nLab semi-simplicial object

Idea

A semi-simplicial object is like a simplicial object, but without degeneracy maps. In Set it is a semi-simplicial set.

Definition

For $\mathcal{C}$ a category or (∞,1)-category, a semi-simplicial object $X$ in $\mathcal{C}$ is a functor or (∞,1)-functor

$$X \colon \Delta_+^{op} \to \mathcal{C}$$

from $\Delta_+$, the wide subcategory of the simplex category on the injective functions (the co-face maps).

Properties

Directedness

As opposed to the simplex category $\Delta$, the subcategory $\Delta_+$ is a direct category.

Examples

• in Sets: semisimplicial set:

• in TopologicalSpaces: semi-simplicial topological space

• in SimplicialSets: semi-Segal space (if extra conditions are met)

• in type theory: semi-simplicial types

Related concepts

• simplicial object

  • simplicial set

  • simplicial object in an (∞,1)-category

• semi-simplicial object

  • semi-simplicial set

  • semi-simplicial type

• semi-category

  • semi-Segal space

References

For more references see also at semi-simplicial set, semi-Segal space, and semi-simplicial type.

Semi-simplicial bundles

Discussion of semi-simplicial fiber bundles is in

• M. Barratt, V. Gugenheim and J. C. Moore, On semisimplicial fibre bundles, Amer. J. Math. 81 (1959), 639-657.

• S. Weingram, The realization of a semisimplicial bundle map is a $k$-bundle map (pdf)