nLab minimal simplicial circle

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:

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

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 $S$ has the group of integers in degrees 0 and 1, and all differentials are zero:

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

Related concepts

• circle, unit circle, circle type, circle group

• cyclic category

  cyclic object, cyclic set, cyclic space

  cyclic homology, cyclic cohomology

• inertia groupoid

References

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

• Jean-Louis Loday, Cyclic Spaces and $S^1$-Equivariant Homology (doi:10.1007/978-3-662-21739-9_7)

  Chapter 7 in: Cyclic Homology, Grundlehren 301, Springer 1992 (doi:10.1007/978-3-662-21739-9)