Daftar istilah teori kategori

Glosarium ini merupakan daftar sifat-sifat dan konsep-konsep di dalam teori kategori dalam ilmu matematika.

Artikel ini menggunakan notasi dan mengikuti konvensi berikut:

[ n ] = {0, 1, 2, …, n }. Persamaan ini dilihat sebagai sebuah kategori dengan menuliskan $i\to j\Leftrightarrow i\leq j$ .
Cat (dari bahasa Inggris: category), kategori dari kategori-kategori (kecil), objek-objeknya merupakan beberapa kategori (yang kecil terhadap suatu semesta) dan fungtor morfismenya.
Fct ( C, D ), kategori fungtor: kategori fungtor dari kategori C ke kategori D .
Set, kategori himpunan (kecil).
s Set, kategori himpunan simplisial.
Jika salah satu dari kata "lemah" dan kata "ketat" dapat menyifati sebuah istilah, maka yang dimaksud adalah yang "lemah"; misalnya "kategori-n" berarti "kategori-n yang lemah" dan bukan yang ketat.
Menyebutkan kategori ∞ secara umum bermaksud secara khusus membicarakan tentang kuasi-kategori — model kategori ∞ yang paling populer.
Angka nol 0 adalah bilangan asli.

A

abelian

Sebuah kategori dikatakan abelian jika memiliki nol obyek, memiliki pullbacks and pushouts, serta semua monomorphisms dan epimorphisms normal.

accessible

1. Given a cardinal number κ, an object X in a category is κ-accessible (or κ-compact or κ-presentable) if

\operatorname {Hom} (X,-)

commutes with κ-filtered colimits.

2. Given a regular cardinal κ, a category is κ-accessible if it has κ-filtered colimits and there exists a small set S of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in S.

additive

A category is additive if it is preadditive (to be precise, has some pre-additive structure) and admits all finite coproducts. Although "preadditive" is an additional structure, one can show "additive" is a property of a category; i.e., one can ask whether a given category is additive or not.^[1]

adjunction

An adjunction (also called an adjoint pair) is a pair of functors F: C → D, G: D → C such that there is a "natural" bijection

\operatorname {Hom} _{D}(F(X),Y)\simeq \operatorname {Hom} _{C}(X,G(Y))

;

F is said to be left adjoint to G and G to right adjoint to F. Here, "natural" means there is a natural isomorphism

\operatorname {Hom} _{D}(F(-),-)\simeq \operatorname {Hom} _{C}(-,G(-))

of bifunctors (which are contravariant in the first variable.)

algebra for a monad

Given a monad T in a category X, an algebra for T or a T-algebra is an object in X with a monoid action of T ("algebra" is misleading and "T-object" is perhaps a better term.) For example, given a group G that determines a monad T in Set in the standard way, a T-algebra is a set with an action of G.

amnestic

A functor is amnestic if it has the property: if k is an isomorphism and F(k) is an identity, then k is an identity.

B

seimbang: Sebuah kategori dikatakan seimbang jika setiap bimorphism-nya (mono and juga epi) isomorphism.
Beck's theorem: Beck's theorem characterizes the category of algebras for a given monad.
bicategory: A bicategory is a model of a weak 2-category.
bifunctor: A bifunctor from a pair of categories C and D to a category E is a functor C × D → E. For example, for any category C, $\operatorname {Hom} (-,-)$ is a bifunctor from C^op and C to Set.
bimonoidal: A bimonoidal category is a category with two monoidal structures, one distributing over the other.
bimorphism: A bimorphism is a morphism that is both an epimorphism and a monomorphism.
Bousfield localization: See Bousfield localization.

^ Remark 2.7. of https://ncatlab.org/nlab/show/additive+category

[1] Remark 2.7. of https://ncatlab.org/nlab/show/additive+category

[1]