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.

The lattice of subtoposes

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

This situation is familiar from $\Delta$ but in the latter case there are fewer Cauchy-complete subcategories available, since an object $[n]$ having non-trivial idempotents its inclusion automatically requires the presence of all $[m]$ with $m\,$<$\,n$ in the subcategory whence there are countably many subtoposes corresponding to the $n$-truncated subcategories on the objects $[0],\dots ,[n]$ (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 $j$ we call the presheaves in the image of the latter inclusion $j$-skeleta. Recall that the Aufhebung of such a level topology $j$ is defined as the largest topology $\hat{j}$ such that all $j$-skeleta are $\hat{j}$-sheaves (if such a $\hat{j}$ 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 $j_w$ on $Set^{\Delta_+^{op}}$ are indexed by infinite words $w$ over the alphabet $\{0,1\}$. They are partially ordered by $j_w\leq j_z$ iff $w_i\leq z_i$ for all letter positions $i=0,1,2,\dots$, where (unsurprisingly) letter 0 counts as smaller than letter 1. We will abbreviate infinite periods $000\dots$ (resp. $111\dots$) by $\bar{0}$ (resp. $\bar{1}$), finite periods of length $m$ by $0^m$ (resp. $1^m$). It is sometimes useful to assume that a finite length of $m=0$ denotes a block of length 0 of that letter (=its absence in that position) e.g. $1^m\bar{0}\;, m=0,1,\dots$ denotes all words that start with a possibly empty finite block of 1s followed by an infinite sequence of 0s (and sometimes by abuse also $\bar{1}$ if $m=\infty$ i.e. $1^\infty$ preempts the occurrence of the second infinite block).

The information contained in the $i$-th letter of $w$ is whether the closure operator corresponding to $j_w$ adds $i$-dimensional semi-simplices to subobjects ($w_i=1$) or not ($w_i=0$) e.g. $j_{\bar{1}}$ adds all semi-simplices of every dimension implying that the empty subobject is $j_\bar{1}$-dense in every object thereby preventing all non-terminal objects from being $j_\bar{1}$-sheaves i.e. $j_{111\dots}=j_max$ while $j_\bar{0}$ where no semi-simplex of any dimension is added and every presheaf is a $j_\bar{0}$-sheaf corresponds to $j_min$. Another way to view the $i$-th letter in $w$ is as specifying whether $[i]$ is ($w_i=0$) or is not ($w_1=1$) contained in the image of the subcategory inclusion $\Delta_{j_w}\hookrightarrow\Delta_+$ corresponding to $j_w$.

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

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

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

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

Since $[k]$ is $q(k)$-skeletal but not $n$-precise (i.e. has precisely one $n$-semi-simplex for any possible scaffold of $n-1$-semi-simplices) for any $0\leq n\leq k+1$, at least if $k\neq 0$, we find that $\hat{q}(k)=o(k+1)$ where $o(m):= j_{0^m\bar{1}}$ are the open topologies to be described below.

The Lawvere-Tierney topologies $j_w$ on simplicial sets can similarly described by infinite words over $\{0,1\}$ with the same interpretation of the letters but there $w$ has to start with a (possibly empty, possibly infinite) block of zeros expressing that the respective sheaf categories consist of $m$-truncated simplicial sets, where $m+1$ is the length of the block of zeros with which $w$ starts e.g. $j_{00\bar{1}}$ corresponds to the 1-truncation, the topos of reflexive graphs.

In $Set^{\Delta_+^{op}}$ these topologies $j_w$ with $w=0^m\bar{1},\quad m=0,\dots ,\infty$ correspond precisely to the open subtoposes. In order to see this, consider the terminal object 1 of $Set^{\Delta_+^{op}}$: 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 $U_n$ around this time with $U_n([k])=\empty\, ,\, k\geq n+1$. By generalities, the corresponding open subtopos is equivalent to $Set^{\Delta_+^{op}}/U_n$ but clearly this is equivalent to the full subcategory of semi-simplicial sets $X$ with $X([k])=\empty\, ,\, k\geq n+1$.

These are precisely the $o(m)$-skeleta of the topologies $o(m)$ defined above whereas the $o(m)$-sheaves are presheaves that have unique filling $n$-semi-simplices for $n\geq m$. Hence, a $o(m)$-skeletal presheaf $X$ is an $o(m+1)$-sheaf and in general not a $j_w$-sheaf for any $j_w$ with $w_{m+1}=1$ since $X[m+1]=\empty\,$. Accordingly, $\hat{o}(m)=o(m+1)\,$: the “Hegelian Aufhebung of $\empty$” starting with $j_{max}=o(0)$ passes stepwise $o(m)\to o(m+1)$ exactly through the open subtoposes except the “subtopos at $\infty$” $o(\infty)=j_min$ for which $\hat{o}(\infty)=o(\infty)$. This contrasts somewhat with the situation in SSet where $\hat{j}_{0^m\bar{1}}=j_{0^{2m-1}\bar{1}}$ for $m\geq 2$ (cf. Aufhebung for more on this case).

Before determining the topologies $c(m)$ for the closed subtoposes, let’s have a look at the lattice operations: the join $j_w\vee j_z$ of two topologies is given by the topology $j_{w\vee z}$ with $(w\vee z)_i=w_i$ in case $w_i=z_i$ or else $(w\vee z)_i=1$. As usual, the sheaf topos corresponding to $j_w\vee j_z$ is given by the intersection of the two sheaf toposes. The meet $j_w\wedge j_z$ is given by $j_{w\wedge z}$ with $(w\wedge z)_i=w_i$ in case $w_i=z_i$ or else $(w\wedge z)_i=0$. Hence the complement of a topology $j_w$ is given by $j_{\overline{w}}$ with $\overline{w}_i\neq w_i$ for all $i$ e.g. for an open topology $o(m) =j_{0^m\bar{1}}$ the closed complement $c(m)=\neg o(m)$ is given by $j_{1^m\bar{0}}$.

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

Let us call a topology $j_w$ locally closed when the corresponding sheaf topos is locally closed i. e. $j_w$ 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 $j_{1^n0^m\bar{1}}$.

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 $\Delta$-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

• simplicial object

  • simplicial set

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

• semi-simplicial object

• semicategory

• semi-Segal space

• 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)

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)