Shahada ya kwanza (juu)Category TheoryImepangwa

Adjunctions

Adjoint functors F ⊣ G via natural bijections Hom(FA, B) ≅ Hom(A, GB), units and counits, free–forgetful adjunctions, Galois connections, currying and the tensor–hom adjunction, and the preservation of limits by right adjoints.

math.cat.adjunctions

Onyesha kila kinachohitajika kufikia mada hii

Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.

Inajengwa juu ya

Natural transformations Universal properties

Masharti ya awali ya lazima 19 (hadi chini kabisa)

Utajifunza

Verify adjunctions such as free monoid ⊣ forgetful, − × B ⊣ (−)^B in Set, and the tensor–hom adjunction.
Pass between the hom-set and the unit–counit definitions of an adjunction using the triangle identities.
Prove that right adjoints preserve limits and left adjoints preserve colimits, and use this to show that some functors have no adjoint.

Fasili

F : C → D is left adjoint to G : D → C, written F ⊣ G, if there are bijections Hom_D(FA, B) ≅ Hom_C(A, GB) natural in A and B.
The unit η : id_C ⇒ GF and counit ε : FG ⇒ id_D of an adjunction satisfy the triangle identities εF ∘ Fη = id_F and Gε ∘ ηG = id_G.
A Galois connection is an adjunction between posets regarded as categories.

Nadharia

A left (or right) adjoint of a functor is unique up to unique natural isomorphism.
Right adjoints preserve all limits that exist, and left adjoints preserve all colimits that exist.
Every adjunction F ⊣ G induces a monad GF.
In Set, − × B ⊣ (−)^B: functions A × B → C correspond to functions A → C^B (currying).

Mawazo potofu ya kawaida

Thinking every functor has an adjoint.
Mixing up left and right; in a free–forgetful adjunction the free functor is the left adjoint.
Assuming an adjunction is an equivalence; its unit and counit need not be isomorphisms.

Inaelekea kwa

Monads Categorical logic Kan extensions

Ugunduzi wa hiari

The Yoneda lemma A functor G : D → C has a left adjoint if and only if, for each object A of C, the functor Hom_C(A, G(−)) : D → Set is representable; the Yoneda lemma then assembles the representing objects into a functor.Njia ya Maajabu →
The λ-calculus Currying, the adjunction − × B ⊣ (−)^B, is the basis of function types in the typed λ-calculus.
← Fields and Galois theory
← Limits and colimits
← Modules

Njia za kujifunza

Graduate preparation

Usomaji zaidi

Saunders Mac Lane, Categories for the Working Mathematician (1971)
Emily Riehl, Category Theory in Context (2016)
Tom Leinster, Basic Category Theory (2014)