nLab
minimal simplicial circle

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The smallest simplicial set whose homotopy type in the classical homotopy category is that of the circle has precisely two non-degenerate simplices, one in degree 0 – a single vertex – and one in degree 1 – a single edge which is a loop on that vertex:

SΔ[1]/Δ[1]. S \;\coloneqq\; \Delta[1]/\partial \Delta[1] \,.

Despite or maybe because its simplicity, the minimal simplicial circle plays a central role in many constructions, notably in the context of cyclic homology (e.g. Loday 1992, 7.1.2).

Properties

Example

The normalized chain complex of the free simplicial abelian group of the minimal simplicial circle SS has the group of integers in degrees 0 and 1, and all differentials are zero:

N (S)[ 0 0 ][1]. N_\bullet \circ \mathbb{Z}(S) \;\simeq\; \left[ \array{ \vdots \\ \big\downarrow \\ 0 \\ \big\downarrow \\ \mathbb{Z} \\ \big\downarrow {}^{\mathrlap{ 0 }} \\ \mathbb{Z} } \;\; \right] \;\simeq\; \mathbb{Z} \oplus \mathbb{Z}[1] \,.

References

Discussion in relation to the cyclic category and cyclic sets/cyclic spaces:

Created on July 12, 2021 at 14:24:10. See the history of this page for a list of all contributions to it.