In mathematics, especially

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, the core of a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

''C'' is the category whose objects are the objects of ''C'' and whose morphisms are the invertible morphisms in ''C''.Pierre Gabriel, Michel Zisman, § 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967

/ref> In other words, it is the largest

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...

subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...

. As a

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

C \mapsto \operatorname(C)

, the core is a

right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

to the inclusion of the category of (small) groupoids into the category of (small) categories. On the other hand, the left adjoint to the above inclusion is the fundamental groupoid functor. For ∞-categories,

\operatorname

is defined as a right adjoint to the inclusion ∞-Grpd

\hookrightarrow

∞-Cat. The core of an ∞-category

C

is then the largest

∞-groupoid In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category (mathematics), category of simplicial sets (with the standa ...

contained in

C

. The core of ''C'' is also often written as

C^

. The left adjoint to the above inclusion is given by a

localization of an ∞-category In mathematics, specifically in higher category theory, a localization of an ∞-category is an ∞-category obtained by inverting some maps. An ∞-category is a presentable ∞-category if it is a localization of an ∞-presheaf category in the ...

. In ''Kerodon'', the subcategory of a

2-category In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation between functors. ...

''C'' obtained by removing non-invertible morphisms is called the pith of ''C''. It can also be defined for an (∞, 2)-category ''C'';https://kerodon.net/tag/01XA namely, the pith of ''C'' is the largest simplicial subset that does not contain non-thin 2-simplexes.

References

Further reading