalization

  • Lawvere's fixed-point theorem
  • Theorem in category theory

    important corollaries of this are: Cantor's theorem Cantor's diagonal argument Diagonal lemma Russell's paradox Gödel's first incompleteness theorem Tarski's

    Lawvere's fixed-point theorem

    Lawvere's_fixed-point_theorem

  • Diagonal argument
  • Topics referred to by the same term

    argument (the earliest) Cantor's theorem Russell's paradox Curry's paradox Diagonal lemma Gödel's first incompleteness theorem Tarski's undefinability theorem

    Diagonal argument

    Diagonal_argument

  • Gödel's incompleteness theorems
  • Limitative results in mathematical logic

    directly, the existence of at least one such statement follows from the diagonal lemma, which says that for any sufficiently strong formal system and any statement

    Gödel's incompleteness theorems

    Gödel's_incompleteness_theorems

  • Hilbert–Bernays-Löb provability conditions
  • ones) and the diagonal lemma hold for Peano arithmetics; once these are established the proof can be easily formalized. Using the diagonal lemma, there is

    Hilbert–Bernays-Löb provability conditions

    Hilbert–Bernays-Löb_provability_conditions

  • Yoneda lemma
  • Embedding of categories into functor categories

    The Yoneda lemma is a fundamental result in category theory, a branch of mathematics. It is an abstract result on functors of the type morphisms into

    Yoneda lemma

    Yoneda_lemma

  • Kurt Gödel
  • Mathematician and philosopher (1906–1978)

    with ZFC Axiom of constructibility Compactness theorem Condensation lemma Diagonal lemma Dialectica interpretation Ordinal definable set Slingshot argument

    Kurt Gödel

    Kurt Gödel

    Kurt_Gödel

  • Lemma (mathematics)
  • Theorem for proving more complex theorems

    local lemma Nakayama's lemma Noether normalization lemma Poincaré's lemma Riesz's lemma Schur's lemma Schwarz's lemma Sperner's lemma Urysohn's lemma Vitali

    Lemma (mathematics)

    Lemma_(mathematics)

  • Rosser's trick
  • Method in mathematical logic

    two numbers, as well as to include some first-order logic.) Using the diagonal lemma, let ρ {\displaystyle \rho } be a formula such that T {\displaystyle

    Rosser's trick

    Rosser's_trick

  • Quine (computing)
  • Self-replicating program

    quine in the strict programming sense. Computer programming portal Diagonal lemma Droste effect Fixed point combinator Self-modifying code Self-interpreter

    Quine (computing)

    Quine (computing)

    Quine_(computing)

  • Fixed-point theorem
  • Condition for a mathematical function to map some value to itself

    fixed-point theorem Caristi fixed-point theorem Diagonal lemma, also known as the fixed-point lemma, for producing self-referential sentences of first-order

    Fixed-point theorem

    Fixed-point_theorem

  • Zorn's lemma
  • Mathematical proposition equivalent to the axiom of choice

    Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for

    Zorn's lemma

    Zorn's lemma

    Zorn's_lemma

  • Gödel's β function
  • β function lemma makes use of the Chinese remainder theorem. Gödel numbering for sequences Gödel's incompleteness theorems Diagonal lemma H. E. Rose,

    Gödel's β function

    Gödel's_β_function

  • Self-reference
  • Sentence, idea or formula that refers to itself

    reference – Series of references where the last object references the first Diagonal lemma – Statement in mathematical logic Droste effect – Recursive visual effect

    Self-reference

    Self-reference

    Self-reference

  • Knower paradox
  • Self-reference paradox

    contradiction that (K) is both not known and known. Since, given the diagonal lemma, every sufficiently strong theory will have to accept something like

    Knower paradox

    Knower_paradox

  • Use–mention distinction
  • Distinction between using a word and mentioning it

    this concept appears in Gödel's incompleteness theorem, where the diagonal lemma plays a crucial role. Stanisław Leśniewski extensively employed this

    Use–mention distinction

    Use–mention_distinction

  • List of lemmas
  • Abhyankar's lemma Fundamental lemma (Langlands program) Five lemma Horseshoe lemma Nine lemma Short five lemma Snake lemma Splitting lemma Yoneda lemma Matrix

    List of lemmas

    List_of_lemmas

  • Löb's theorem
  • Provability logic

    arithmetic, then the existence of modal fixed points follows from the diagonal lemma. In addition to the existence of modal fixed points, we assume the following

    Löb's theorem

    Löb's_theorem

  • List of mathematical logic topics
  • choice Zorn's lemma Boolean algebra (structure) Boolean-valued model Burali-Forti paradox Cantor's back-and-forth method Cantor's diagonal argument Cantor's

    List of mathematical logic topics

    List_of_mathematical_logic_topics

  • Saul Kripke
  • American philosopher and logician (1940–2022)

    informal self-referential meaning, and this idea – manifested by the diagonal lemma – is the basis for Tarski's theorem that truth cannot be consistently

    Saul Kripke

    Saul Kripke

    Saul_Kripke

  • Kleene's recursion theorem
  • Theorem in computability theory

    lambda calculus for the same purpose as the first recursion theorem. Diagonal lemma a closely related result in mathematical logic. Ershov, Yuri L. (1999)

    Kleene's recursion theorem

    Kleene's_recursion_theorem

  • Fodor's lemma
  • Concept in mathematical set theory

    mathematics, particularly in set theory, Fodor's lemma (or the pressing-down lemma) states: Fodor's lemma—If κ {\displaystyle \kappa } is a regular, uncountable

    Fodor's lemma

    Fodor's_lemma

  • Hilbert–Bernays paradox
  • since no number is identical with its successor. Since, given the diagonal lemma, every sufficiently strong theory will have to accept something like

    Hilbert–Bernays paradox

    Hilbert–Bernays_paradox

  • Matrix determinant lemma
  • In linear algebra

    In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic

    Matrix determinant lemma

    Matrix_determinant_lemma

  • Autogram
  • Self-describing sentence

    2's, 6 3's, 3 4's, 1 5, 2 6's, 1 7, 2 8's, and 1 9. Quine (computing) Diagonal lemma Sallows, L., In Quest of a Pangram, Abacus, Vol 2, No 3, Spring 1985

    Autogram

    Autogram

  • Diagonal subgroup
  • X, G acts k-transitively on X n. Burnside's lemma can be proved using the action of the twofold diagonal subgroup. Diagonalizable group Sahai, Vivek;

    Diagonal subgroup

    Diagonal_subgroup

  • Ultrafilter on a set
  • Maximal proper filter

    of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the other hand, there exist models of ZF

    Ultrafilter on a set

    Ultrafilter on a set

    Ultrafilter_on_a_set

  • Whitehead's lemma
  • matrix whose diagonal block is 1 {\displaystyle 1} and i j {\displaystyle ij} -th entry is s {\displaystyle s} . The name "Whitehead's lemma" also refers

    Whitehead's lemma

    Whitehead's_lemma

  • Program equilibrium
  • Game theory

    program game to be given access to their own source code. By the diagonalization lemma, one can use quining to enable programs to refer to their source

    Program equilibrium

    Program_equilibrium

  • Adjoint functors
  • Relationship between two functors abstracting many common constructions

    notion of a limit. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and

    Adjoint functors

    Adjoint_functors

  • Diagonal intersection
  • P(κ)/INS a κ+-complete Boolean algebra, when equipped with diagonal intersections. Club set Fodor's lemma Thomas Jech, Set Theory, The Third Millennium Edition

    Diagonal intersection

    Diagonal_intersection

  • Commutative diagram
  • Collection of maps which give the same result

    include those typically given for the five lemma, the snake lemma, the zig-zag lemma, and the nine lemma. In higher category theory, one considers not

    Commutative diagram

    Commutative diagram

    Commutative_diagram

  • Mostowski collapse lemma
  • Result in mathematics and set theory

    In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by Andrzej

    Mostowski collapse lemma

    Mostowski_collapse_lemma

  • Algebraic stack
  • Generalization of algebraic spaces or schemes

    groupoids "Lemma 92.10.11 (045G)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29. "Section 78.5 (046I): Bootstrapping the diagonal—The Stacks

    Algebraic stack

    Algebraic_stack

  • Ken Brown's lemma
  • Mathematical concept in homotopy theory

    In mathematics, specifically in homotopy theory, Ken Brown's lemma gives a sufficient condition for a functor on a category of fibrant objects to preserve

    Ken Brown's lemma

    Ken_Brown's_lemma

  • Catalan number
  • Recursive integer sequence

    {2n+1}{n}}\,,} which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurrence relations C 0 = 1

    Catalan number

    Catalan number

    Catalan_number

  • Operator theory
  • Mathematical study of linear operators

    operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward

    Operator theory

    Operator_theory

  • Natural transformation
  • Central object of study in category theory

    completely known and easy to describe; this is the content of the Yoneda lemma. Saunders Mac Lane, one of the founders of category theory, is said to have

    Natural transformation

    Natural_transformation

  • Eigenvalues and eigenvectors
  • Concepts from linear algebra

    entries only along the main diagonal are called diagonal matrices. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider

    Eigenvalues and eigenvectors

    Eigenvalues_and_eigenvectors

  • Hawkins–Simon condition
  • Result in mathematical economics on existence of a non-negative equilibrium output vector

    Kotelyanskiĭ, as it is referred to by Felix Gantmacher as Kotelyanskiĭ lemma. Diagonally dominant matrix Perron–Frobenius theorem Sylvester's criterion Hawkins

    Hawkins–Simon condition

    Hawkins–Simon_condition

  • Glossary of logic
  • alternative logical systems that deviate from classical logic. diagonalization lemma A lemma used in the proof of Gödel's incompleteness theorems, stating

    Glossary of logic

    Glossary_of_logic

  • Isomorphism
  • In mathematics, invertible homomorphism

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Isomorphism

    Isomorphism

    Isomorphism

  • Abelian category
  • Category with direct sums and certain types of kernels and cokernels

    stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions

    Abelian category

    Abelian_category

  • Reverse mathematics
  • Branch of mathematical logic

    weak weak Kőnig's lemma if and only if for every set X there is a set Y that is 1-random relative to X. DNR (short for "diagonally non-recursive") adds

    Reverse mathematics

    Reverse_mathematics

  • Jacobi's formula
  • Formula for the derivative of a matrix determinant

    (A)\,dA)_{jj}=\operatorname {tr} (\operatorname {adj} (A)\,dA).\ \square } Lemma 1. det ′ ( I ) = t r {\displaystyle \det '(I)=\mathrm {tr} } , where det

    Jacobi's formula

    Jacobi's_formula

  • Pullback (category theory)
  • Most general completion of a commutative square given two morphisms with same codomain

    case E1 × E2 is a fiber bundle over B × B, and pulling back along the diagonal map B → B × B gives a space homeomorphic (diffeomorphic) to E1 ×B E2, which

    Pullback (category theory)

    Pullback_(category_theory)

  • Functor
  • Mapping between categories

    morphism to itself. The identity functor is an endofunctor. Diagonal functor The diagonal functor is defined as the functor from D {\displaystyle D} to

    Functor

    Functor

  • Topos
  • Mathematical category

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Topos

    Topos

  • Schur–Horn theorem
  • Characterizes the diagonal of a Hermitian matrix with given eigenvalues

    Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations

    Schur–Horn theorem

    Schur–Horn_theorem

  • Diagonal functor
  • In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}

    Diagonal functor

    Diagonal_functor

  • Gibbard–Satterthwaite theorem
  • Impossibility result for ranked-choice voting systems

    lemma statement, and let y=Best(Pz,Menu2(x)). Assume by contradiction that Px prefers z to y. By Lemma 1, f(Px,Pz)=Best(Pz, Menu2(x)) = y. By Lemma 3

    Gibbard–Satterthwaite theorem

    Gibbard–Satterthwaite_theorem

  • Category (mathematics)
  • Collection of objects and morphisms

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Category (mathematics)

    Category (mathematics)

    Category_(mathematics)

  • Morphism
  • Map (arrow) between two objects of a category

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Morphism

    Morphism

  • Rasiowa–Sikorski lemma
  • Mathematical lemma

    In axiomatic set theory, the Rasiowa–Sikorski lemma named after Helena Rasiowa and Roman Sikorski is one of the most fundamental facts used in the technique

    Rasiowa–Sikorski lemma

    Rasiowa–Sikorski_lemma

  • Condensation lemma
  • Lemma in constructibility theory

    In set theory, a branch of mathematics, the condensation lemma is a result about sets in the constructible universe. It states that if X is a transitive

    Condensation lemma

    Condensation_lemma

  • Ramsey's theorem
  • Statement in mathematical combinatorics

    either case the proof is complete. Lemma 1 implies that any R(r,s) is finite. The right hand side of the inequality in Lemma 2 expresses a Ramsey number for

    Ramsey's theorem

    Ramsey's_theorem

  • Inverse limit
  • Construction in category theory

    non-empty finite sets is non-empty. This is a generalization of Kőnig's lemma in graph theory and may be proved with Tychonoff's theorem, viewing the

    Inverse limit

    Inverse_limit

  • Pushout (category theory)
  • Most general completion of a commutative square given two morphisms with same domain

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Pushout (category theory)

    Pushout_(category_theory)

  • List of mathematical proofs
  • simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's lemma Kőnig's theorem (set theory) Kőnig's theorem (graph theory) Lagrange's

    List of mathematical proofs

    List_of_mathematical_proofs

  • Lift (mathematics)
  • forms Arithmetic geometry: Andrew Wiles (1995) modularity lifting Hensel's lemma Monad (functional programming) uses map functional to lift simple operators

    Lift (mathematics)

    Lift_(mathematics)

  • Polar decomposition
  • Type of matrix representation

    The existence of a polar decomposition is a consequence of Douglas' lemma: Lemma—If A, B are bounded operators on a Hilbert space H, and A*A ≤ B*B, then

    Polar decomposition

    Polar_decomposition

  • Category theory
  • General theory of mathematical structures

    and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes

    Category theory

    Category theory

    Category_theory

  • Monomorphism
  • Injective homomorphism

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Monomorphism

    Monomorphism

    Monomorphism

  • Direct limit
  • Special case of colimit in category theory

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Direct limit

    Direct_limit

  • Moschovakis coding lemma
  • The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible

    Moschovakis coding lemma

    Moschovakis_coding_lemma

  • Perron–Frobenius theorem
  • Theorem in linear algebra

    B), then B = eiφ D AD−1 for some diagonal unitary matrix D (i.e. diagonal elements of D equals to eiΘl, non-diagonal are zero). If some power Aq is reducible

    Perron–Frobenius theorem

    Perron–Frobenius_theorem

  • Limit (category theory)
  • Mathematical concept

    may be thought of as the category of all diagrams of shape J in C. The diagonal functor Δ : C → C J {\displaystyle \Delta :{\mathcal {C}}\to {\mathcal

    Limit (category theory)

    Limit_(category_theory)

  • Monoidal category
  • Category admitting tensor products

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Monoidal category

    Monoidal_category

  • Set (mathematics)
  • Collection of mathematical objects

    Other equivalent forms are described in the following subsections. Zorn's lemma is an assertion that is equivalent to the axiom of choice under the other

    Set (mathematics)

    Set (mathematics)

    Set_(mathematics)

  • Full and faithful functors
  • Functors which are surjective and injective on hom-sets

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Full and faithful functors

    Full_and_faithful_functors

  • Cartesian closed category
  • Type of category in category theory

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Cartesian closed category

    Cartesian_closed_category

  • Bertrand's ballot theorem
  • Election result probability theorem

    Bertrand's ballot theorem is related to the cycle lemma. They give similar formulas, but the cycle lemma considers circular shifts of a given ballot counting

    Bertrand's ballot theorem

    Bertrand's_ballot_theorem

  • Applied category theory
  • Applications of category theory

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Applied category theory

    Applied_category_theory

  • Spectral theorem
  • Result about when a matrix can be diagonalized

    result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because

    Spectral theorem

    Spectral_theorem

  • Initial and terminal objects
  • Special objects used in (mathematical) category theory

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Initial and terminal objects

    Initial_and_terminal_objects

  • Higher category theory
  • Generalization of category theory

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Higher category theory

    Higher_category_theory

  • Outline of category theory
  • Overview of and topical guide to category theory

    Abelian category Exact sequence Exact functor Snake lemma Nine lemma Five lemma Short five lemma Mitchell's embedding theorem Injective cogenerator Derived

    Outline of category theory

    Outline_of_category_theory

  • Woodbury matrix identity
  • Theorem of matrix ranks

    original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the

    Woodbury matrix identity

    Woodbury_matrix_identity

  • Additive category
  • Type of category in category theory

    projection morphisms, and ik will denote the injection morphisms. The diagonal morphism is the canonical morphism ∆: A → A ⊕ A, induced by the universal

    Additive category

    Additive_category

  • Glossary of category theory
  • f^{*}} and is denoted by f ∗ {\displaystyle f_{*}} . Ken Brown's lemma Ken Brown's lemma gives a sufficient condition for a functor to preserve weak equivalences

    Glossary of category theory

    Glossary_of_category_theory

  • Kan extension
  • Category theory constructs

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Kan extension

    Kan_extension

  • Turing's proof
  • Proof by Alan Turing

    requires the use of formal logic to prove a first lemma, followed by a brief word-proof of the second: Lemma 1: If S1 [symbol "0"] appears on the tape in some

    Turing's proof

    Turing's_proof

  • Equivalence of categories
  • Abstract mathematics relationship

    Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels

    Equivalence of categories

    Equivalence_of_categories

  • Cyclic quadrilateral
  • Quadrilateral whose vertices lie on a circle

    which E divides one diagonal equals that of the other diagonal. This is known as the intersecting chords theorem since the diagonals of the cyclic quadrilateral

    Cyclic quadrilateral

    Cyclic quadrilateral

    Cyclic_quadrilateral

  • Epimorphism
  • Surjective homomorphism

    → Y is not surjective, let y ∈ Y − fX. Since fX is closed, by Urysohn's Lemma there is a continuous function g1 : Y → [0, 1] such that g1 is 0 on fX and

    Epimorphism

    Epimorphism

  • Opposite category
  • Mathematical category formed by reversing morphisms

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Opposite category

    Opposite_category

  • Universal property
  • Characterizing property of mathematical constructions

    × C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}} and define the diagonal functor Δ : C → C × C {\displaystyle \Delta :{\mathcal {C}}\to {\mathcal

    Universal property

    Universal property

    Universal_property

  • Subcategory
  • Category whose objects and morphisms are inside a bigger category

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Subcategory

    Subcategory

  • Exact sequence
  • Sequence of homomorphisms such that each kernel equals the preceding image

    snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case. The five lemma gives

    Exact sequence

    Exact sequence

    Exact_sequence

  • Elementary topos
  • diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Elementary topos

    Elementary_topos

  • Busemann function
  • of Γ[a,b]. The generalisation of Morse's lemma to CAT(-1) spaces is often referred to as the Morse–Mostow lemma and can be proved by a straightforward generalisation

    Busemann function

    Busemann_function

  • Kleisli category
  • Category theory

    diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful

    Kleisli category

    Kleisli_category

  • Symmetric matrix
  • Matrix equal to its transpose

    for every }}i,j,\quad a_{ji}=a_{ij}.} Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different

    Symmetric matrix

    Symmetric matrix

    Symmetric_matrix

  • 2-category
  • Generalization of category

    n-category Doctrine (mathematics) Pseudofunctor String diagram 2-Yoneda lemma Pasting theorem Ehresmann 1965 Bénabou 1967 Kelly & Street 1974, § 1.2.

    2-category

    2-category

  • Quasi-category
  • Generalization of a category

    but space-valued (for example, for a correct formulation of the Yoneda lemma). The homotopy hypothesis says that one can take an ∞-groupoid, concretely

    Quasi-category

    Quasi-category

  • Diagram (category theory)
  • Indexed collection of objects and morphisms in a category

    each diagram to its colimit. The universal functor of a diagram is the diagonal functor; its right adjoint is the limit and its left adjoint is the colimit

    Diagram (category theory)

    Diagram_(category_theory)

  • Product (category theory)
  • Generalized object in category theory

    category C × C . {\displaystyle \mathbf {C} \times \mathbf {C} .} The diagonal functor Δ : C → C × C {\displaystyle \Delta :\mathbf {C} \to \mathbf {C}

    Product (category theory)

    Product_(category_theory)

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