Pseudo Double Functor Overview

• Pseudo double functor is a mapping between pseudo double categories that strictly preserves vertical composition and weakly preserves horizontal composition via coherent isomorphisms.
• It plays a central role in homotopy theory, algebraic quantum field theory, and higher categorical models by ensuring compositional coherence in complex structures.
• Applications include constructing model structures on double categories and strictifying weak morphisms in contexts such as AQFT and $(\infty,n)$-category theory.

A pseudo double functor is a structure-preserving morphism between pseudo double categories, generalizing the concept of a 2-functor by allowing the preservation of horizontal composition only up to coherent isomorphism (or invertible square), while maintaining strict preservation of vertical composition and identity. Pseudo double functors formalize the interplay between two compositions in categorical structures where strict functoriality cannot be expected in all directions, and they play a central role in the homotopy theory of double categories, the categorical formulation of algebraic quantum field theory (AQFT), and the modeling of higher structures (Moser et al., 2020, Komalan, 12 Jan 2026, Paoli, 2016).

1. Formal Definition and Data

Given pseudo double categories or double categories (internal categories in $\mathbf{Cat}$) $C$ and $D$, a pseudo double functor $F: C \to D$ consists of the following data (Paoli, 2016, Komalan, 12 Jan 2026):

• On objects and vertical arrows: A functor $F_0: C_0 \to D_0$ between object/vertical categories, strictly preserving vertical composition and identities.
• On horizontal arrows and squares: A functor $F_1: C_1 \to D_1$ between the arrow categories (i.e., horizontal arrows and squares), making the source/target squares commute: $d_i \circ F_1 = F_0 \circ d_i$ for $i = 0,1$.
• Pseudo (i.e., up to coherent isomorphism) on horizontal composition: For all composable horizontal arrows $h: A \to B$, $k: B \to C$ in $C$,

$$\Phi_h(h,k):\ F_1(h) \circ F_1(k) \Longrightarrow F_1(h \circ k)$$

is an isomorphism (invertible square).

• Pseudo on horizontal units: For each object $A\in C_0$,

$$\Phi_v(A):\ \mathrm{id}_{F_0(A)} \Longrightarrow F_1(\mathrm{id}_A)$$

is an invertible square.

• Coherence conditions: The invertible squares $\Phi_h$ and $\Phi_v$ must satisfy the associativity pentagon and unit triangles (see §3 below).

In the common case discussed by Moser–Sarazola–Verdugo, and in model-categorical contexts, only the horizontal direction is pseudo: vertical structure is strictly preserved, while nontrivial invertible squares account for horizontal associativity and unit constraints (Moser et al., 2020).

2. Coherence, Axioms, and Commutative Diagrams

The essential feature of a pseudo double functor is encoded in coherence diagrams that generalize Mac Lane’s coherence for monoidal categories to the double categorical setting (Paoli, 2016, Komalan, 12 Jan 2026):

• Associativity pentagon: For any four horizontally composable 1-cells, the five ways of rebracketing under repeated application of $\Phi_h$ and associators commute.
• Unit triangles: For any composable horizontal arrow and any object, the two possible composites (using unitors and $\Phi_v$) agree.
• Naturality: $\Phi_h$ and $\Phi_v$ are compatible with squares (2-cells) in the source category, preserving the vertical composition and strict interchange law for squares.

These axioms ensure that all diagrams built from the pseudo-structure commute, enforcing a well-behaved notion of composition that is not strictly functorial in the horizontal direction but coherent up to specified isomorphisms.

3. Classification: Strict, Pseudo, and Horizontally Pseudo Functors

The distinction between strict and pseudo double functors relates to whether the structure is preserved strictly or up to invertible squares:

• Strict double functor: Preserves both vertical and horizontal composition and identities on the nose.
• Pseudo double functor: Typically, weakens preservation in the horizontal direction only (composition/identity up to coherent isomorphism), with strict preservation of vertical structure and squares.
• Horizontally pseudo double functor: Terminology introduced by Moser–Sarazola–Verdugo (Moser et al., 2020) emphasizing that only the horizontal operations are weakened, matching the 2-categorical analogy with pseudo-functors.

All definitions reduce to a common pattern: two functors $F_0, F_1$ between the vertical and arrow categories, plus invertible natural transformations encoding weakening of horizontal associativity and unitality.

4. Role in Homotopy Theory and Whitehead-Type Theorems

Pseudo double functors are fundamental in the homotopy theory of double categories. In the model structure on the category $\mathrm{DblCat}$, weak equivalences—called double biequivalences—are precisely the strict double functors admitting a horizontally pseudo double functor as quasi-inverse up to horizontal pseudo natural equivalence (Moser et al., 2020). This is the double categorical analogue of the Whitehead theorem for 2-categories: a functor is a weak equivalence if and only if it is invertible up to (pseudo) natural equivalence.

Key results:

• Whitehead theorem (Moser–Sarazola–Verdugo, Theorem 4.13): For $A, B$ strict double categories with vertical categories as disjoint unions of free $\{0 \to 1\}$, a strict double functor $F: A \to B$ is a weak equivalence if and only if there exists a normal (i.e., strictly identity-preserving on horizontal units) horizontally pseudo double functor $G: B \to A$ and horizontal pseudo natural equivalences $\eta: \mathrm{id}_A \simeq GF$, $\epsilon: FG \simeq \mathrm{id}_B$.

This characterizes the correct notion of homotopy inverse for double categories and underpins the model structures that recover 2-categorical and bicategorical homotopy theory (Moser et al., 2020).

5. Applications: Algebraic Quantum Field Theory and Higher Categories

Pseudo double functors formalize structure in operator-algebraic AQFT by encoding both “vertical” inclusions (e.g., region embeddings) and “horizontal” bimodule/fusion structure (Komalan, 12 Jan 2026):

• A pseudo double functor $$\mathcal F_\mathcal A: \mathbf{Mink}(M) \to \mathbf{vNA}$$ (from the spacetime double category to the von Neumann algebra double category) describes AQFT data: vertical arrows model inclusion of observable algebras (Haag–Kastler nets, strict), while horizontal arrows correspond to correspondences (Connes fusion, pseudo).
• Coherence isomorphisms arise from the unitary associativity and unitality inherent in Connes fusion. Functoriality of restriction and compatibility with fusion are encoded in squares.
• Explicit examples: Diffeomorphism-covariant nets and massive scalar field theories admit strict vertical functoriality and pseudo horizontal structure, precisely matching the pseudo double functor paradigm.

In the theory of higher categories, pseudo double functors play a crucial role in constructing and strictifying weakly globular $n$-fold categories: Paoli (Paoli, 2016) explicitly constructs pseudo double functors as pseudo-morphisms of internal categories in $\mathbf{Cat}$, whose strictification (e.g., via Power–Lack) yields a weakly globular double category or $n$-fold category. The pseudo data is essential for capturing higher categorical weak morphisms, central to modern approaches to $(\infty,n)$-categories.

6. Examples and Strictification

Canonical examples include:

• Span$\to$Prof: The pseudo double functor from the double category of spans in $\mathbf{Set}$ to the double category of profunctors in small categories, where horizontal composition is only preserved up to the canonical coend isomorphisms, and identities correspond only up to isomorphism (Paoli, 2016).
• Path-object construction: In model structures on double categories, the path object $\mathrm{Path}(A) := [\mathbb E, A]_\mathrm{pseudo}$ uses pseudo double functors from the “free-living horizontal adjoint equivalence” double category $\mathbb E$, providing the correct homotopy-theoretic structure (Moser et al., 2020).
• Strictification: Every pseudo double functor admits strictification (Paoli–Power–Lack), producing a weakly globular double category and an equivalent strict double functor, preserving the essential weak morphism data in a strict setting (Paoli, 2016).

7. Significance and Structural Properties

Pseudo double functors enable:

• The formulation of model structures on double categories compatible with the classical model structures on $2$-categories and bicategories, recovering their homotopy theory via full subcategory embeddings (Moser et al., 2020).
• Rigorous double-categorical semantics in AQFT, resolving the obstruction to genuine functoriality in the presence of multiple compatible compositions (Komalan, 12 Jan 2026).
• Essential data for the study and classification of higher categorical morphisms, strictification procedures, and, by extension, models of $(\infty, n)$-categories (Paoli, 2016).

A plausible implication is that in domains where multiple systems of composition interact but are not strictly compatible, pseudo double functors provide the only viable categorical foundation for capturing such weakly functorial structures. Their coherence data are indispensable for both the mathematical and physical applications demanding this flexibility.