nLab
circle object

Contents

Context

2-Category Theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The semantics of the circle type in (2,1)-category theory.

Definition

In a weak (2,1)-category CC with terminal object **, a circle object is an object S 1S^1 in CC with a global element β:*S 1\beta:* \to S^1 and an equivalence λ:ββ\lambda:\beta \cong \beta such that for every other object AA in CC with a global element β A:*A\beta_A:* \to A and an equivalence λ A:β Aβ A\lambda_A:\beta_A \cong \beta_A, there is a functor f:S 1Af:S^1 \cong A and a functor f :(ββ)(β Aβ A)f^{'}:(\beta \cong \beta) \to (\beta_A \cong \beta_A) and equivalences p:fββ Ap:f \circ \beta \cong \beta_A and q:f λλ Af q:f^{'} \circ \lambda \cong \lambda_A \circ f^{'} satisfying coherence laws.

Properties

The loop space object of the object S 1S^1 with global element β\beta the circle object is equivalent to an integers object ZZ.

See also

Created on May 14, 2022 at 00:30:53. See the history of this page for a list of all contributions to it.