Categories of Games in Category Theory

• Category of games is a framework that formalizes game structures using objects and morphisms to retain moves, strategies, and invariants.
• It connects combinatorial, logical, and computational models, enabling systematic analysis of game composition and equivalence.
• This approach has broad applications including semantics in programming languages, network game theory, and analysis of equilibria.

A category of games formalizes combinatorial, economic, logical, or computational games within the framework of category theory, making explicit the objects (games or game-like structures) and the morphisms (maps that preserve the essential features of games, such as moves, strategies, player roles, or structural invariants). The concept plays a central role in connecting game theory, logic, semantics, and algebraic methods, offering a common language to paper composition, equivalence, and transformation of games across diverse research paradigms.

1. Structural Paradigms for Categories of Games

A wide range of categorical frameworks for games exist, reflecting differences in the underlying mathematical model (sequential or concurrent play, combinatorial versus economic focus), the formalization of strategies and morphisms, and the foundational role played by resources and logics.

• Arenas and Event Structures: In classical and concurrency-aware game semantics, games are modeled as arenas (or event structures), encoding possible moves (as events), their polarity (Player/Opponent), causality relations, and sometimes symmetry (to express uniformity in replicated resources). Categories such as Tcg (thin concurrent games) or Cho (concurrent Hyland-Ong games) are constructed by taking objects as these structured arenas and morphisms as strategies respecting the event, causality, and symmetry structures (Castellan et al., 2014).
• Graph-based Models: In the modeling of network games or impartial combinatorial games, games may be represented as graphs, rulegraphs, or gamegraphs, with morphisms as (option, strategy, or payoff) preserving maps, generalizing notions found in universal algebra (Bašić et al., 2023, Lavore et al., 2020).
• Coalgebras and Recursive Structure: Categories of games can also be defined via coalgebraic structures, e.g., identifying games with recursive coalgebras of a finitary powerset functor, yielding a locally finitely presentable, symmetric monoidal closed category equipped with internal homs, sum, and product operations (Hora, 27 Oct 2025).
• Signature Games and Comonads: For games over relational structures (e.g., Spoiler-Duplicator games in finite model theory), the category is built from comonads in a bicategory of signature games (event structures with logical winning conditions), and morphisms are formulated as concurrent strategies or spans, depending on game polarity (Montacute et al., 18 May 2024).
• Interaction Systems: Some frameworks dispense with strategies, instead taking as morphisms simulation relations between systems, leading to a strongly synchronous symmetric monoidal closed category with models of linear logic (0905.4062).

2. Morphisms: Strategies, Simulations, and Structure-Preserving Maps

The notion of morphism depends crucially on the category’s focus.

• Strategies: In classical game semantics, morphisms are commonly (winning, possibly innocent, possibly history-free) strategies, seen as sets or presheaves of plays or interaction sequences, with composition defined via pullbacks, Kan extensions, or interaction/hiding constructions (Eberhart et al., 2017, Castellan et al., 2014).
• Option-Preserving Maps: For digraph-based models (rulegraphs/gamegraphs), morphisms are functions that respect the option structure: $\alpha : \mathsf{G} \to \mathsf{H}$ satisfies $\Opt(\alpha(p)) = \alpha(\Opt(p))$ (Bašić et al., 2023).
• Simulations: In simulation-based frameworks, morphisms are relations $r \subseteq S_1 \times S_2$ satisfying path-lifting/simulation conditions rather than specific win criteria or chronological play (0905.4062).
• Coalgebra Homomorphisms: In coalgebraic categories, morphisms are coalgebra homomorphisms preserving allowed transitions, often with additional properties such as path-lifting or recursive solution preservation (Hora, 27 Oct 2025).
• Synchronous Correlations: In categories modeling nonlocal games, morphisms are synchronous correlations between input and output sets (e.g., stochastic matrices), with categorical properties (monomorphism/epimorphism) characterized in terms of matrix singularity (Lackey et al., 2018).

3. Algebraic and Categorical Features

Categories of games typically exhibit rich algebraic structure, with features dependent on their foundational definitions:

Property	Manifestation in Categorical Game Models	Reference Examples
Symmetric monoidal closed	Tensor product (sum, parallel play); internal homs (function/strategy space)	(Hora, 27 Oct 2025, Castellan et al., 2014)
Compact closed structure	Important for diagrammatic representation and feedback/tracing discipline	(Watanabe et al., 2021, Castellan et al., 2014)
Locally finitely presentable	All small (co)limits exist; filtered colimits; finite games finitely presentable	(Hora, 27 Oct 2025, Bašić et al., 2023)
Subobject classifier/Epi-mono fact.	Enables internal logic, inclusion of subgames, and logical analysis	(Hora, 27 Oct 2025)
Cartesian closed/Extensive	Internal exponentials; well-behaved coproducts; crucial for modeling higher types and infinite games	(Duzi et al., 7 Jan 2024)

Monoidal and closed structure is realized via sums/products of games (disjoint union, synchronous play), internal homs as game spaces (function types), and categorical distributivity and composition reflect the operational semantics of sequential and parallel play.

4. Functoriality, Compositionality, and Equivalence

A central feature is the paper of functoriality and compositional semantics:

• Compositionality: Functors from syntactic categories (e.g., pregroup grammars, open graph categories) to semantic categories of games support modular construction and analysis of complex games by composition of game fragments (Lavore et al., 2020, Hedges et al., 2018).
• Adjunctions and Comonads: The systematic construction of game comonads via adjunctions canonically lifts structures from relational or algebraic settings to game-theoretic contexts (Montacute et al., 18 May 2024).
• Equivalences: Categories such as impartial rulegraphs, gamegraphs, and recursive coalgebras are shown to admit equivalence up to maximal congruence or minimal quotient, justifying the traditional recursive set-of-options model (Bašić et al., 2023). Categories of infinite games admit arboreal, functorial, and metric equivalences, e.g., to pruned trees, presheaves, or ultrametric spaces (Duzi et al., 7 Jan 2024).

5. Categorical Analysis of Equilibria and Logical Invariants

The categorical viewpoint enables generalizations and systematic treatment of equilibrium and property preservation:

• Game-Theoretic Invariants: Isomorphisms in categories of extensive-form games are characterized as bijections/homeomorphisms preserving nodes, information, players, and ordinal utility. Nash and subgame-perfect equilibria, strategy sets, perfect-information, and no-absentmindedness are all invariant under categorical isomorphism (Streufert, 2021).
• Congruence and Quotients: Categorical analogues of universal algebra support manipulations of games by congruence relations, quotients, and isomorphism theorems (Bašić et al., 2023).
• Logical Equivalences: Comonadic categories systematize classical Spoiler-Duplicator games, generalizing logical equivalence, bisimulation, and combinatorial invariants such as treewidth and chromatic number (Montacute et al., 18 May 2024).
• Equilibria-Preserving Morphisms: Specialized subcategories (e.g., games/morphisms preserving Nash equilibria) capture strategic invariance under transformation, supporting the modular paper of game-theoretic phenomena (Tohmé et al., 2023).

6. Applications and Theoretical Impact

The categorical analysis of games has broad impact, encompassing:

• Finite Model Theory and Logic: Categorical frameworks unify and generalize sophisticated equivalences and combinatorial invariants arising from Spoiler-Duplicator and Ehrenfeucht-Fraïssé games, introducing new tools for finite model theory (Montacute et al., 18 May 2024).
• Semantics of Programming Languages: Categories of games provide denotational models for linear logic, simply-typed and untyped lambda calculi, and dependent type theory, modeling complex computational phenomena including concurrency, state, and quantum computation (Castellan et al., 2014, Hora, 27 Oct 2025, Abramsky et al., 2015, Abramsky et al., 9 Apr 2024).
• Network Game Theory: Functorial and compositional approaches support scalable, modular construction and analysis of games played on distributed structures, including computation of equilibria via network decomposition (Lavore et al., 2020).
• Topological and Infinite Games: Categories of infinite games (as trees/presheaves/metric spaces) generalize Banach-Mazur games and provide new perspectives on classic covering/convergence dualities in topology (Duzi et al., 7 Jan 2024).
• Nonlocal and Quantum Games: Categories of nonlocal games with morphisms as quantum/correlated strategies allow for resource-theoretic reasoning and precise mathematical control over transformation, equivalence, and decomposition (Lackey et al., 2018).

7. Summary Table: Key Categorical Categories of Games

Reference	Category Type/Name	Objects	Morphisms/Maps	Structural Highlights
(Hora, 27 Oct 2025)	$\mathbf{Games}$ (recursive coalgebras)	Games as finite digraphs	Coalgebra homomorphisms	Locally finitely presentable, monoidal closed, subobject classifier
(Castellan et al., 2014)	Tcg/Cho (event-structural)	(Thin) event structures, HO arenas	Symmetry-respecting, thin, single-threaded strategies	Compact/cartesian closed, concurrency, symmetry
(Bašić et al., 2023)	$\mathbf{GGph}$, $\mathbf{RGph}$	Digraphs (gamegraphs/rulegraphs)	Option-preserving maps, congruence quotients	Universal algebra analogies, enumeration
(Montacute et al., 18 May 2024)	$_\delta$, bicat. of sig. games	Relational structures/signature games	Comonad-based strategies, coKleisli maps, spans	Systematic comonadic construction
(Streufert, 2021)	$\mathbf{Gm}$	Extensive-form games (trees + info)	Node-preserving, continuous, utility-order preserving maps	Invariants: isomorphism, Nash/subgame equilibrium
(Duzi et al., 7 Jan 2024)	$\mathbf{Game}_{A}$, $\mathbf{Game}_{B}$	Pruned trees + winning sets	Chronological, outcome-preserving maps	Complete, cartesian closed, Banach-Mazur universality
(Eberhart et al., 2017)	Game settings	Arenas/play categories	(Presheaves of) plays/interactions	Abstract categorical framework, innocence
(Lackey et al., 2018)	FinSet$^S$ categories	Finite sets	Synchronous correlations (quantum/etc)	Matrix algebra, detailed factorization

This structural diversity reflects the category of games' central position in contemporary logic, computer science, and game theory, as a tool for unification, abstraction, and modular analysis.