Shehtman Complexes in Simplicial Sets

• Shehtman complexes are simplicial sets where any compatible pair of codimension-1 faces extends to an n-simplex, defined via a two-face extension property.
• They serve as an intermediate notion between Kan complexes and more relaxed structures by satisfying Beck–Chevalley conditions, especially in dimensions above two.
• Explicit examples like K(ℕ,2) illustrate that while every Kan complex is a Shehtman complex, the converse fails, highlighting subtle differences in higher-categorical settings.

Searching arXiv for the specified paper to ground the article in the published source. Shehtman complexes are simplicial sets characterized by a two-face extension property that the paper "Beck-Chevalley Conditions in Simplicial Sets" isolates as a simplicial-set analogue of a Beck–Chevalley lifting condition, motivated by work of Valentin Shehtman around 2022 (Chakhvashvili, 21 Jul 2025). For a simplicial set $S$, the condition requires that for every $n>1$ and every pair $0\le p<q\le n$, any compatible pair of codimension-1 faces extends to an $n$-simplex. In the paper’s formulation, Shehtman complexes occupy a strictly intermediate position among standard simplicial notions: every Kan complex is a Shehtman complex, but the converse fails (Chakhvashvili, 21 Jul 2025).

1. Definition and basic formulation

The paper first introduces the more general notion of a Shehtman fibration. For a simplicial map $f:E\to B$, and for every $n>1$ and every pair $0\le p<q\le n$, let $\romb_{p,q}[n]\subseteq \Delta[n]$ denote the union of the $p$-th and $q$-th faces of the standard simplex $n>1$0. Then $n>1$1 is a Shehtman fibration if every commutative square

$n>1$2

admits a lift (Chakhvashvili, 21 Jul 2025).

A simplicial set $n>1$3 is a Shehtman complex when the unique map $n>1$4 to the terminal simplicial set is a Shehtman fibration. Equivalently, $n>1$5 is a Shehtman complex if and only if for every $n>1$6 and every $n>1$7, whenever one is given $n>1$8-simplices

$n>1$9

such that

$0\le p<q\le n$0

there exists an $0\le p<q\le n$1-simplex $0\le p<q\le n$2 satisfying

$0\le p<q\le n$3

These are the Beck–Chevalley conditions $0\le p<q\le n$4 of the paper (Chakhvashvili, 21 Jul 2025).

The definition can therefore be summarized as follows: a simplicial set is Shehtman if any compatible pair of codimension-1 faces in dimensions $0\le p<q\le n$5 extends to an $0\le p<q\le n$6-simplex. The compatibility equation $0\le p<q\le n$7 is precisely the simplicial identity needed for the two prescribed faces to meet consistently along their common codimension-2 face.

2. Beck–Chevalley interpretation

The paper explains that the Shehtman condition is equivalent to a weak pullback condition on face maps. For each $0\le p<q\le n$8 and $0\le p<q\le n$9, the square

$n$0

is required to be a weak pullback square, meaning that the induced map

$n$1

is surjective (Chakhvashvili, 21 Jul 2025).

This reformulation expresses the extension property entirely in terms of face maps. Given a compatible pair of $n$2-simplices, one asks for surjectivity onto the fiber product determined by their common restriction to dimension $n$3. The paper identifies this as the Beck–Chevalley-style formulation and notes its relation to Beck–Chevalley conditions for bifibrations in descent theory (Chakhvashvili, 21 Jul 2025).

A plausible implication is that Shehtman complexes are best understood not merely as weakened horn-fillers, but as simplicial sets in which selected face-map squares satisfy a systematic exactness property. That perspective distinguishes them from definitions phrased only by lifting against specific cofibrations.

3. Relation to Kan conditions

The paper assumes standard simplicial-set notation: $n$4 denotes the standard $n$5-simplex, represented as $n$6, and $n$7 denotes the $n$8-th horn, the union of all faces except the $n$9-th face (Chakhvashvili, 21 Jul 2025). A simplicial map $f:E\to B$0 is a Kan fibration if every horn inclusion $f:E\to B$1 has the right lifting property with respect to $f:E\to B$2, and a Kan complex is a simplicial set whose map to the point is a Kan fibration (Chakhvashvili, 21 Jul 2025).

The Shehtman condition is weaker than the Kan condition because it tests lifting only for $f:E\to B$3, the union of two faces, rather than for a full horn inclusion (Chakhvashvili, 21 Jul 2025). Nonetheless, the paper establishes a close relationship in low dimensions. Every Shehtman complex satisfies the Kan conditions in dimensions $f:E\to B$4, and the Beck–Chevalley conditions in dimensions $f:E\to B$5 coincide with the usual Kan conditions in those dimensions (Chakhvashvili, 21 Jul 2025).

More precisely, Proposition $f:E\to B$6 states that the conjunction of all Kan conditions $f:E\to B$7 for $f:E\to B$8 is equivalent to the conjunction of all Beck–Chevalley conditions $f:E\to B$9 for $n>1$0. For $n>1$1, the correspondence is

$n>1$2

This shows that the distinction between Kan and Shehtman behavior begins only in dimensions strictly above $n>1$3 (Chakhvashvili, 21 Jul 2025).

A central proposition then states that every Kan fibration is a Shehtman fibration; in particular, every Kan complex is a Shehtman complex (Chakhvashvili, 21 Jul 2025). The proof is immediate from the fact that $n>1$4 is a trivial cofibration for $n>1$5: both are contractible, so a Kan fibration lifts against it. The converse, however, fails, and establishing that failure is one of the paper’s main purposes.

4. Ordinary categorical nerves

For an ordinary small category $n>1$6, the paper proves that the nerve $n>1$7 is a Shehtman complex if and only if it is a Kan complex (Chakhvashvili, 21 Jul 2025). Since the nerve of a category is Kan exactly when the category is a groupoid, this yields the stated criterion: $n>1$8

The proof uses the low-dimensional equivalence from Proposition $n>1$9. In a nerve of a category, the only nontrivial 2-dimensional Beck–Chevalley conditions reduce to invertibility of morphisms, so the Shehtman condition already forces groupoidness (Chakhvashvili, 21 Jul 2025). Thus, within ordinary category theory, the weakening from Kan to Shehtman produces no genuinely new class of nerves.

This result precludes a possible misconception that Shehtman complexes enlarge the class of categorical nerves in dimension $0\le p<q\le n$0. The paper instead shows that non-Kan Shehtman behavior must be sought in higher-categorical constructions.

5. Duskin nerves and the 2-categorical translation

To construct non-Kan examples, the paper studies the Duskin nerve $0\le p<q\le n$1 of a small 2-category $0\le p<q\le n$2 (Chakhvashvili, 21 Jul 2025). It recalls that an $0\le p<q\le n$3-simplex consists of objects $0\le p<q\le n$4 for $0\le p<q\le n$5, 1-morphisms $0\le p<q\le n$6, and 2-morphisms $0\le p<q\le n$7, subject to the usual tetrahedral coherence for every $0\le p<q\le n$8: $0\le p<q\le n$9 (Chakhvashvili, 21 Jul 2025).

The paper also recalls the criterion that the Duskin nerve of a small 2-category is a Kan complex if and only if every 1-morphism is invertible up to invertible 2-morphisms and every 2-morphism is invertible (Chakhvashvili, 21 Jul 2025). Against this background, it characterizes the Shehtman condition in explicitly 2-categorical terms: for the Duskin nerve $\romb_{p,q}[n]\subseteq \Delta[n]$0, the Beck–Chevalley condition $\romb_{p,q}[n]\subseteq \Delta[n]$1 is equivalent to the existence of suitable 2-morphisms $\romb_{p,q}[n]\subseteq \Delta[n]$2 filling all the missing coherence diagrams, with explicit equations (1)–(6) listed in the paper (Chakhvashvili, 21 Jul 2025).

These equations are presented as the 2-categorical analogue of the simplicial Beck–Chevalley lifting property. This suggests that, in the Duskin setting, the Shehtman condition is controlled by a bounded family of coherence fillers rather than by full invertibility data. That distinction is what makes it possible for Duskin nerves to be Shehtman without being Kan.

6. The family $\romb_{p,q}[n]\subseteq \Delta[n]$3

The paper specializes the Duskin-nerve construction to a concrete family arising from commutative monoids. For a commutative monoid $\romb_{p,q}[n]\subseteq \Delta[n]$4, it defines a simplicial set $\romb_{p,q}[n]\subseteq \Delta[n]$5 as the Duskin nerve of a certain 2-category built from $\romb_{p,q}[n]\subseteq \Delta[n]$6 (Chakhvashvili, 21 Jul 2025). Its $\romb_{p,q}[n]\subseteq \Delta[n]$7-simplices are families

$\romb_{p,q}[n]\subseteq \Delta[n]$8

of elements of $\romb_{p,q}[n]\subseteq \Delta[n]$9 satisfying the cocycle-like equalities

$p$0

for every $p$1 (Chakhvashvili, 21 Jul 2025).

In this setting, the Beck–Chevalley condition becomes an explicit algebraic system. Given data $p$2 with $p$3, there must exist elements

$p$4

satisfying the six relations

$p$5

The paper uses this explicit formulation to build examples and counterexamples (Chakhvashvili, 21 Jul 2025).

This algebraization is significant because it translates a simplicial lifting problem into solvability of equations internal to the monoid. The resulting objects resemble cocycle-type simplicial sets built from monoids, and the Beck–Chevalley conditions become algebraic compatibility constraints rather than abstract lifting statements (Chakhvashvili, 21 Jul 2025).

7. Explicit examples and strictness of the inclusion

The paper gives two complementary examples. For $p$6 under multiplication, it shows that $p$7 is not a Shehtman complex (Chakhvashvili, 21 Jul 2025). It tests the condition $p$8 by choosing $p$9 with

$q$0

and all other relevant $q$1. These values satisfy the cocycle equations because in each equation at most one term is $q$2. However, the Beck–Chevalley lifting would require equations including

$q$3

which cannot all hold simultaneously in $q$4 (Chakhvashvili, 21 Jul 2025).

The decisive example is $q$5 under addition. The paper proves that $q$6 is not Kan, because by the earlier proposition the Duskin nerve of a 2-category is Kan only when 1-morphisms and 2-morphisms are invertible, and $q$7 under addition is not a group (Chakhvashvili, 21 Jul 2025). At the same time, $q$8 is a Shehtman complex. The proof uses the embedding

$q$9

Since $n>1$00 is a Kan complex, it is also Shehtman. Given a Beck–Chevalley problem in $n>1$01, one solves it in $n>1$02, then shifts all relevant integers by a common integer $n>1$03 so that

$n>1$04

all lie in $n>1$05 while still satisfying the same equations (Chakhvashvili, 21 Jul 2025). This proves the existence of Beck–Chevalley fillers in $n>1$06.

The example $n>1$07 establishes the strict inclusion

$n>1$08

It is the paper’s explicit example of a Shehtman complex that is not a Kan complex (Chakhvashvili, 21 Jul 2025).

8. Position within simplicial and higher-categorical context

The paper places Shehtman complexes relative to several standard classes. It states that Kan complexes form a strict subclass of Shehtman complexes; that for nerves of ordinary categories, Shehtman coincides with Kan and hence with nerves of groupoids; and that for Duskin nerves of 2-categories, the Shehtman condition becomes an explicit family of equations on 2-morphisms (Chakhvashvili, 21 Jul 2025). It also notes that the $n>1$09 examples resemble cocycle-type simplicial sets built from monoids, with Beck–Chevalley equations appearing as algebraic compatibility conditions (Chakhvashvili, 21 Jul 2025).

The historical framing is that these simplicial sets were introduced by Valentin Shehtman around 2022 in connection with associating a simplicial set to a first-order theory where $n>1$10-simplices correspond to $n>1$11-types of the theory, and the Beck–Chevalley conditions were important in that context (Chakhvashvili, 21 Jul 2025). The paper’s aim is to isolate this structure in simplicial-set language, prove its basic relation to Kan complexes, and provide a counterexample to the converse.

The paper does not claim equivalence with quasi-categories or discuss them directly (Chakhvashvili, 21 Jul 2025). This is an important boundary on interpretation: Shehtman complexes are introduced as a distinct simplicial condition defined by two-face extension and weak-pullback surjectivity, not as a reformulation of more familiar higher-categorical models. A plausible implication is that their main interest lies in the specific Beck–Chevalley exactness they encode, rather than in serving as a replacement for existing notions of weak higher categories.