# 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:

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

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:

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

## References

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