Ruptured Kan Complexes

• Ruptured Kan complexes are enriched simplicial sets that classify horn fill status into coherent, gap-witnessed, and open, redefining horn extension.
• They generalize classical Kan complexes by treating failure of horn filling as a positive, certified obstruction rather than mere absence of a filler.
• The framework supports modeling of type theories like Open Horn Type Theory by explicitly representing transport horns and associated obstructions.

Ruptured Kan complexes are simplicial structures introduced in connection with Open Horn Type Theory (OHTT) to model types in which horn extension phenomena are not exhausted by the classical dichotomy of fillable versus unfillable. In the formulation of OHTT, a horn may be coherently fillable, gap-witnessed, or open, and ruptured Kan complexes provide the corresponding semantic environment by enriching simplicial sets with explicit coherence and gap data subject to an exclusion law (Poernomo, 30 Dec 2025). Relative to ordinary Kan complexes, the key shift is that failure of horn filling is represented by positive structure rather than by mere absence of a filler: a gap is a certified obstruction. This permits the expression of constructions such as the transport horn, where a term and a path cohere but transport along that path is witnessed as gapped, a phenomenon stated to be inexpressible in Homotopy Type Theory (HoTT) because HoTT’s Kan condition guarantees successful transport (Poernomo, 30 Dec 2025).

1. Formal definition

A ruptured Kan complex is built from a ruptured simplicial set. A simplicial set is recalled as a functor $X:\Delta^{\rm op}\to\mathbf{Set}$ with face and degeneracy operators, and its $n$-simplices are $X_n$. An ordinary $(n,k)$-horn $\Lambda^n_k\hookrightarrow\Delta^n$ is the sub-simplicial set of the standard $n$-simplex missing its $k$th face (Poernomo, 30 Dec 2025).

A ruptured simplicial set is a triple

$\mathcal X=(X,\;Coh,\;Gap)$

where $X$ is an ordinary simplicial set, $Coh_n\subseteq X_n$ is the subset of $n$0-simplices that are witnessed coherent, and for each horn shape $n$1,

$n$2

is the set of gap-witnessed horns (Poernomo, 30 Dec 2025).

These data satisfy the Exclusion condition: no horn in $n$3 admits a coherent filler in $n$4. This condition makes coherence and gap mutually exclusive at the level of horn extension. The construction therefore separates three notions that are conflated in the classical setting: actual coherent extension, witnessed obstruction, and lack of either witness.

The paper’s definition of a ruptured Kan complex is deliberately weak in the classical Kan-theoretic sense. A ruptured simplicial set is a ruptured Kan complex precisely because no further filling requirement is imposed; one merely records, for each horn, whether it is coherently filled, gap-witnessed, or open (Poernomo, 30 Dec 2025). This contrasts with the ordinary Kan condition, where every horn must admit a filler.

2. Horn trichotomy and the role of openness

For a horn $n$5, exactly one of three states holds: $n$6 and the three cases are disjoint by Exclusion (Poernomo, 30 Dec 2025).

The first case is coherent fill: there exists $n$7 extending the horn. The second is gap: the horn itself lies in $n$8. The third is open: neither coherent fill nor gap witness is present. The open state is not reducible to negation in the classical or intuitionistic sense; in OHTT, it reflects the possibility that a horn is neither coherently resolved nor positively obstructed.

This trichotomy generalizes the binary derivable/underivable distinction emphasized in the description of OHTT. A plausible implication is that ruptured Kan complexes are intended not only as a geometric generalization of Kan complexes but also as a semantic infrastructure for logics and type theories where undecidedness and obstruction must be distinguished. In the source description, this is tied directly to the primitive judgments of coherence and gap, together with the fact that judgments may be open (Poernomo, 30 Dec 2025).

The paper’s informal summary characterizes a ruptured Kan complex as a Kan complex “with holes” turned into first-class witnesses. That formulation is interpretive but faithful to the stated purpose: “holes” are not simply absent fillers but tracked objects in the semantic structure (Poernomo, 30 Dec 2025).

3. Relation to classical Kan complexes and simplicial sets

Ordinary Kan complexes arise as a special case. If $n$9 is an ordinary Kan complex, then

$X_n$0

is a ruptured Kan complex in which no horn is gap-witnessed or open (Poernomo, 30 Dec 2025). In this sense, classical Kan complexes embed fully faithfully into ruptured Kan complexes as the fully coherent, no-gap objects.

The same formula also yields what the paper calls a “trivially ruptured” Kan complex for any ordinary simplicial set $X_n$1: $X_n$2 This exhibits $X_n$3 as a full subcategory of $X_n$4, the category of ruptured simplicial sets (Poernomo, 30 Dec 2025).

The relation to classical totality is central. Classical Kan complexes impose that every horn must admit a coherent filler. Ruptured Kan complexes drop that totality requirement while preserving a structured accounting of the status of each horn: fillable, obstructed, or undetermined (Poernomo, 30 Dec 2025). The paper also notes a dual extreme, namely “fully gapped” simplicial sets in which every horn is declared gap-witnessed, while adding that most interesting examples lie between the fully coherent and fully gapped extremes.

Within OHTT, HoTT is recovered as the coherent fragment obtained by imposing totality (Poernomo, 30 Dec 2025). This places ruptured Kan complexes in a semantic hierarchy: ordinary Kan complexes model the all-coherent limit, while ruptured Kan complexes model systems in which coherence is no longer universal.

4. Categorical and structural constructions

Morphisms of ruptured simplicial sets are ordinary simplicial maps that preserve both coherence and gaps: $X_n$5 These objects and morphisms form a category $X_n$6, equipped with a forgetful functor

$X_n$7

to ordinary simplicial sets (Poernomo, 30 Dec 2025).

Products are defined componentwise on coherent simplices and disjunctively on gap structure. If

$X_n$8

then

$X_n$9

This yields closure under products in the stated sense (Poernomo, 30 Dec 2025).

The coherent core $(n,k)$0 is defined as the smallest simplicial subset containing all of $(n,k)$1. The paper states that horns in the core which are coherently fillable in $(n,k)$2 are again fillable in $(n,k)$3, and that $(n,k)$4 is the maximal ordinary sub-Kan complex carried by $(n,k)$5 (Poernomo, 30 Dec 2025). This construction identifies the ordinary Kan-complex content already present inside a ruptured structure.

The source further states that usual notions such as Kan fibration, mapping space, and homotopy can be extended to the ruptured setting by recording each lifting or extension problem as coherent, gapped, or open. In particular, a “ruptured Kan fibration” $(n,k)$6 allows transport horns as gapped liftings (Poernomo, 30 Dec 2025). The statement that ruptured Kan complexes are closed, conjecturally, under usual simplicial constructions such as mapping spaces and limits is presented with the qualification “provided one equips the result with the ‘obvious’ ruptured structure,” so it remains partly prospective rather than fully established in the cited material.

5. Transport horns and low-dimensional intuition

The central configuration emphasized in OHTT is the transport horn: a term and a path both cohere, while transport along the path is witnessed as gapped (Poernomo, 30 Dec 2025). This is presented as a semantic obstruction that HoTT cannot express because HoTT’s Kan condition guarantees that all transport succeeds. Ruptured Kan complexes are designed to host exactly such configurations.

A two-dimensional illustration is given by the usual triangular horn

$(n,k)$7

where $(n,k)$8 and $(n,k)$9 are coherently witnessed, while the third edge $\Lambda^n_k\hookrightarrow\Delta^n$0 is gap-witnessed by $\Lambda^n_k\hookrightarrow\Delta^n$1 (Poernomo, 30 Dec 2025). In a classical Kan complex, such a horn must admit a filler. In a ruptured Kan complex, it may instead be coherent, gapped, or open.

This low-dimensional picture provides the geometric intuition for the theory’s broader claims. The horn does not merely fail to fill; rather, the missing component may itself be the object of positive witness. This suggests a reinterpretation of obstruction theory in simplicial semantics: the absence of a simplex can be typed and certified rather than inferred indirectly from failed extension.

6. Semantic function within Open Horn Type Theory

Open Horn Type Theory extends dependent type theory with two primitive judgment forms, coherence and gap, subject to a mutual exclusion law (Poernomo, 30 Dec 2025). Gap is not defined via implication and is not treated as classical or intuitionistic negation. Instead, it is a primitive witness of non-coherence. Ruptured simplicial sets and ruptured Kan complexes are introduced as the semantics corresponding to this judgmental architecture.

The semantics supports a trichotomy of judgments: coherent, gapped, and open. The coherent fragment recovers HoTT under totality, while the non-total setting supports obstructions that are explicitly unavailable in ordinary HoTT semantics (Poernomo, 30 Dec 2025). The paper develops three classes of such obstructions in detail: topological, semantic, and logical.

These classes are listed as follows:

Class	Examples named in the source
Topological	monodromy, holonomy, characteristic classes
Semantic	polysemy, meaning fibrations
Logical	resource-sensitive derivability, substructural failure

The source states that in each case the gap witness is positive structure, “not absence of proof, but certified obstruction” (Poernomo, 30 Dec 2025). This formulation is significant because it separates ruptured semantics from frameworks in which negative information is represented only by non-derivability. A plausible implication is that ruptured Kan complexes are intended to support semantics in which obstruction data carries independent inferential content.

7. Interpretation, scope, and limitations

Ruptured Kan complexes generalize classical Kan complexes by relaxing total horn filling while preserving a precise classification of horn status. The generalization is technically controlled by the Exclusion condition and by the distinction between coherent simplices and gap-witnessed horns (Poernomo, 30 Dec 2025). The embedding of ordinary Kan complexes as the all-coherent, no-gap objects ensures backward compatibility with established simplicial semantics.

At the same time, the construction is not simply a weakening of the Kan condition. The addition of gap witnesses changes the ontology of the model: obstruction becomes part of the primitive structure. The open state introduces a further distinction between certified failure and as-yet-unwitnessed status. This suggests that ruptured Kan complexes are best understood not as defective Kan complexes but as enriched simplicial objects tailored to a type theory with primitive non-coherence.

Several claims in the source are explicitly established, including the categorical definition of $\Lambda^n_k\hookrightarrow\Delta^n$2, closure under products, the coherent core construction, and the embedding of classical Kan complexes and simplicial sets (Poernomo, 30 Dec 2025). Other extensions, including closure under “the usual simplicial constructions” such as mapping spaces and limits, are described as conjectural when equipped with the “obvious” ruptured structure. That qualification marks an important boundary: the framework is presented with a clear foundational core and a program for further development, rather than as a fully completed replacement for ordinary simplicial homotopy theory.

Within that program, ruptured Kan complexes function as the semantic counterpart to OHTT’s central thesis: horn extension should not be forced into a binary regime of success versus failure, because some obstructions are themselves mathematically meaningful objects. In that sense, they provide a model of types where coherence, obstruction, and indeterminacy coexist in a single simplicial formalism (Poernomo, 30 Dec 2025).