Core of a category

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In mathematics, especially category theory, the core of a category C is the category whose objects are the objects of C and whose morphisms are the invertible morphisms in C.[1][2][3] In other words, it is the largest groupoid subcategory.

As a functor C ↦ core ⁡ ( C ) {\displaystyle C\mapsto \operatorname {core} (C)} {\displaystyle C\mapsto \operatorname {core} (C)}, the core is a right adjoint to the inclusion of the category of (small) groupoids into the category of (small) categories.[1] On the other hand, the left adjoint to the above inclusion is the fundamental groupoid functor.

For ∞-categories, core {\displaystyle \operatorname {core} } {\displaystyle \operatorname {core} } is defined as a right adjoint to the inclusion ∞-Grpd ↪ {\displaystyle \hookrightarrow } {\displaystyle \hookrightarrow } ∞-Cat.[4] The core of an ∞-category C {\displaystyle C} {\displaystyle C} is then the largest ∞-groupoid contained in C {\displaystyle C} {\displaystyle C}. The core of C is also often written as C ≃ {\displaystyle C^{\simeq }} {\displaystyle C^{\simeq }}. The left adjoint to the above inclusion is given by a localization of an ∞-category.

In Kerodon, the subcategory of a 2-category C obtained by removing non-invertible morphisms is called the pith of C.[5] It can also be defined for an (∞, 2)-category C;[6] namely, the pith of C is the largest simplicial subset that does not contain non-thin 2-simplexes.

References

  1. Pierre Gabriel, Michel Zisman, § 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967)
  2. "Construction 1.3.5.4". Kerodon.
  3. core groupoid at the nLab
  4. § 3.5.2. and Corollary 3.5.3. of Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
  5. "Construction 2.2.8.9 (The Pith of a 2-Category)". Kerodon.
  6. "5.4.5 The Pith of an (∞,2)-Category". Kerodon.

Further reading