Inner Horn Fillers in Higher Category Theory

• Inner horn fillers are combinatorial extensions that complete partial diagrams, defining composition and coherence in higher category theory.
• They are employed in various presheaf models—such as simplicial, bisimplicial, Γ-sets, and Θ₂-sets—to classify and compare algebraic structures.
• Their role in ensuring unique or homotopically unique fillers underpins applications in ∞-categories, symmetric monoidal structures, and Floer homotopy theory.

Inner horn fillers are a fundamental mechanism in the homotopical and higher-categorical characterization of algebraic structures via presheaf models, defining compositional and coherence properties in terms of extension problems for presheaf-valued diagrams. This concept, originating with the study of quasicategories and nerve constructions, generalizes into a unifying framework for the identification, classification, and comparison of higher-categorical objects across a variety of presheaf categories, such as simplicial sets, bisimplicial sets, Γ-sets, Θ₂-sets, and semi-simplicial spaces. The analogy extends to $\infty$-categories, symmetric monoidal structures, double categories, and more, with unique or homotopically unique horn-filling conditions providing the key combinatorial control over the higher-dimensional composition and coherence laws.

1. Definition and Construction of Inner Horns

Given a category $\Delta$ of finite ordinals and order-preserving maps, the standard $n$-simplex $\Delta[n]$ is the representable object $\mathrm{Hom}_\Delta(-,[n])$ in the category of simplicial sets. Its boundary $d\Delta[n]$ is the union of all proper faces, obtained via coface maps $\partial^i: \Delta[n-1] \to \Delta[n]$. For $0 < k < n$, the $k$th inner horn $\Lambda^n_k \subset \Delta[n]$ is the union of all $i$th faces except the $k$th:

$$\Lambda^n_k = \bigcup_{i \neq k}\, \partial^i(\Delta[n-1]) \subset \Delta[n],$$

with the inclusion $i_k: \Lambda^n_k \hookrightarrow \Delta[n]$. A map $\Lambda^n_k \to X$ provides a partial $n$-simplex in $X$, and a filler is an extension to a map $\Delta[n] \to X$ completing the horn.

This extends naturally to semi-simplicial settings, with $$\Delta^n_s = \mathrm{Hom}_{\Delta_{\mathrm{inj}}}(-,[n])$$ and horns defined analogously but without degeneracies. In generalized presheaf categories $C = \Delta, \Delta^2, \Gamma, \Theta_2$, the notion of horn replaces the missing face in a family of compositional data, and inner horns are those omitting a face indexed by an interior position.

2. Inner Kan Condition and Unique Fillers

The inner Kan condition requires that for every inner horn inclusion $i_k: \Lambda^n_k \hookrightarrow \Delta[n]$, the induced map

$$\mathrm{Map}_C(\Delta[n], X) \to \mathrm{Map}_C(\Lambda^n_k, X)$$

is surjective, i.e., every inner horn has a filler. When this map is bijective (or, in higher settings, a homotopy equivalence), fillers are unique (up to contractible choice). For simplicial sets, $X$ is inner Kan if every inner horn $\Lambda^n_k \to X$ ($02-reduced inner Kan, requiring uniqueness for all $n \ge 3$.

Duskin’s theorem establishes that a simplicial set $X$ is isomorphic to the nerve of a $(2,1)$-category if and only if it is inner Kan and has unique fillers for inner horns of dimension at least $3$; these are termed 2-reduced inner Kan complexes (Watson, 2014). This structure encodes the composition and all associated coherence data required for bicategories with invertible 2-morphisms.

In the context of semi-simplicial spaces, existence of fillers up to homotopy (rather than strict uniqueness) suffices for modeling $\infty$-categories, reflecting the inherently weak nature of higher categorical composition (Oldervoll, 16 Jan 2026).

3. Nerve Functors and Correspondence with Algebraic Structures

Nerve constructions realize concrete algebraic objects as presheaves with specified horn-filling properties. For standard categories:

• The nerve $N(\mathcal{C})$ of a category $\mathcal{C}$ is a simplicial set with unique fillers for all inner horns.
• For bicategories (Duskin nerve $N(\mathcal{B})$), unique fillers above dimension $2$ encode the invertibility of 2-morphisms and coherence.

Similar correspondences are established in other presheaf categories:

• Bisimplicial sets $(\Delta^2)$ and Verity double categories: Nerves $N(D)$ correspond bijectively to 2-reduced inner Kan bisimplicial sets.
• $\Gamma$-sets and symmetric monoidal groupoids: The $\Gamma$-nerve realizes symmetric monoidal groupoids as 2-reduced inner Kan $\Gamma$-sets, with horn-filling for Segal spines.
• $\Theta_2$-sets and fancy bicategories: The $\Theta_2$-nerve maps fancy bicategories (bicategories plus thin sub-bicategory) to 2-reduced inner Kan $\Theta_2$-sets (Watson, 2014).

This framework generalizes to higher $\Theta_n$-sets or presheaves on $\Gamma \times \Delta^k$, where unique horn fillers in appropriate codimensions serve as quasicategorical models for weak higher-categorical data.

4. Composition, Coherence, and Higher-Dimensional Generalizations

Composition in these models is defined via unique horn fillers. In the bicategory case:

• Composition of 1-arrows corresponds to the unique 2-simplex filling an inner horn $\Lambda^2_1$.
• Associativity and coherence (pentagon, triangle identities) are enforced by unique fillers for 3- and 4-dimensional horns.

Generalization to higher dimensions is achieved by defining “inner horns” in any presheaf category with a dimension-grading and suitable coface structure. Presheaves with unique inner horn fillers above a certain dimension serve as quasicategory analogues. For example, inner Kan $\Gamma$-sets model $E_\infty$-spaces, and inner Kan $\Theta_n$-sets conjecturally model $(\infty,n)$-categories (Watson, 2014).

5. Inner Horn Fillers up to Homotopy and Quasi-Unitality

Relaxing strict uniqueness to uniqueness up to homotopy is crucial for $\infty$-category models in semi-simplicial settings. In this context, the inner Kan condition is formulated in terms of the right lifting property with respect to inner horn inclusions $j^k_{n,s}$, requiring that the corresponding mapping spaces are surjective on $\pi_0$ (existence up to homotopy), with uniqueness up to contractible choice recovered via Reedy-fibrancy and coherence arguments (Oldervoll, 16 Jan 2026).

A salient feature in the semi-simplicial setting is the absence of canonical degeneracies, prompting the introduction of weak or “quasi-unitality” conditions as properties rather than structure:

• Existence of idempotent equivalences for each object (after Steimle–Tanaka).
• Marking of all inner-Kan equivalences (after Henry).
• Presence of coherent outer degeneracies (after Ayala–Blumberg).

All three avatars of quasi-unitality are shown to be equivalent in this framework. The semi-simplicial approach is especially powerful in geometric or Floer-theoretic contexts, where explicit degeneracies are unavailable but geometric extension by inner horn fillers can be constructed (Oldervoll, 16 Jan 2026).

6. Applications and Connections with Homotopical Algebra

The inner horn filler paradigm enables a unified treatment of algebraic and higher categorical structures in homotopical settings. Applications include:

• The model-theoretic description of $\infty$-categories via complete semi-Segal spaces and marked semi-simplicial spaces (IK$_+$), with an equivalence

$IK_+ \simeq CSS \simeq \Cat_\infty,$

where $CSS$ denotes the $\infty$-category of complete semi-Segal spaces (Oldervoll, 16 Jan 2026).

• The modeling of symmetric monoidal $(\infty,0)$-categories as inner Kan $\Gamma$-sets, expected to be Quillen-equivalent to Segal’s special $\Gamma$-spaces and hence to $E_\infty$-spaces (Watson, 2014).
• In Floer homotopy theory, the category of flow categories arises naturally from inner Kan semi-simplicial sets with quasi-unitality verified via geometrically constructed idempotent flows (Oldervoll, 16 Jan 2026).

7. Summary Table: Inner Horn Fillers Across Presheaf Models

Presheaf Category	Algebraic Structure Modeled	Horn-Filling Condition
$\Delta$-sets	Categories/$(2,1)$-categories	Unique fillers, $n\geq3$
$\Delta^2$-sets	Verity double categories	Unique bi-horn fillers, $m+n>2$
$\Gamma$-sets	Symmetric monoidal groupoids/$E_\infty$	Unique fillers for Segal spines
$\Theta_2$-sets	Fancy bicategories	Unique tree-shaped horn fillers
Semi-simplicial	$\infty$-categories	Fillers up to homotopy, quasi-unitality

In all cases, presheaves with unique inner horn fillers in appropriate dimensions are precisely nerves of corresponding higher algebraic structures, providing a combinatorial and homotopical foundation for higher category theory and related fields (Watson, 2014, Oldervoll, 16 Jan 2026).