nLab semi-simplicial set

Semi-simplicial sets

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

Semi-simplicial sets

Idea

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

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

Definition

Let Δ\Delta denote the simplex category, which is a skeleton of the category of inhabited finite totally ordered sets. Let Δ +\Delta_+ denote the wide subcategory of Δ\Delta containing only the injective functions (it is sometimes written Δ inj\Delta_{inj}, Δ i\Delta_i, Δ^\widehat{\Delta}, Δ\Delta' or Δ¯\overline{\Delta}). Thus, Δ +\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:Δ opSetX\colon \Delta^{op}\to Set. Similarly, a semi-simplicial set is a presheaf X:Δ + opSetX\colon \Delta_+^{op} \to Set.

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

Δ injjΔ \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

Xη Xj *j !X 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 XX \in SimplicialSets, the geometric realization of the underlying semi-simplicial set j *Xj^\ast X is the fat geometric realization of XX (see there):

    X|j *X| \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):

    |j *X| whe|X|. \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 *Xϵ XX 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.

The lattice of subtoposes

Since the maps in Δ +\Delta_+ are all strictly monotone, any object [n][n] receives only (finitely many) morphisms from objects [m][m] with mnm\leq n whence all slices Δ +/[n]\Delta_+/[n] are finite. This is sufficient for all Grothendieck topologies on Δ +\Delta_+ to be rigid whence all subtoposes of Set Δ + opSet^{\Delta_+^{op}} are essential and of presheaf type and their lattice is isomorphic to the lattice of Cauchy-complete full subcategories of Δ +\Delta_+.

This situation is familiar from Δ\Delta but in the latter case there are fewer Cauchy-complete subcategories available, since an object [n][n] having non-trivial idempotents its inclusion automatically requires the presence of all [m][m] with mm\,<n\,n in the subcategory whence there are countably many subtoposes corresponding to the nn-truncated subcategories on the objects [0],,[n][0],\dots ,[n] (plus the two trivial subcategories).

Recall that an essential localization of a topos embeds the corresponding subtopos as a reflective subcategory (i.e. sheaves for the corresponding topology) as well as a coreflective subcategory. For a given (essential) topology jj we call the presheaves in the image of the latter inclusion jj-skeleta. Recall that the Aufhebung of such a level topology jj is defined as the largest topology j^\hat{j} such that all jj-skeleta are j^\hat{j}-sheaves (if such a j^\hat{j} exists). This differs somewhat from the definition given at Aufhebung but will suffice in the present context where all subtoposes are levels.

The Lawvere-Tierney topologies j wj_w on Set Δ + opSet^{\Delta_+^{op}} are indexed by infinite words ww over the alphabet {0,1}\{0,1\}. They are partially ordered by j wj zj_w\leq j_z iff w iz iw_i\leq z_i for all letter positions i=0,1,2,i=0,1,2,\dots, where (unsurprisingly) letter 0 counts as smaller than letter 1. The information contained in the ii-th letter of ww is whether the closure operator corresponding to j wj_w adds ii-dimensional semi-simplices to subobjects (w i=1w_i=1) or not (w i=0w_i=0) e.g. j 111j_{111\dots} adds all semi-simplices of every dimension implying that the empty subobject is j 111j_{111\dots}-dense in every object thereby preventing all non-terminal objects from being j 111j_{111\dots}-sheaves i.e. j 111=j maxj_{111\dots}=j_max while j 000j_{000\dots} where no semi-simplex of any dimension is added and every presheaf is a j 000j_{000\dots}-sheaf corresponds to j minj_min. Another way to view the ii-th letter in ww is as specifying whether [i][i] is (w i=0w_i=0) or is not (w 1=1w_1=1) contained in the image of the subcategory inclusion Δ j wΔ +\Delta_{j_w}\hookrightarrow\Delta_+ corresponding to j wj_w.

A topology j wj_w restricts to a topology j w| nj_w|_n on the n-truncation Δ +| n\Delta_+|_n by taking the prefix of length nn+1 e.g. j 00,j 01,j 10,j 11j_{00},j_{01},j_{10},j_{11} are the four topologies on the 1-truncation, the topos of directed graphs.

The initial object 0Set Δ + op0\in Set^{\Delta_+^{op}} is a j wj_w-sheaf for the topology j wj_w precisely when w 0=0w_0=0 since otherwise every j wj_w-sheaf contains exactly one vertex hence these j 0wj_{0w'} correspond to the dense subtoposes of Set Δ + opSet^{\Delta_+^{op}} .

There is a countably infinite number of co-atoms j wj_w where ww contains exactly one 0. These correspond to different copies of SetSet induced by the inclusion of the one-morphism category on the objects [n][n] in Δ +\Delta_+. In particular, the dense double negation copy j ¬¬j_{\neg\neg} corresponds to j 0111j_{0111\dots}, the j ¬¬j_{\neg\neg}-sheaves consisting of semi-simplicial sets with an arbitrary set of 00-semi-simplexes (= vertices) and exactly one nn-semi-simplex in the higher dimensions for every configuration of nn-1-semi-simplices which can bound an nn-semi-simplex. Since 0 is the only j maxj_{max}-skeletal presheaf we see incidentally that j^ max=j ¬¬\hat{j}_{max}=j_{\neg\neg}.

We denote the corresponding topologies by j k 0j^0_k where kk indicates the position where the 0 occurs e.g. j 0 0=j ¬¬j^0_0=j_{\neg\neg}. Since the [k][k] corresponds precisely to the groupoidal subcategories of Δ +\Delta_+ the subtoposes Sh j k 0(Set Δ + op)Sh_{j^0_k}(Set^{\Delta_+^{op}}) are the only Boolean subtoposes of Set Δ + opSet^{\Delta_+^{op}}. The endofunctor of the j k 0j^0_k-skeletal comonad on Set Δ + opSet^{\Delta_+^{op}} is given by X X[k][k]X\mapsto \coprod_{X[k]}[k] (in particular, the j k 0j^0_k-skeleta are those semi-simplicial sets XX with X X[k][k]X\simeq \coprod_{X[k]}[k] ). This basically replaces XX by its “set” of kk-semi-simplices in the disconnected form of copies of the standard kk-semi-simplex and discards everything else.

The Lawvere-Tierney topologies j wj_w on simplicial sets can similarly described by infinite words over {0,1}\{0,1\} with the same interpretation of the letters but there ww has to start with a (possibly empty, possibly infinite) block of zeros expressing that the respective sheaf categories consist of mm-truncated simplicial sets, where m+1m+1 is the length of the block of zeros with which ww starts e.g. j 00111j_{00111\dots} corresponds to the 1-truncation, the topos of reflexive graphs.

In Set Δ + opSet^{\Delta_+^{op}} these topologies j wj_w with w=0 m111,m=0,,w=0^m111\dots,\quad m=0,\dots ,\infty correspond precisely to the open subtoposes. To see, this consider the terminal object 1 of Set Δ + opSet^{\Delta_+^{op}}: in contrast to SSet where the degeneracies prevent this, we can truncate 1 at arbitrary dimensions by simply discarding the higher dimensional semi-simplices. Hence there is a countably infinite number of non-trivial subterminal objects U nU_n around this time with U n([k])=,kn+1U_n([k])=\empty\, ,\, k\geq n+1. By generalities, the corresponding open subtopos is equivalent to Set Δ + op/U nSet^{\Delta_+^{op}}/U_n but clearly this is equivalent to the full subcategory of semi-simplicial sets XX with X([k])=,kn+1X([k])=\empty\, ,\, k\geq n+1.

These are precisely the j U m oj_{U_m}^o-skeleta of j U m o:=j 0 m111j_{U_m}^o:=j_{0^m111\dots} whereas the j U m oj_{U_m}^o-sheaves are presheaves that have unique filling nn-semi-simplices for nmn\geq m. Hence, a j U m oj_{U_m}^o-skeletal presheaf XX is an j U m+1 oj_{U_{m+1}}^o-sheaf and in general not a j wj_w-sheaf for any j wj_w with w m+1=1w_{m+1}=1 since X[m+1]=X[m+1]=\empty\,. Accordingly, j^ U m o=j U m+1 o\hat{j}_{U_m}^o=j_{U_{m+1}}^o\,: the “Hegelian Aufhebung of \empty” starting with j max=j U 1 oj_{max}=j_{U_{-1}}^o passes stepwise j U m oj U m+1 oj_{U_{m}}^o\to j_{U_{m+1}}^o exactly through the open subtoposes except the “subtopos at \inftyj U o=j minj_{U_\infty}^o=j_min for which j^ U o=j U o\hat{j}_{U_\infty}^o=j_{U_\infty}^o. This contrasts somewhat with the situation in SSet where j^ 0 m111=j 0 2m1111\hat{j}_{0^m111\dots}=j_{0^{2m-1}111\dots} for m2m\geq 2 (cf. Aufhebung for more on this case).

Before determining the topologies j U m cj_{U_m}^c for the closed subtoposes, let’s have a look at the lattice operations: the join j wj zj_w\vee j_z of two topologies is given by the topology j wzj_{w\vee z} with (wz) i=w i(w\vee z)_i=w_i in case w i=z iw_i=z_i or else (wz) i=1(w\vee z)_i=1. As usual, the sheaf topos corresponding to j wj zj_w\vee j_z is given by the intersection of the two sheaf toposes. The meet j wj zj_w\wedge j_z is given by j wzj_{w\wedge z} with (wz) i=w i(w\wedge z)_i=w_i in case w i=z iw_i=z_i or else (wz) i=0(w\wedge z)_i=0. Hence the complement of a topology j wj_w is given by j w¯j_{\overline{w}} with w¯ iw i\overline{w}_i\neq w_i for all ii e.g. for an open topology j U m o=j 0 n111j_{U_m}^o =j_{0^n111\dots} the closed complement j U m cj_{U_m}^c is given by j 1 n000j_{1^n000\dots}.

The open and closed topologies are far from being the only complemented topologies - the letter flipping operation ww¯w\to\overline{w} provides a complement j w¯j_\overline{w} for every topology j wj_w. Hence, the lattice of topologies is not only as usually a Heyting algebra but a Boolean algebra, in fact, the product algebra 2\prod_\mathbb{N}\mathbf{2}.

Let us call a topology j wj_w locally closed when the corresponding sheaf topos is locally closed i. e. j wj_w is the join of an open and a closed topology. The locally closed topologies that are neither open nor closed themselves are then necessarily of the form j 1 n0 m111j_{1^n0^m111\dots}.

For further details on the Lawvere-Tierney topologies, closure operators and sheaves involved in both cases see Rosset-Hansen-Endrullis (2024).

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 Δ\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 Δ\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. Δ +\Delta_+ emphasizes that it is the subcategory of Δ\Delta that raise the degree when Δ\Delta is seen as a Reedy category but the ++ may also ambiguously suggests that it adds something to Δ\Delta. The notation Δ inj\Delta_{inj} emphasizes that it is the subcategory of injective morphisms of Δ\Delta. Similarly for Δ i\Delta_i though less explicitly. The notation Δ^\widehat{\Delta} (e.g. in Friedman) has the risk of introducing a confusion for readers used with the hat notation for presheaves. The notation Δ\Delta' and Δ¯\overline{\Delta} (e.g. in Sina Hazratpour) express that it is a variant of Δ\Delta but without giving precisions.

References

See also the references at semi-simplicial object and:

In homotopy type theory:

On the model structure on semi-simplicial sets:

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:

on the lattice of Lawvere-Tierney topologies:

  • A. Rosset, H. H. Hansen, J. Endrullis, Characterisation of Lawvere-Tierney Topologies on Simplicial Sets, Bicolored Graphs, and Fuzzy Sets, arXiv:2407.04535 (2024). (abstract)

Last revised on November 21, 2024 at 11:13:31. See the history of this page for a list of all contributions to it.