Ruptured Simplicial Sets Overview

• Ruptured simplicial sets are non-standard constructs where the desingularization functor collapses repeated vertices, converting singular simplices into degenerate ones.
• The iterative enforced collapse applies a transfinite sequence of elementary quotients to systematically remove pathological identifications within simplicial structures.
• This process relates to broader concepts like weak pullbacks and horn failures, underscoring its significance in homotopy theory and the study of non-singular simplicial sets.

Searching arXiv for papers on desingularization, non-singular simplicial sets, and related weakened simplicial structures. “Ruptured simplicial sets” is not a standard technical term in the cited literature. The closest precise notion is the passage from an arbitrary simplicial set to a non-singular simplicial set by the desingularization functor $D$, which canonically removes singular behavior carried by non-degenerate simplices with repeated vertices by forcing them to become degenerate in a universal quotient (Fjellbo, 2020). In that sense, “rupture” is best understood not as deleting faces or weakening simplicial identities, but as a controlled collapse of pathological identifications inside simplices. Closely related work situates non-singular simplicial sets inside a Quillen-equivalent homotopy theory (Fjellbo, 2020), while other nearby literatures use different mechanisms—weak pullback conditions, horn failures, or simplicial blow-ups—to capture other forms of local defect or weakened compositional structure (Constantin et al., 2021, Horiuchi, 2020, Chataur et al., 2012).

1. Terminological status and conceptual scope

The phrase “ruptured simplicial sets” does not occur as a standard definition in the relevant arXiv literature. The most precise match is provided by the theory of desingularization of simplicial sets, where singularities are understood as failures of non-degenerate simplices to be embedded (Fjellbo, 2020). A simplicial set is

$sSet = Fun(\Delta^{op}, Set),$

with $\Delta$ the category of finite ordinals

$[n]=\{0<1<\cdots<n\}$

and order-preserving maps. For $X\in sSet$, one writes $X_n=X([n])$, and an operator $\alpha:[m]\to[n]$ acts on a simplex $x\in X_n$ from the right by $x\alpha\in X_m$. Face operators are injective operators in $\Delta$, and degeneracy operators are surjective ones (Fjellbo, 2020).

This usage places “rupture” on the side of regularization by quotienting. It does not refer to semisimplicial objects, deleted-face objects, punctured simplices, or missing horn structures. Several of the related papers explicitly distinguish their subjects from such interpretations. The lower-bound theorem of Avvakumov and Karasev studies ordinary simplicial sets with contractible closed faces, not punctured or defective simplicial structures (Avvakumov et al., 2021). Ivanov’s treatment of horns, skeleta, and Postnikov systems concerns fibrant simplicial sets and their truncations, not a theory of ruptured objects (Ivanov, 2020). Chataur–Saralegi-Aranguren–Tanré use simplicial blow-up in filtered face sets, which is a replacement of local singular models rather than a deletion of simplicial data (Chataur et al., 2012).

A second nearby usage appears in the paper on weak cartesian properties of simplicial sets. There the relevant phenomenon is not singular vertices inside a simplex but the weakening of pullback conditions to weak pullbacks, so that fillers exist without uniqueness. That work suggests a different metaphor of “rupture,” namely controlled failure of exact cartesianity, formalized through completeness conditions, span complete simplicial sets, and inner span complete simplicial sets (Constantin et al., 2021). A third nearby usage appears in Horiuchi’s example of a simplicial monoid whose underlying simplicial set is not a quasi-category; there the “rupture” is a failure of inner horn filling, not a failure of embeddedness (Horiuchi, 2020).

2. Non-singularity and the desingularization reflector

The central formal notion is that of a non-singular simplicial set. For a simplex $sSet = Fun(\Delta^{op}, Set),$0, Yoneda gives a representing map

$sSet = Fun(\Delta^{op}, Set),$1

A simplicial set is called non-singular if the representing map of each non-degenerate simplex is degreewise injective (Fjellbo, 2020). Equivalently, each non-degenerate simplex is embedded. Concretely, for every simplicial degree $sSet = Fun(\Delta^{op}, Set),$2, the map

$sSet = Fun(\Delta^{op}, Set),$3

is injective. The cited work repeatedly uses the equivalent criterion that a non-degenerate simplex is embedded if and only if its vertices are pairwise distinct (Fjellbo, 2020). The same criterion is emphasized in the companion homotopy-theoretic treatment of non-singular simplicial sets (Fjellbo, 2020).

Degeneracy is measured via the Eilenberg–Zilber lemma: every simplex has a unique factorization

$sSet = Fun(\Delta^{op}, Set),$4

with $sSet = Fun(\Delta^{op}, Set),$5 non-degenerate and $sSet = Fun(\Delta^{op}, Set),$6 a degeneracy operator (Fjellbo, 2020). A simplex is degenerate if it is a proper degeneracy $sSet = Fun(\Delta^{op}, Set),$7 for some proper degeneracy operator $sSet = Fun(\Delta^{op}, Set),$8; otherwise it is non-degenerate. The pathology relevant to “rupture” arises when a simplex is non-degenerate yet not embedded, equivalently when it has repeated vertices (Fjellbo, 2020, Fjellbo, 2020).

These simplicial sets form a reflective full subcategory

$sSet = Fun(\Delta^{op}, Set),$9

Its left adjoint is the desingularization functor

$\Delta$0

The original definition recalled in Fjellbo’s paper is

$\Delta$1

where the product ranges over all quotient maps $\Delta$2 with $\Delta$3 non-singular. Corestricting the canonical map gives

$\Delta$4

The universal property is that every simplicial map from $\Delta$5 to a non-singular simplicial set factors uniquely through $\Delta$6; equivalently,

$\Delta$7

and

$\Delta$8

for every simplicial set $\Delta$9 and non-singular simplicial set $[n]=\{0<1<\cdots<n\}$0 (Fjellbo, 2020).

This reflector is the precise mathematical content behind the “rupture” metaphor. It does not delete offending simplices. Instead, it identifies simplices just enough that every non-embedded non-degenerate simplex becomes degenerate in the quotient. The resulting object is the least drastic non-singular quotient of the original simplicial set (Fjellbo, 2020).

3. Enforcers and iterative enforced collapse

The main advance of “Iterative Desingularization” is an explicit transfinite construction of $[n]=\{0<1<\cdots<n\}$1 by repeated elementary quotients (Fjellbo, 2020). The elementary step is the enforced collapse

$[n]=\{0<1<\cdots<n\}$2

For a non-degenerate simplex $[n]=\{0<1<\cdots<n\}$3, define a relation on the vertex set $[n]=\{0<1<\cdots<n\}$4 by

$[n]=\{0<1<\cdots<n\}$5

Then define a reflexive relation $[n]=\{0<1<\cdots<n\}$6 if there exists $[n]=\{0<1<\cdots<n\}$7 with $[n]=\{0<1<\cdots<n\}$8 and $[n]=\{0<1<\cdots<n\}$9, and let $X\in sSet$0 be the equivalence relation generated by $X\in sSet$1. The paper proves that if $X\in sSet$2 and $X\in sSet$3, then $X\in sSet$4. Hence the quotient $X\in sSet$5 inherits a total order and is canonically isomorphic to some $X\in sSet$6. The associated degeneracy operator

$X\in sSet$7

is called the enforcer of $X\in sSet$8 (Fjellbo, 2020).

The enforcer is the least drastic degeneracy operator compatible with the equalities among the vertices of $X\in sSet$9. If $X_n=X([n])$0 is already embedded, $X_n=X([n])$1 is the identity; if not, $X_n=X([n])$2 is proper. Proposition 3.3 in the paper shows that enforcers necessarily appear in any map from $X_n=X([n])$3 to a non-singular target. In particular, every non-embedded non-degenerate simplex is already forced, in $X_n=X([n])$4, to factor through its enforcer (Fjellbo, 2020).

The enforced collapse $X_n=X([n])$5 is defined as the pushout along all enforcers of all non-degenerate simplices. Writing

$X_n=X([n])$6

with maps

$X_n=X([n])$7

one forms the pushout

$X_n=X([n])$8

Equivalently,

$X_n=X([n])$9

where $\alpha:[m]\to[n]$0 is the simplicial congruence generated degreewise by identifying simplices whose preimages in $\alpha:[m]\to[n]$1 are identified by $\alpha:[m]\to[n]$2 (Fjellbo, 2020).

A single enforced collapse need not suffice. The paper therefore defines a transfinite sequence

$\alpha:[m]\to[n]$3

with

$\alpha:[m]\to[n]$4

At a limit ordinal $\alpha:[m]\to[n]$5, one takes the colimit of the earlier stages. The crucial lemma is that if $\alpha:[m]\to[n]$6 is not embedded, then its image in $\alpha:[m]\to[n]$7 is degenerate. Thus every singular non-degenerate simplex present at stage $\alpha:[m]\to[n]$8 is broken at the next stage, although new singular non-degenerate simplices can appear as images of earlier simplices (Fjellbo, 2020).

The main theorem states that for every simplicial set $\alpha:[m]\to[n]$9 there is an ordinal $x\in X_n$0 such that the canonical map

$x\in X_n$1

is the composition of the $x\in X_n$2-sequence

$x\in X_n$3

The stabilization argument is transfinite and cardinality-based: one cannot indefinitely choose distinct original non-degenerate simplices of $x\in X_n$4 whose images remain singular forever, because there are only set-many such simplices (Fjellbo, 2020).

4. Examples, calculations, and homotopical effects

The examples in the desingularization literature show that the operation is combinatorially natural but can be homotopically drastic. The cited paper explicitly states that desingularization has “homotopically destructive tendencies” (Fjellbo, 2020). The sharpest example is the simplicial sphere model

$x\in X_n$5

For $x\in X_n$6,

$x\in X_n$7

Thus desingularization can collapse a standard sphere model to a point (Fjellbo, 2020). The companion paper on homotopy theory of non-singular simplicial sets repeats this example and uses it to show that desingularization alone is not homotopically safe (Fjellbo, 2020).

The simplest extreme case is when $x\in X_n$8 is a singleton. Then any non-singular quotient must have all higher-dimensional simplices degenerate, and one obtains

$x\in X_n$9

(Fjellbo, 2020). A more structured example is the pushout

$x\alpha\in X_m$0

with

$x\alpha\in X_m$1

Here the non-degenerate $x\alpha\in X_m$2-simplex $x\alpha\in X_m$3 is singular, its enforcer is

$x\alpha\in X_m$4

and the paper computes

$x\alpha\in X_m$5

This is an explicit one-step desingularization in which collapsing the identified $x\alpha\in X_m$6-face forces the $x\alpha\in X_m$7-simplex to degenerate down to dimension $x\alpha\in X_m$8 (Fjellbo, 2020).

Subdivision can significantly change the outcome. For

$x\alpha\in X_m$9

one has

$\Delta$0

After one subdivision,

$\Delta$1

After two subdivisions,

$\Delta$2

the suspension of a $\Delta$3-gon. More generally, for $\Delta$4,

$\Delta$5

This indicates that repeated subdivision can make desingularization preserve much more of the intended homotopy-theoretic content (Fjellbo, 2020).

The need for transfinite iteration is genuine. Fjellbo constructs a $\Delta$6-dimensional simplicial set $\Delta$7 such that for every finite $\Delta$8,

$\Delta$9

is still singular. This rules out any general finite-step bound and explains why the reflector $sSet = Fun(\Delta^{op}, Set),$00 must be described transfinitely (Fjellbo, 2020).

A related but different regularity phenomenon appears in the lower-bound theorem for simplicial sets with contractible faces. Avvakumov and Karasev emphasize that in a simplicial set an $sSet = Fun(\Delta^{op}, Set),$01-simplex may have repeated vertices, top-dimensional simplices may be degenerate, and the realization of the simplicial subset generated by one simplex need not be an honest $sSet = Fun(\Delta^{op}, Set),$02-ball. Their hypothesis that all closed faces are contractible is used precisely to exclude such pathologies when deriving the bound

$sSet = Fun(\Delta^{op}, Set),$03

under the presence of cohomology classes

$sSet = Fun(\Delta^{op}, Set),$04

with non-zero product (Avvakumov et al., 2021). This does not define “ruptured simplicial sets,” but it shows that local simplex pathologies have global combinatorial consequences.

5. Homotopy theory of non-singular simplicial sets

Although desingularization alone can destroy homotopy type, non-singular simplicial sets nevertheless support the same homotopy theory as ordinary simplicial sets after appropriate regularization (Fjellbo, 2020). The category

$sSet = Fun(\Delta^{op}, Set),$05

admits a proper cofibrantly generated model structure right-induced from simplicial sets. In that structure, a map $sSet = Fun(\Delta^{op}, Set),$06 in $sSet = Fun(\Delta^{op}, Set),$07 is a weak equivalence if and only if $sSet = Fun(\Delta^{op}, Set),$08 is a weak equivalence, and a fibration if and only if $sSet = Fun(\Delta^{op}, Set),$09 is a fibration (Fjellbo, 2020).

The crucial Quillen adjunction is

$sSet = Fun(\Delta^{op}, Set),$10

This is a Quillen equivalence (Fjellbo, 2020). In particular, the unit

$sSet = Fun(\Delta^{op}, Set),$11

is a weak equivalence for every simplicial set $sSet = Fun(\Delta^{op}, Set),$12. The result shows that non-singular simplicial sets present the same homotopy theory as all simplicial sets, but only after double subdivision followed by desingularization (Fjellbo, 2020).

This is closely tied to the combinatorics of embedded simplices. The paper recalls the Barratt nerve

$sSet = Fun(\Delta^{op}, Set),$13

where $sSet = Fun(\Delta^{op}, Set),$14 is the poset of non-degenerate simplices of $sSet = Fun(\Delta^{op}, Set),$15, and the natural degreewise surjective map

$sSet = Fun(\Delta^{op}, Set),$16

One has

$sSet = Fun(\Delta^{op}, Set),$17

Thus non-singular simplicial sets are exactly those for which subdivision already agrees with the Barratt-nerve construction built from non-degenerate simplices (Fjellbo, 2020).

The paper also identifies categorical control mechanisms for quotient singularities. It introduces eden and abyss as special full simplicial subsets that behave like sieve- and cosieve-type regions under quotients, and Strøm maps as degreewise injective maps with eden/abyss factorizations and simplicial deformation retraction data (Fjellbo, 2020). These notions are used to prove that $sSet = Fun(\Delta^{op}, Set),$18 is bicomplete and fits into a square of Quillen equivalent model categories together with simplicial sets, small categories, and posets (Fjellbo, 2020).

A common misconception is therefore that “rupturing” simplices into non-singular ones is merely cosmetic. The homotopy-theoretic results show the opposite: desingularization by itself may be destructive, while the corrected functor $sSet = Fun(\Delta^{op}, Set),$19 is homotopically appropriate. The distinction between $sSet = Fun(\Delta^{op}, Set),$20 and $sSet = Fun(\Delta^{op}, Set),$21 is essential (Fjellbo, 2020).

6. Related notions of rupture: weak cartesianity, horn failure, and simplicial blow-up

Several adjacent theories capture different kinds of local defect or weakened structure and are best distinguished from desingularization.

One line of work weakens exact cartesianity rather than embeddedness. A commutative square in $sSet = Fun(\Delta^{op}, Set),$22 is a weak pullback if compatible elements admit a lift, without uniqueness. For a simplicial set $sSet = Fun(\Delta^{op}, Set),$23, the class $sSet = Fun(\Delta^{op}, Set),$24 consists of squares in $sSet = Fun(\Delta^{op}, Set),$25 that $sSet = Fun(\Delta^{op}, Set),$26 sends to weak pullbacks (Constantin et al., 2021). This leads to span complete simplicial sets, where $sSet = Fun(\Delta^{op}, Set),$27 contains all balanced squares of coface maps, and inner span complete simplicial sets, where $sSet = Fun(\Delta^{op}, Set),$28 contains all pushout squares of coface maps. These classes are characterized by filler existence for families of acyclic configurations. Every quasicategory is inner span complete, every Kan complex is span complete, and examples include simplicial sets of metrics, relational databases, and joint probability distributions (Constantin et al., 2021). Here the “rupture” is a loss of uniqueness of fillers, not a failure of simplices to be embedded.

A second line concerns explicit horn-filling failure. Horiuchi constructs the simplicial commutative monoid

$sSet = Fun(\Delta^{op}, Set),$29

whose underlying simplicial set is not a quasi-category (Horiuchi, 2020). In low degrees,

$sSet = Fun(\Delta^{op}, Set),$30

with face maps

$sSet = Fun(\Delta^{op}, Set),$31

The inner horn $sSet = Fun(\Delta^{op}, Set),$32 with visible faces

$sSet = Fun(\Delta^{op}, Set),$33

has no filler, since any filler would require

$sSet = Fun(\Delta^{op}, Set),$34

in $sSet = Fun(\Delta^{op}, Set),$35, which is impossible (Horiuchi, 2020). This is a failure of higher compositional coherence, not a desingularization problem.

A third line is simplicial blow-up in filtered face sets. For a filtered simplex

$sSet = Fun(\Delta^{op}, Set),$36

the blow-up replaces it by

$sSet = Fun(\Delta^{op}, Set),$37

with an explicit map $sSet = Fun(\Delta^{op}, Set),$38 (Chataur et al., 2012). This construction is used to define Thom–Whitney cochains, perverse degrees, intersection cohomology, and perverse CDGAs on filtered face sets. It is a replacement of singular local models by cone-product models, not a removal of singularities by quotienting (Chataur et al., 2012).

Ivanov’s work on bounded cohomology adds yet another nearby perspective. Horns $sSet = Fun(\Delta^{op}, Set),$39, boundaries $sSet = Fun(\Delta^{op}, Set),$40, skeleta $sSet = Fun(\Delta^{op}, Set),$41, and Postnikov truncations $sSet = Fun(\Delta^{op}, Set),$42 provide controlled omissions or quotients of simplicial data, while the “unraveling”

$sSet = Fun(\Delta^{op}, Set),$43

preserves bounded cohomology (Ivanov, 2020). These constructions are structurally adjacent to “rupture” metaphors, but they belong to fibrant simplicial homotopy theory rather than the desingularization of repeated-vertex simplices.

Taken together, these literatures indicate that “ruptured simplicial sets” has no single established meaning. In the most precise sense supported by the cited papers, it denotes simplicial sets that have been desingularized into non-singular ones by the reflector

$sSet = Fun(\Delta^{op}, Set),$44

possibly through an iterative transfinite sequence of enforced collapses (Fjellbo, 2020). In broader nearby senses, it can also evoke simplicial sets with weakened pullback exactness (Constantin et al., 2021), horn-filling defects (Horiuchi, 2020), or singular simplices replaced by blow-up models adapted to intersection cohomology (Chataur et al., 2012).