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数学

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Andrius Kulikauskas

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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

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Software

Book.Topos istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2019 balandžio 16 d., 12:40 atliko AndriusKulikauskas -
Pridėtos 72-74 eilutės:

Readings
* [[http://www.math.harvard.edu/~lurie/papers/HTT.pdf | Higher Topos Theory]] Jacob Lurie
2019 balandžio 06 d., 11:27 atliko AndriusKulikauskas -
Pridėta 75 eilutė:
* [[https://www.youtube.com/watch?v=vmcbm5FxRJE | Colin McLarty: Nonetheless one should learn the language of topos]]
2018 rugsėjo 30 d., 20:59 atliko AndriusKulikauskas -
Ištrinta 71 eilutė:
* [[https://en.wikipedia.org/wiki/Yoneda_lemma | Wikipedia: Yoneda lemma]]
2018 rugsėjo 30 d., 20:57 atliko AndriusKulikauskas -
Ištrintos 45-71 eilutės:

%center%Attach:YonedaLemma.png

'''Yoneda Lemma'''

Key points: Every object has a morphism to itself. Every set function has at least one element in its range.

What: Set of properties. Why: Set of relationships as dictaded by its relationships but especially its relationship with itself (its essence).

What is left unspoken: Sets are labelled - elements are labels. Categories are unlabelled only structure. Is this related to model theory?

The properties of an entity correspond to the analogous stances: "This is the essence of the entity, the property that makes it what it is."

Hom(A,–) gives the perspectives from A, including perspectives on perspectives.

If F is the empty functor, or if F(A) is empty, then there are no natural transformations.

F(A) is a set of labels.

* The Yoneda Lemma asserts that {$C^{op}$} embeds in {${\textbf{Set}}^C$} as a full subcategory.
* The functor category {$D^C$} has as objects the functors from ''C'' to ''D'' and as morphisms the natural transformations of such functors. The Yoneda lemma describes representable functors in functor categories.

Let F be an arbitrary functor from C to Set. Then Yoneda's lemma says that: For each object A of C, the natural transformations from {$h^A$} to F are in one-to-one correspondence with the elements of F(A). That is,

{${Nat} (h^{A},F)\cong F(A)$}

Moreover this isomorphism is natural in A and F when both sides are regarded as functors from SetC x C to Set.
2018 rugsėjo 30 d., 20:56 atliko AndriusKulikauskas -
Pakeistos 1-2 eilutės iš
See: [[Category theory]]
į:
See: [[Category theory]], [[Yoneda Lemma]]
Ištrintos 3-4 eilutės:

* Explain how substitution of eigenvalues into symmetric functions can add information. And relate this to the Yoneda lemma where associativity adds information when we go from objects (node-variables) to morphisms (matrix edges).
2018 birželio 22 d., 12:17 atliko AndriusKulikauskas -
Pridėtos 32-33 eilutės:

"Sketches of an Elephant: A Topos Theory Compendium 2 Volume Set"
2018 birželio 22 d., 12:07 atliko AndriusKulikauskas -
Pridėtos 30-31 eilutės:

Combinatorics - create objects "for children" - then count them - the formulas appear in other contexts - gives clues for related toposes.
2018 birželio 22 d., 11:20 atliko AndriusKulikauskas -
Pridėtos 28-29 eilutės:

Sheaves are not interesting but rather, what we see from our external point of view, and what needs to be removed to see the interesting internal point of view.
2018 birželio 22 d., 10:58 atliko AndriusKulikauskas -
Pridėtos 26-27 eilutės:

Can define objects and math before logic.
2018 birželio 22 d., 10:49 atliko AndriusKulikauskas -
Pridėtos 24-25 eilutės:

Example: internal view (of complicated curves) are embedded in an external space (and the latter is removed in the topos).
2018 birželio 22 d., 10:46 atliko AndriusKulikauskas -
Pridėtos 22-23 eilutės:

Double negation in the simpler, topos world, almost holds - but when you translate into the richer world - you get a highly nontrivial statement.
2018 birželio 20 d., 19:15 atliko AndriusKulikauskas -
Pridėtos 20-21 eilutės:

Morita equivalence.
2018 birželio 20 d., 18:21 atliko AndriusKulikauskas -
Pakeista 19 eilutė iš:
The logic of geometry is based on local coherence. And the global consequences...?
į:
The logic of geometry is based on local coherence. And the global consequences...? are topology?
2018 birželio 20 d., 18:21 atliko AndriusKulikauskas -
Pridėtos 18-19 eilutės:

The logic of geometry is based on local coherence. And the global consequences...?
2018 birželio 20 d., 17:21 atliko AndriusKulikauskas -
Pakeistos 15-17 eilutės iš
Pullback (and limits) are the basis for making distincions.
į:
Pullback (and limits) are the basis for making distinctions.

What are the adjoints for the push forward?
2018 birželio 20 d., 17:20 atliko AndriusKulikauskas -
Pridėtos 14-15 eilutės:

Pullback (and limits) are the basis for making distincions.
2018 birželio 20 d., 17:18 atliko AndriusKulikauskas -
Pridėtos 12-13 eilutės:

Pullback is crucial because its adjoints are exists and for all.
2018 birželio 20 d., 16:25 atliko AndriusKulikauskas -
Pridėtos 10-11 eilutės:

Presheaf category is the total information. Topos is a weakened, limited set of information.
2018 birželio 20 d., 16:17 atliko AndriusKulikauskas -
Pridėtos 8-9 eilutės:

A Grothendieck topos is a reflective subcategory of a presheaf category such that the reflection functor preserves finite limits.
2018 gegužės 28 d., 13:52 atliko AndriusKulikauskas -
Pakeista 78 eilutė iš:
* Category Theory: Topos Logic by Dr.Marni Sheppeard
į:
* [[https://www.youtube.com/watch?v=XRIJpFKijwM | Category Theory: Topos Logic by Dr.Marni Sheppeard]]
2018 gegužės 28 d., 11:38 atliko AndriusKulikauskas -
Pakeistos 77-78 eilutės iš
į:
Videos
* Category Theory: Topos Logic by Dr.Marni Sheppeard
2018 balandžio 15 d., 12:08 atliko AndriusKulikauskas -
Pridėtos 28-31 eilutės:

What: Set of properties. Why: Set of relationships as dictaded by its relationships but especially its relationship with itself (its essence).

What is left unspoken: Sets are labelled - elements are labels. Categories are unlabelled only structure. Is this related to model theory?
2018 balandžio 15 d., 12:06 atliko AndriusKulikauskas -
Pridėtos 26-27 eilutės:

Key points: Every object has a morphism to itself. Every set function has at least one element in its range.
2018 balandžio 13 d., 11:17 atliko AndriusKulikauskas -
Pridėtos 32-33 eilutės:

F(A) is a set of labels.
2018 balandžio 13 d., 11:06 atliko AndriusKulikauskas -
Pridėtos 30-31 eilutės:

If F is the empty functor, or if F(A) is empty, then there are no natural transformations.
2018 balandžio 13 d., 10:35 atliko AndriusKulikauskas -
Pridėtos 7-9 eilutės:
>><<

-----------
Pridėta 11 eilutė:
-----------
2018 balandžio 13 d., 10:35 atliko AndriusKulikauskas -
Pridėtos 2-5 eilutės:

>>bgcolor=#FFFFC0<<

* Explain how substitution of eigenvalues into symmetric functions can add information. And relate this to the Yoneda lemma where associativity adds information when we go from objects (node-variables) to morphisms (matrix edges).
2018 balandžio 13 d., 10:24 atliko AndriusKulikauskas -
Pridėtos 20-21 eilutės:

Hom(A,–) gives the perspectives from A, including perspectives on perspectives.
2018 balandžio 12 d., 11:33 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Attach:topos.png
2018 balandžio 11 d., 13:12 atliko AndriusKulikauskas -
Pakeista 13 eilutė iš:
Attach:YonedaLemma.png
į:
%center%Attach:YonedaLemma.png
2018 balandžio 11 d., 13:12 atliko AndriusKulikauskas -
Pridėtos 16-17 eilutės:

The properties of an entity correspond to the analogous stances: "This is the essence of the entity, the property that makes it what it is."
2018 balandžio 10 d., 09:08 atliko AndriusKulikauskas -
Pakeista 13 eilutė iš:
į:
Attach:YonedaLemma.png
2018 balandžio 09 d., 09:22 atliko AndriusKulikauskas -
Pakeistos 15-16 eilutės iš
Yoneda Lemma
į:
'''Yoneda Lemma'''
Pakeista 48 eilutė iš:
* [[https://en.wikipedia.org/wiki/History_of_topos_theory | History of topos theory]]
į:
* [[https://en.wikipedia.org/wiki/History_of_topos_theory | Wikipedia: History of topos theory]]
Pakeistos 50-51 eilutės iš
* [[https://en.wikipedia.org/wiki/Category_theory | Category theory]]
į:
* [[https://en.wikipedia.org/wiki/Category_theory | Wikipedia: Category theory]]
* [[https://en.wikipedia.org/wiki/Yoneda_lemma | Wikipedia: Yoneda lemma
]]
2018 balandžio 08 d., 13:41 atliko AndriusKulikauskas -
Pakeista 19 eilutė iš:
Let F be an arbitrary functor from C to Set. Then Yoneda's lemma says that: For each object A of C, the natural transformations from hA to F are in one-to-one correspondence with the elements of F(A). That is,
į:
Let F be an arbitrary functor from C to Set. Then Yoneda's lemma says that: For each object A of C, the natural transformations from {$h^A$} to F are in one-to-one correspondence with the elements of F(A). That is,
2018 balandžio 08 d., 13:40 atliko AndriusKulikauskas -
Pakeista 21 eilutė iš:
{${Nat} (h^{A},F)\cong F(A). \mathrm {Nat} (h^{A},F)\cong F(A)$}
į:
{${Nat} (h^{A},F)\cong F(A)$}
2018 balandžio 08 d., 13:40 atliko AndriusKulikauskas -
Pakeista 21 eilutė iš:
{${Nat} (h^{A},F)\cong F(A).} {\mathrm {Nat}}(h^{A},F)\cong F(A)$}
į:
{${Nat} (h^{A},F)\cong F(A). \mathrm {Nat} (h^{A},F)\cong F(A)$}
2018 balandžio 08 d., 13:39 atliko AndriusKulikauskas -
Pridėtos 18-23 eilutės:

Let F be an arbitrary functor from C to Set. Then Yoneda's lemma says that: For each object A of C, the natural transformations from hA to F are in one-to-one correspondence with the elements of F(A). That is,

{${Nat} (h^{A},F)\cong F(A).} {\mathrm {Nat}}(h^{A},F)\cong F(A)$}

Moreover this isomorphism is natural in A and F when both sides are regarded as functors from SetC x C to Set.
2018 balandžio 08 d., 13:09 atliko AndriusKulikauskas -
Pakeista 35 eilutė iš:
{$R → X × _{X / R}X {\displaystyle R\to X\times _{X/R}X\,\!} R \to X \times_{X/R} X \,\!$}
į:
{$R → X × _{X / R}X$}
2018 balandžio 08 d., 13:09 atliko AndriusKulikauskas -
Pakeista 35 eilutė iš:
{$R → X × X / R X {\displaystyle R\to X\times _{X/R}X\,\!} R \to X \times_{X/R} X \,\!$}
į:
{$R → X × _{X / R}X {\displaystyle R\to X\times _{X/R}X\,\!} R \to X \times_{X/R} X \,\!$}
2018 balandžio 08 d., 13:08 atliko AndriusKulikauskas -
Pridėtos 9-14 eilutės:
A topos is a category that has the following two properties:
* All limits taken over finite index categories exist.
* Every object has a power object.

Pridėtos 18-37 eilutės:

'''Grothendieck topos'''

Let C be a category. A theorem of Giraud states that the following are equivalent:
* There is a small category D and an inclusion C ↪ Presh(D) that admits a finite-limit-preserving left adjoint.
* C is the category of sheaves on a Grothendieck site.
* C satisfies Giraud's axioms, below.

A category with these properties is called a "(Grothendieck) topos". Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf.

Giraud's axioms for a category C are:
* C has a small set of generators, and admits all small colimits. Furthermore, colimits commute with fiber products.
* Sums in C are disjoint. In other words, the fiber product of X and Y over their sum is the initial object in C.
* All equivalence relations in C are effective.

The last axiom needs the most explanation. If X is an object of C, an "equivalence relation" R on X is a map R→X×X in C such that for any object Y in C, the induced map Hom(Y,R)→Hom(Y,X)×Hom(Y,X) gives an ordinary equivalence relation on the set Hom(Y,X). Since C has colimits we may form the coequalizer of the two maps R→X; call this X/R. The equivalence relation is "effective" if the canonical map

{$R → X × X / R X {\displaystyle R\to X\times _{X/R}X\,\!} R \to X \times_{X/R} X \,\!$}

is an isomorphism.
2018 balandžio 08 d., 12:50 atliko AndriusKulikauskas -
Pridėta 11 eilutė:
* The functor category {$D^C$} has as objects the functors from ''C'' to ''D'' and as morphisms the natural transformations of such functors. The Yoneda lemma describes representable functors in functor categories.
2018 balandžio 08 d., 11:33 atliko AndriusKulikauskas -
Pridėta 16 eilutė:
* [[https://en.wikipedia.org/wiki/Category_theory | Category theory]]
2018 balandžio 08 d., 11:33 atliko AndriusKulikauskas -
Pridėta 15 eilutė:
* [[https://en.wikipedia.org/wiki/Glossary_of_category_theory | Wikipedia: Glossary of category theory]]
2018 balandžio 08 d., 11:32 atliko AndriusKulikauskas -
Pakeista 10 eilutė iš:
* The Yoneda Lemma asserts that {$C^{op}$} embeds in {${\textbf{Set}}^C$}SetC as a full subcategory.
į:
* The Yoneda Lemma asserts that {$C^{op}$} embeds in {${\textbf{Set}}^C$} as a full subcategory.
2018 balandžio 08 d., 11:32 atliko AndriusKulikauskas -
Pridėtos 8-10 eilutės:

Yoneda Lemma
* The Yoneda Lemma asserts that {$C^{op}$} embeds in {${\textbf{Set}}^C$}SetC as a full subcategory.
2018 balandžio 08 d., 11:14 atliko AndriusKulikauskas -
Pridėtos 1-13 eilutės:
See: [[Category theory]]

Topos:
* Category that behaves like the category of sheaves of sets on a site (topological space).
* Behaves much like the category of sets in that they possess a notion of localization.
* A generalization of point-set topology.
* Grothendieck topoi used in algebraic geometry. More general topoi used in logic.

Sources
* [[https://en.wikipedia.org/wiki/Topos | Wikipedia: Topos]]
* [[https://en.wikipedia.org/wiki/History_of_topos_theory | History of topos theory]]

Topos


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Puslapis paskutinį kartą pakeistas 2019 balandžio 16 d., 12:40
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