nLab semi-simplicial set

Idea

A semi-simplicial set is like a simplicial set, but without the degeneracy maps: it is a sequence $\{X_n\}_{n \in \mathbb{N}}$ of sets together with functions called face maps between them which encode that an element in $X_{n+1}$ has $(n+1)$ “faces” (boundary segments) which are elements in $X_{n}$.

The semi-simplicial set version of a simplicial complex is also called a Delta set.

Definition

Let $\Delta$ denote the simplex category, which is a skeleton of the category of inhabited finite totally ordered sets. Let $\Delta_+$ denote the wide subcategory of $\Delta$ containing only the injective functions (it is sometimes written $\Delta_{inj}$, $\Delta_i$, $\widehat{\Delta}$, $\Delta'$ or $\overline{\Delta}$). Thus, $\Delta_+$ is equivalent to the category of inhabited finite totally ordered sets and order-preserving injections.

Recall that a simplicial set is a presheaf $X\colon \Delta^{op}\to Set$. Similarly, a semi-simplicial set is a presheaf $X\colon \Delta_+^{op} \to Set$.

More generally, for $\mathcal{C}$ any other category, a functor $\Delta_+^{op} \to \mathcal{C}$ is a semi-simplicial object in $\mathcal{C}$.

Properties

Adjunction with simplicial sets

The forgetful functor from SimplicialSets to the category of semi-simplicial sets is given by precomposition with the opposite functor of the non-full wide subcategory inclusion

$$\Delta_{inj} \xrightarrow{\;j\;} \Delta$$

into the simplex category and hence has both a left adjoint as well as a right adjoint given by left/right Kan extension, respectively:

The adjunction unit of the left adjoint pair

$$X \xrightarrow{ \;\eta_X\; } j^\ast j_! X$$

is a weak homotopy equivalence in the sense that its geometric realization is so (Rourke & Sanderson 71, Rem. 5.8).

Notice that

1. for $X \in$ SimplicialSets, the geometric realization of the underlying semi-simplicial set $j^\ast X$ is the fat geometric realization of $X$ (see there):

   $$\left\Vert X \right\Vert \;\coloneqq\; \left\vert j^\ast X \right \vert$$

2. There is (by this Prop.) a natural weak homotopy equivalence from the fat to the ordinay geometric realization of a simplicial set (which is always “good” when regarded as a simplicial space):

   $$\left\vert j^\ast X \right \vert \;\simeq_{whe}\; \left\vert X \right \vert \,.$$

It follows (see also MO:a/75101) that also the adjunction counit

$$j_! j^\ast X \xrightarrow{\; \epsilon_X \;} X$$

is a simplicial weak equivalence.

Relation to semi-categories

The nerve of a semicategory is a semi-simplicial set (satisfying the Segal conditions) just as the nerve of a category is a simplicial set.

Model category structure

There is a model structure on semi-simplicial sets, transferred along the right adjoint to the forgetful functor from the model structure on simplicial sets.

Related concepts

• simplicial object

  • simplicial set

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

• semi-simplicial object

• semicategory

• semi-Segal space

Historical and terminological remarks

The original paper Eilenberg & Zilber 50 defined both (what we now call) semi-simplicial sets, under the name semi-simplicial complexes, and (what we now call) simplicial sets, under the name complete semi-simplicial complexes. The motivation for the name “semi-simplicial” was that a semi-simplicial set is like a simplicial complex, but lacks the property that a simplex is uniquely determined by its vertices. Then they added the degeneracies and a corresponding adjective “complete.”

Over time it became clear that “complete semi-simplicial complexes” were much more important and useful than the non-complete ones. This seems to have led first to the omission of the adjective “complete,” and then the omission of the prefix “semi” (and at some point the replacement of “complex” by “set”), resulting in the current name simplicial sets.

The concept is essentially the same as that of $\Delta$-set, as used by Rourke & Sanderson 71. Their motivation was from geometric topology.

On the other hand, in other contexts the prefix “semi-” is used to denote absence of identities (such as a semigroup (which is, admittedly, missing more than identities relative to a group) or a semicategory), thus if we start from the modern name “simplicial sets” it makes independent sense to refer to their degeneracy-less variant as “semi-simplicial sets.” This is coincidentally in line with the original terminology of Eilenberg and Zilber, but not of course with the intermediate usage of “semi-simplicial set” for what we now call a “simplicial set.”

Note also the existence of an alternative terminology “presimplicial sets”, or “pre-simplicial sets”, which can be traced back at least to the textbook “Cellular Structures in Topology” by Fritsch and Piccinini in 1990. This terminology is commonly used e.g. in the context of simplicial models for concurrent programs or higher-dimensional automata (see e.g. “First introduction to simplicial sets” by Sina Hazratpour).

Similarly, the subcategory of injective functions of the simplex category was written $\Delta$ at some time of the history (e.g. in Rourke & Sanderson 71) but this is now the standard notation for the simplex category. In the more recent history, different notations can be found but none seems to be widely adopted. $\Delta_+$ emphasizes that it is the subcategory of $\Delta$ that raise the degree when $\Delta$ is seen as a Reedy category but the $+$ may also ambiguously suggests that it adds something to $\Delta$. The notation $\Delta_{inj}$ emphasizes that it is the subcategory of injective morphisms of $\Delta$. Similarly for $\Delta_i$ though less explicitly. The notation $\widehat{\Delta}$ (e.g. in Friedman) has the risk of introducing a confusion for readers used with the hat notation for presheaves. The notation $\Delta'$ and $\overline{\Delta}$ (e.g. in Sina Hazratpour) express that it is a variant of $\Delta$ but without giving precisions.

Related concepts

• semi-simplicial object

• semi-simplicial topological space

References

• Samuel Eilenberg, Joseph Zilber, Semi-Simplicial Complexes and Singular Homology, Annals of Mathematics vol. 51 no. 3 (1950), 499–513 (jstor:1969364)

• Daniel Kan, Is an ss complex a css complex?, Advances in Mathematics

  Volume 4, Issue 2, April 1970, Pages 170-171 (doi:10.1016/0001-8708(70)90021-6)

• Colin Rourke, Brian Sanderson, Δ-Sets I: Homotopy Theory. The Quarterly Journal of Mathematics 22: 321–338 (1971) (doi:10.1093/qmath/22.3.321, pdf)

• S. Buoncristiano, Colin Rourke, Brian Sanderson, A geometric approach to Homology Theory, LMS Lect. Notes 18, (1976)

• Peter Hilton, On a generalization of nilpotency to semi-simplicial complexes (pdf)

• Alex Heller, Homotopy resolutions of semi-simplicial complexes, Transactions of the American Mathematical Society Vol. 80, No. 2 (Nov., 1955), pp. 299-344 (JSTOR)

• James McClure, On semisimplicial sets satisfying the Kan condition (arXiv:1210.5650).

• Greg Friedman, An elementary illustrated introduction to simplicial sets, Rocky Mountain J. Math. 42(2): 353-423 (2012) (arXiv:0809.4221).

• Andrew Ranicki, Algebraic L-Theory and Topological Manifolds, Cambridge University Press 2002.

See also the references at semi-simplicial object and:

• Wikipedia, Delta sets

• MO, Semi-simplicial versus simplicial sets (and simplicial categories) Degeneracies for semi-simplicial Kan complexes

In homotopy type theory:

• Thorsten Altenkirch, Paolo Capriotti, Nicolai Kraus, Extending Homotopy Type Theory with Strict Equality, CSL 2016, arXiv

On the model structure on semi-simplicial sets:

• Benno van den Berg, A note on semisimplicial sets, 2013 (pdf)

as a weak model category:

• Simon Henry, Theorem 5.5.6 of: Weak model categories in classical and constructive mathematics, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958. (arXiv:1807.02650, tac:35-24)

as a semi-model category:

• Jan Rooduijn, A right semimodel structure on semisimplicial sets, Amsterdam 2018 (pdf, mol:4787)

as a fibration category and cofibration category:

• Christian Sattler, Constructive homotopy theory of marked semisimplicial sets (arXiv:1809.11168)