Uemura's Bi-Initial Characterisation

• The bi-initial characterisation provides a universal property framework for describing generalized algebraic theories and pseudo-functor models via unique 1-cell and 2-cell mappings.
• It enables modular construction methods, such as reducing arbitrary GATs to two-sorted theories with adjoint relationships and precise coreflections between model categories.
• Key applications include establishing bi-representations, bi-adjunctions, and weighted bi-limits, offering foundational insights for syntactic and semantic translations in higher-categorical contexts.

Uemura’s bi-initial characterisation provides a universal property framework for understanding generalized algebraic theories (GATs) and their models, as well as the representation theory of 2-categorical and double-categorical pseudo-functors. The essential insight is that certain structures (notably categories of GATs or elements of pseudo-functors) can be described via bi-initial objects in specifically structured 2-categories. This approach supports modular construction and reduction results, such as the reduction of arbitrary GATs to two-sorted theories with adjoint relationships at the level of models (Clingman et al., 2020, Avrillon et al., 27 Jan 2026). The following sections elaborate the categorical foundations, formal definitions, main theorems, proof strategies, and key applications of Uemura's bi-initial characterisation.

1. Bi-Initial Objects in 2-Categories

A bi-initial object in a 2-category $\mathcal{K}$ is an object $I$ such that for every $A\in \mathcal{K}$, the hom-category $\mathcal{K}(I,A)$ is equivalent to the terminal category. Concretely, this means there exists at least one 1-cell $I\to A$, and between any two such 1-cells, there is a unique invertible 2-cell. This generalizes the usual notion of initial object in 1-category theory, accommodating the presence of nontrivial 2-cells in higher-categorical contexts (Clingman et al., 2020, Avrillon et al., 27 Jan 2026).

The formal bi-initiality property thereby provides powerful uniqueness (up to equivalence) and existence conditions for morphisms out of $I$ into any other object.

2. Double and 2-Categorical Bi-Initiality

In the setting of double categories, a double bi-initial object $I$ is characterized by the universality of horizontal arrows and squares emanating from $I$. Precisely, for every object $A$ in a double category $\mathbb{A}$, there is a horizontal arrow $I\rightarrow A$, and for every vertical $u:A\Rightarrow B$ and any pair of such horizontals, there exists a unique square filling the diagram.

The equivalence of double bi-initiality and bi-initiality in associated 2-categories is made precise: $I$ is double bi-initial in a double category $\mathbb{A}$ if and only if $I$ is bi-initial in the horizontal 2-category $H\mathbb{A}$ and the vertical identity $\mathrm{vid}_I$ is bi-initial in the 2-category $V\mathbb{A}$. This transfer is mediated by a functor $V$ that extracts the relevant 2-categorical structure from a double category (Clingman et al., 2020).

3. Uemura’s Universal Property for GATs

Let $\mathrm{CartExp}$ denote the 2-category of pairs $(\mathcal{C},p)$, where $\mathcal{C}$ is finitely complete and $p:Y\to X$ is an exponentiable morphism. Uemura established that $(\mathrm{FinGAT},\,p_\mathrm{GAT})$—where $\mathrm{FinGAT}$ is Cartmell's category of finite GATs, and $p_\mathrm{GAT}$ is the canonical projection—is bi-initial in $\mathrm{CartExp}$ (Avrillon et al., 27 Jan 2026). This means:

• For any $(\mathcal{C},p)$ in $\mathrm{CartExp}$, there exists a finite limit-preserving $F:\mathrm{FinGAT}\to\mathcal{C}$ with $F(p_\mathrm{GAT})\cong p$ that preserves pushforwards along $p_\mathrm{GAT}$.
• Any two such functors are uniquely isomorphic via an invertible 2-cell.

This categorifies the initiality property of algebraic theories and extends it to the context of GATs indexed by exponentiable structure.

4. Applications: Bi-Representations, Adjunctions, and Limits

Uemura’s bi-initial framework enables categorical characterizations of representability, bi-adjunctions, and bi-limits:

• Bi-representation of pseudo-functors: Given a 2-category $\mathcal{C}$ and a normal pseudo-functor $F:\mathcal{C}^{op}\to\mathrm{Cat}$, a bi-representation is a pair $(I,\rho)$ with $I\in \mathrm{Ob}(\mathcal{C})$ and a pseudo-natural adjoint equivalence $\rho: \mathcal{C}(-,I)\simeq F$. Such a bi-representation exists precisely when there is a bi-initial object $(I,i)$ in the relevant 2-category of elements $\mathrm{El}(F)$. This equivalence extends to double categories via the correspondence between double bi-initial and bi-initial objects (Clingman et al., 2020).
• Bi-adjunctions: A normal pseudo-functor $L:\mathcal{C}\to\mathcal{D}$ has a right bi-adjoint if for every $D\in\mathcal{D}$, the object $(R D, \varepsilon_D)$ is bi-terminal (the dual of bi-initial) in an appropriate slice category.
• Weighted bi-limits: A weighted cone $(X,\lambda)$ on a pseudo-functor $F$ with weight $W$ is a bi-limit if it is bi-terminal in a pseudo-slice double category; this reduces to a statement about bi-initiality under certain tensorial conditions.

5. GAT Reduction and Section-Retraction Coreflection

The bi-initial property of $(\mathrm{FinGAT}, p_\mathrm{GAT})$ is used to establish that any generalized algebraic theory (GAT) can be reduced canonically to a two-sorted GAT via a functorial, semantically well-behaved process. By slicing over an appropriate family GAT and invoking bi-initiality in the sliced category, one constructs a translation $\mathrm{FinGAT} \to \mathrm{FinGAT}/\mathrm{FamG}$ corresponding to "insert $U, El$" in syntax (Avrillon et al., 27 Jan 2026).

This gives rise to a strict coreflection between models: $$\mathrm{Mod}(\mathrm{GAT}) \xrightarrow{L} \mathrm{Mod}(\widehat{\mathrm{GAT}}) \xrightarrow{R} \mathrm{Mod}(\mathrm{GAT}), \quad RL = \mathrm{id},$$ where every model of the original GAT admits a unique extension to a two-sorted model and retracts back identically under $R$. The key to this construction is the functorial uniqueness and existence guaranteed by bi-initiality.

6. Significance and Modularity

The universal property encapsulated by bi-initiality supports a modular approach to the semantics and translation theory for GATs and higher-categorical algebraic structures. Instead of requiring recitation of low-level syntactic details, semantic properties follow from the categorical universal property: $\mathrm{FinGAT}$ as the free finitely complete category with an exponentiable arrow. Common constructions—such as model categories, initial models, sort reduction, and translations—derive from this property via formal categorical machinery. This provides an abstract foundation for the semantics of type theories and related algebraic systems (Avrillon et al., 27 Jan 2026).

A plausible implication is that extensions to bi-terminality and dualization yield analogous results for co-algebraic theory, bi-colimits, and op-functorial constructions, particularly given the uniformity of the bi-initial/bi-terminal framework in double categorized and 2-categorical contexts.

7. Diagrammatic Summary

Key constructions facilitated by the bi-initial characterization include the following:

Construction	Category/Functor	Universal Property Role
Model functor	$\mathrm{FinGAT}\to\mathrm{Cat}$	Sends $p_\mathrm{GAT}$ to $\mathrm{PtdSet}\to\mathrm{Set}$ uniquely up to iso
Two-sortification	$\mathrm{FinGAT}\to\mathrm{FinGAT}/\mathrm{FamG}$	Reduces GAT to two-sorted form via slice bi-initiality
Coreflection on models	$$\mathrm{Mod}(\widehat{\mathrm{GAT}})\to\mathrm{Mod}(\mathrm{GAT})$$	Induces section-retraction by pullback and adjunction

The bi-initial characterisation thus organizes much of the foundational algebraic structure required for semantic and syntactic manipulation of generalized theories and higher categorical constructs (Clingman et al., 2020, Avrillon et al., 27 Jan 2026).