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Higher Categorical Cryptoeconomics: HoTT Blockchains, Petri Net
Computad Ledgers, Coregulator DAOs, and Cohomology ZKPs
Melanie Swan1, Takashi Kido2, Renato P. dos Santos3
1University College London
2Teikyo University
3academicum.ai
melanie@DIYgenomics.org, kido.takashi@gmail.com, info@academicum.ai
Blockchains are formal systems for equipping objects with value, transacting their
exchange, and creating domain-specific event histories. Categorical
cryptoeconomics is the application of category-theoretic methods to blockchain
study with formalisms which pertain to blockchains and generalize to the
programmable computational infrastructure more broadly. Section 1 provides an
overview of twenty categorical cryptoeconomic primitives (in algebraic topology
(persistent cohomology, semitopology), logic, sheaves, set theory, group theory,
optics, and blockchain Petri nets) and their use in consensus, ledger construction,
mining, and smart contract platforms. Section 2 introduces four progressively
higher categorical cryptoeconomic formulations: HoTT (homotopy type theory)
blockchains, Petri net computad ledgers, coregulator DAOs (decentralized
autonomous organizations), and cohomology ZKPs (zero-knowledge proofs). The
progression is first, nodes as themselves simplicial sets, fibrations, and 2-Segal
spaces, second, nodes switched as gradients, third, time-modulated node and path
propagation, and fourth, physics-agnostic node and path multiplexing. A 2-
category of smart network technologies is envisioned with object instances of
blockchains, AI, deep learning, robotics, autonomous vehicles, and digital biology
health twins, and morphisms as structure-preserving functors.
Keywords: blockchain, cryptoeconomics, formal systems, model theory, type theory
1 Introduction: Universal Categorical Primitives
Category theory provides a rich conceptual frontier with standardized tools for deployment to
various areas of digital transformation as follows. Modern technologies in the computational
infrastructure must treat all physical scales: relativistic, classical, and quantum (e.g. quantum-
secure cryptography, relativistic quantum ZKPs), and are aided by categorical formulations
[Gorard, 2023; Baez, 2006]. The computational infrastructure entails the digitization of fields of
enterprise into computable formats, with the standard categorical primitive of symmetric
monoidal categories (SMCs), a tensor product-based structure. Diagrammatic methods allow the
design of computation-ready systems, including with monads (many inputs to one output) and
PROPs (many inputs to many outputs). Computational infrastructure technologies treat data and
math, hence graph-theoretic, network science, and database management methods imply the
standard categorical prims of optics, lenses, Frobenius formulations (combining and splitting),
and Hausdorff spaces (separated spaces, non-overlapping neighborhoods). Sheaves and
presheaves are a standard “glocal” primitive linking the local and global levels of a system
(glueing together local spaces to form a global structure). Category theory is an approach to
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dynamics with rate change analytics, system evolution by Hebbian and Markovian processes,
and open systems engaging with the environment. Systems are complex and multiscalar, possibly
having different physics regimes at different scale tiers.
Category theory is a system of interfaces which provides a duality viewpoint in that it
may be more convenient to look at a system projected in the formal method of another system
(e.g. AdS/CFT). Category theory was invented to bring algebra and topology together in
algebraic topology [Eilenberg & MacLane, 1945]. With a duality lens, it is possible to transport
information about a topological space into the language of algebraic invariance, which is the
only way to study certain spaces, and then take those inferences, and project them back into
topology, and create new spaces by manipulating the invariants.
Economic design principles are pervasive in category theory with resource theories used
in various processes of identifying needs, routing resources, assessing the cost, tacking
transaction ledgers and event histories. Thus, categorical cryptoeconomics is implied in the
formal digital instantiation of objects equipped with value. Petri nets are the standard primitive
of two-tier networks with propagation and dynamics at different levels per token balances at
nodes which can be updated system wide. Categorical formulations see systems in terms of
information, energy, and entropy. A slate of quantum-ready formulations based on symmetry
(higher Galois theory, space-time branching) incorporate operads ready to act on systems.
2 Categorical Cryptoeconomics
Categorical cryptoeconomics is the application of category-theoretic methods to problems in
blockchains including cryptoeconomics (economic design principles instantiated in blockchain
networks) [Swan, 2024]. There is a growing area of formal methods synthesis as categorical
methods are being applied to blockchains in various areas such as consensus, ledger design,
smart contract interoperability, escrows, and mining networks (Table 1). As in other
computational infrastructure technologies (e.g. AI, deep learning, quantum computing), the
motivations for categorical formulations are both on the practical side of more efficient secure
deployment and on the theoretical attempt to elucidate foundations. Theoretical foundations are
pursued through attempts to express blockchains with first-order logic and an examination of
algebraic structure. Another theme is topology, with methods such as persistent homology
(topology-associated data), cohomology, semitopology, sheaves, and topoi. Finally, set theory,
group theory, and optics; all methods of data instantiation and manipulation are engaged.
Table 1. Categorical Cryptoeconomic Primitives (representative list).
Blockchain Topic Innovation Categorical Methods Reference
1
Consensus
Topos
-
based C
onsensus
P
rotocols
Copresheaf toposes,
Heyting
a
lg
ebras
Lambert
2
Consensus
Proofchains as Sheaves
First
-
order logic, global coherence
Murfet
3
Consensus
Poincar
é
Protocol Manifolds
Cohomology, type theory
M
eldman
4
Consensus
Distributed Collaborative Action
Semitopology
Gabbay
5
UT
x
O
Ledgers
Blockchain Algebras
Alg
top, homomorphism, type th
eory
Gabbay
6
Ledger
design
DLT Resource
Convertibility
SMC d
iagrams
,
proarrow equipments
Nester
7
Ledger design
Contract legal
-
execution automata
Temporal logic, abstract query lang
B
ottoni
8
Chain construction
DLT Monoidal Equational Theory
PROPs, DAG algebras
Fiore
9
Chain construction
Blockchain Axiomatization
First
-
order fo
rmal
l
ogic
Goncharov
10
Mining networks
Blockchain
Symmetry Group
Group theory/Algebraic cosets
Zhao
11
Escrows
Profunctor Escrows
Optics/Comonoids
Genovese
12
Price movement
ChainNet: Bitcoin price dynamics
Persistent homology (topology)
Abay
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13
Interoperability
JAM (join
-
accumulate machine)
Power sets/
Set
t
heory
Wood
UTxO (unspent transaction output), SMC (symmetric monoidal category), DLT (distributed ledger technology)
Consensus. The first main application area of categorical cryptoeconomics is consensus.
Consensus protocols are the automated algorithmic process by which decentralized network
nodes come to agreement about a state change in the ledger. There are various blockchain
consensus models. The most well-known are Bitcoin’s proof-of-work (accounting (mining)
nodes provably engage in cryptographic puzzle-solving in order to propose a new transaction
block), Ethereum’s proof-of-stake (nodes commit resources with a large penalty for malicious
behavior and are randomly assigned to block-making which is validated by other nodes),
Filecoin’s proof-of-storage (time-stamped hourly hash proofs are provided by network nodes
attesting to the amount of storage space provided), and validators (on enterprise (private) chains,
a trusted tier of nodes (e.g. universities or other third parties) is randomly assigned to package
and validate new blocks).
As in many categorical formulations, categorical cryptoeconomics also makes the
argument that topological shapes convey logic that can be studied and deployed is a basic
assumption. The categorical formulations of blockchain consensus generally aim towards a
provably formal instantiation that is computationally implementable (often via symmetric
monoidal categories) and scales to more abstract levels.
One project proposes topos-based consensus protocols, using copresheaf toposes and
Heyting algebras [Lambert 2021]. The concept of “estimate safety” within consensus protocols is
introduced, in abstract correct-by-construction protocols as a forcing statement in the internal
logic of a given topos. The modal semantics advance is illustrated in the setting of copresheaf
toposes and Heyting algebras (logical version of Boolean algebras) in which a statement
(consensus state) and its negation cannot both be safe (true) simultaneously.
Second, another project establishes connections between blockchain consensus and
Grothendieck topoi, modeling distributed ledgers as coherent systems of local models [Murfet,
2019]. The project proposes the concept of Proofchains as Sheaves in which a proofchain
implements logical order to reach global coherence (distributed consensus). A proofchain is a
consensus protocol modulated with first-order logic.
A third project proposes a blockchain cohomology formulation with algebraic topology
for scalable verification in consensus protocols [Meldman, 2018]. The concept of blockchain
sheaves is introduced to model the distribution and interaction of data across different parts of
the blockchain network. The notion of a Protocol Topology defined on blockchain sheaves is
extended to propose a Protocol Manifold in the form of Poincaré consensus protocol manifolds.
The work uses a homological formulation to demonstrate that the topological framework can be
implemented within a type system for monadic (computable) execution.
Fourth, “glocal” (global-local) consensus in distributed systems has many applications
beyond the computational infrastructure. A blockchain-based semitopology consensus process is
proposed for decentralized decision-making with actionable coalitions as a foundational
cryptoeconomic primitive. An actionable coalition is a group of participants in a distributed
system that can collaborate to update their local state by taking a distributed collaborative action
[Gabbay & Losa, 2024]. The concept is implemented in a semitopological framework which
generalizes point-set topology by removing the restriction that intersections of open sets need to
be open. An actionable coalition depends on the actions being modeled.
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Ledger Design and Operation. The second main application area of categorical cryptoeconomics
is ledger design and operations. One project provides a rigorous algebraic framework for
modeling UTxO blockchains (using monoids, groups, and homomorphisms) [Gabbay, 2021].
Ledger architectures are either account-based state machines (e.g. Ethereum) or UTxO (unspent
transaction outputs) models (e.g. Bitcoin) in which subsequent transactions are composed from
the available balances (outputs) of previous transactions). The EUTxO (extended UTxO in which
outputs have additional data and logic (e.g. Cardano)) model is generalized into a blockchain
algebra representing the system of equations as four type equations (input, output, transaction,
and validator (code-script check of data-key pairs) and as an algebra denoted as an abstract
chunk system (chunks are blockchain segments which have monoidal structure) [Gabbay, 2021].
The concrete type equations and the abstract algebra are formalized as categories, with weakly
equivalent (adjoint) functors mapping between them in each direction. Systems such as EUTxO
arise as a loop of embeddings cycling between the concrete and abstract algebras.
A second ledger design project uses diagrammatic reasoning and compositional theories
of resource convertibility to design blockchain ledgers [Nester, 2020]. The string diagrams
format allows multi-tier reasoning about resource convertibility and ownership dynamics in
blockchains, and is computationally implementable through symmetric monoidal categories. The
method offers a rigorous mathematical representation of the material history and ownership of
virtual assets through augmented string diagrams. The project is further extended to capture
concurrency with resource transduction formalized by proarrow equipments (2-categories with
additional categorical structure (proarrows, e.g. adjunctions, monads, Kan extensions) within the
2-category) [Nester, 2023]. Bookkeeping ledgers have long been recognized as a distributed
system concurrency problem and other categorical formulations have been proposed such as
bicategories in a spanning graphs, directional value, and symmetric monoidal bicategory
instantiation of partita doppia (double-entry bookkeeping) [Katis et al., 2008].
Smart contract ledger design presents a distinct and complex challenge compared to
cryptocurrency blockchains, especially in terms of blockchain security. One team uses formal
temporal logic to formulate automated smart contracts distinguishing between the two tiers of
legal automata and execution automata [Bottoni et al., 2021]. Smart contract platforms (e.g.
Ethereum) function as dynamic ledgers that retain complete transaction histories. The platforms
have potential for task automation, however the ad-hoc complexity of their current
implementations often results in vulnerability. The proposed approach models contracts as finite-
state automata to clearly establish resource allocations and legal obligations, enhancing
reliability through stronger connections to legal semantics. Temporal logic forms the foundation
of a new abstract query language, which queries contract execution progress over time. The
model conceptualizes resources and actors within contracts as elements that are tracked through
states and transitions, enabling the automation of contract execution within the ledger’s
dynamics, including event modeling with tokens representing events that impact contract states.
The comprehensive temporal logic distinguishes between legal automata, which represent formal
obligations, and execution automata, which represent actions taken to fulfill contract terms.
Chain Construction and Mining Networks. Three projects propose categorical methods for the
formalization of chain construction (how the chain of blocks is automatically assembled) and
optimal mining network structure. The first chain construction project applies PROPs (product
and permutation) algebras to formalize DLT DAG (directed acyclic graph) input-output
operations [Fiore & Campos, 2013]. DAG DLTs is a ledger model used in smart city IoT
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infrastructure. One such project is IOTA, a DLT relying on users for the mining function in that
submitting a transaction first requires validating two-other randomly assigned transactions.
Rather than the centralizing method of “select longest chain” used by cryptocurrency chains to
converge on one account of reality, IOTA is highly decentralized as multiple messages can be
attached by different processing nodes in parallel. The ledger tangle growth architecture provides
an interesting mathematical problem which researchers model as a directed acyclic graph
(DAG), growing in multiple directions with newly arriving blocks as vertices. A categorical
interpretation of the Tangle could be explored with PROP (product and permutation) algebras as
the interface for DAG input-output operations [Fiore & Campos, 2013]. The second chain
construction project axiomatizes the “select longest chain” ledger construction method with first-
order logic [Goncharov & Nechesov, 2023]. Another project analyzes how the mining network is
self-configured and the logic of the longest change is investigated in category theoretic projects.
A blockchain group theory is proposed as a formal structure describing mining pool selection
[Zhao, 2020]. Mining nodes are instantiated as symmetric groups (groups consisting of all
permutations of a set) studying their algebraic properties in coset structures and element orders.
Escrows, Price Dynamics, and Interoperability. Categorical formulations aid other blockchain
formalism projects as well. One project deploys optics and lenses as categorical formulations
comprised of two-way data accessors used for bidirectional transformation (updating) in
functional programming. Lenses are a kind of optics which enable the ability to “focus” or zoom
in on a subfield within a larger area to view and update its state (for example using ViewStreet
and UpdateStreet functions). Since they involve two-way exchange, optics are suggested as a
formalization of cryptographic escrows (trading protocols in a trustless environment) [Genovese
et al., 2024]. Optics are defined on a monoid and comonoid for cryptographic escrows as they
can simultaneously model and manage the flow of information and value in a secure and
structured manner. A monoid is an algebraic structure with a single associative binary operation
and an identity element. In a cryptographic escrow, the monoid structure represents the
aggregation of transactions or data. For example, the combination of multiple payments into a
single escrow account can be modeled using a monoid, where the binary operation is the addition
of payments, and the identity element is the zero payment (no transaction). A comonoid is the
dual of a monoid, with operations that can be thought of as splitting or copying data. In a
cryptographic escrow, the comonoid structure can represent the distribution of funds or data from
the escrow account to multiple recipients. The comultiplication operation models the splitting of
the escrowed amount into parts, and the counit represents the finalization or release of the
escrowed funds. The algebraic properties of monoids and comonoids ensure that the operations
are well-defined and consistent, which is crucial for maintaining the integrity and security of the
escrow process. Specific morphisms (Lock, Unlock, Envelope) are used to execute the
cryptographic escrows in a Tambara module (profunctor equipped with additional structure to
interact with the action of a monoidal category). Categorical escrows provide a flexible update to
the Bitcoin scripting language version (Hash Time Locked Contracts) used for conditional
payments in the Lightning Network.
Another project focuses on cryptocurrency pricing dynamics. As market pricing
strategies are of considerable importance to traders and other blockchain users, another project
has applied a topology approach with persistent homology to better predict Bitcoin price
dynamics [Abay, 2019]. The project develops a neural network learning graph for Bitcoin price
dynamics called ChainNet. ChainNet offers a price prediction platform based on persistent
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homology, a topological data analysis (TDA) method. The project develops a learning graph for
Bitcoin price dynamics based on topological features using the two approaches of graph fibration
and Betti sequences. The fibration approach is based on graph filtration in that the Bitcoin
transaction network is filtered with increasing thresholds of Bitcoin amounts to create multiple
realizations of the network. These realizations are merged to train a neural network model. The
Betti sequences approach uses topological summaries to capture persistent features in terms of
Betti sequences and Betti derivatives. The topological features of the blockchain learning graph
can be analyzed with persistent homology. The results over the full Bitcoin network show that
for less than seven-days ahead prediction, the Betti models bring a prediction gain of almost 40%
over baseline approaches.
Finally, interoperability between blockchains is a key need for the domain to become
more full-fledged in the computational infrastructure. Blockchains are in an AOL-Compuserve
moment in which different chains are not interoperable. One project, Polkadot, envisions a
universal platform for both on-platform blockchains (parachains) and other blockchains. A next-
generation VM, the JAM (join-accumulate machine) protocol joins transactions from multiple
chains or parachains and accumulates data into the evolving overall state of the ledger) [Wood,
2024]. The ledger format is service accounts which can perform various computations and
maintain a state, code, and balance. Different service accounts can pass the results of
computations to one another, effectively creating a network of interdependent services. A set
theoretic model is used to coordinate parachain groupings, state transition rules, and collator
(validator) selection. A category theoretic model could extend this by interpreting the JAM
architecture in a limits-colimits structure with limits to represent situations in which multiple
services that rely on a common state converge into a cohesive state, characterizing the formation
of work packages and colimits to describe the aggregation of work reports into a consolidated
state, illustrating how individual outputs are transformed into a single service state.
Petri Nets. Petri nets are emerging as a foundational primitive in category theory and beyond for
the modeling of complex dynamical systems. In the basic setup, Petri nets consist of three
elements: places (vertices), transitions (vertices), arcs (edges between places and transitions).
However, Petri nets have the additional innovation of token balances stored at nodes which can
accumulate and trigger an event, for example firing a chemical reaction. The token system has
three aspects: tokens (resource balances at nodes), markings (overall network configurations of
token balances), and firing events (processes for updating network token balances). The token
system creates a second system level (L2) at which dynamics can be studied directly. The token
instantiation and multi-level systems dynamics of Petri nets lend themselves to blockchain
interpretations as many projects take advantage of the 2-tier network structure (Table 2).
Table 2. Petri Nets and Blockchains (representative list).
Blockchain Topic Innovation Categorical Methods Reference
1
Security
Platform for attack analysis
Petri nets (attack vectors)
Shahriar
2
Security
/Dynamics
Btc address
-
transaction Petri nets
2
-
tier net: Pre
-
arcs & Post
-
arcs
Pinna
3*
Security/Collaboration
Multi
-
domain workflows
2
-
tier net: contract
-
program
Cushing
4*
Contracts/Transport
Blockchain
-
based multi
-
modal
2
-
tier net: transport
-
information
Chu
5
Model Development
R
e
-
enterable colored Petri net
s
Topological tag switching
Zaitsev
6
Security
Blockchain forensic Petr nets
Embedded Petri net transitions
Wu
7
Security/
AMMs
Contract Metaspecification
2
-
tier net: technical
-
economic
Sorensen
*Blockchain primitives for beyond blockchain use. Btc: Bitcoin, AMMs: Automatic Market Makers
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also One project uses Petri nets to develop a rigorous blockchain security analysis system
[Shahriar et al., 2020]. The system models the dynamism of a dozen different kinds of attacks
(e.g. 51%, sybil, eclipse, DDOS, DNS, selfish-mining) to identify attack vectors and
vulnerabilities. A formalization of blockchain security methods is increasingly necessary as
quantum computing becomes immanent.
A second project uses a Petri net formalism to model the network of Bitcoin addresses
and transactions [Pinna et al., 2017]. A set-theoretic description is obtained for the two
blockchain elements (addresses and transactions) to instantiate the Petri net algebraic
representation of the blockchain. In the model, when Petri net state transition firing occurs,
tokens are absorbed in places (nodes) connected with Pre-arcs and tokens are produced in places
connected with Post-arcs. The Pre matrix contains the number of ingoing arcs to transitions for
each place-transition pair and the Post matrix contains the number of outgoing arcs for each
place-transition pair. The model identifies typical behaviors associated with Bitcoin owners and
recapitulates this behavior through the firing rate of the Petri net.
Another team employs a Petri net for blockchain workflow, interaction, and security
modeling [Cushing et al., 2023]. The approach involves implementing a smart contract-based
workflow coordinator on the blockchain and employing a three-layered architecture to
coordinate off-chain tasks. The involved tokens (“economy tokens”) are modulated with
incentives to foster collaboration among workflow parties. A three-layer architecture enables the
Petri nets to coordinate the on-chain and off-chain tasks involved in incentive-integrated
workflows with e-token assignment (economic tokens) as the primitive.
A fourth project creates a generic primitive for multimodal transportation networks to use
in shipping and logistics [Chu et al., 2023]. The team uses blockchain Petri nets to develop a
single contract system for multimodal transport (e.g. sea-rail logistics), taking advantage of the
automation capability of blockchain and the two-tier Petri net levels to convey the transport
network and the information network.
A fifth team uses colored Petri nets (tokens with a data value) for the efficient simulation
of blockchain networks, especially those with more than one hundred nodes [Zaitsev, et al.,
2024]. Invoking a colored Petri net consensus protocol, the project defines a re-enterable colored
Petri net model representing a general construct for model-driven development. The benefit of
the re-enterable model is the invariance of its structure with respect to the topology of the
network, the number of attached nodes, and their software and hardware parameters, which are
represented as the marking of dedicated places. Topology tags (colored tokens) are associated to
dynamic objects and switched in the operation of the re-enterable model functioning.
Another team proposes a blockchain Petri net forensics model for tracing malicious and
illegal trades [Wu et al., 2021]. Whereas existing methods are a bit primitive regarding Bitcoin
addresses and transaction flow, Petri nets allow other important information to be included such
as transaction structure and behavior features. The blockchain forensic Petri nets allow a better
focus on address clustering and Bitcoin flow analysis.
Finally, not a Petri net project but a two-tier meta-network project models the
technological execution tier of smart contracts and the economic logic of smart contracts to
check for security and well-formedness [Sorensen, 2024]. The customization in programmable
finance means that the security surface for smart contracts is much larger than that of
cryptocurrencies. DeFi protocols allow anyone to launch unaudited and possibly malicious
contracts in which user funds can be stolen or lost through programming error. One crypto
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primitive of concern is automated market makers (AMMs), a standard decentralized exchange
(DEX) functionality. Instead of matching buyers and sellers in an order book, AMMs apply
pricing algorithms to a liquidity pool to swap user funds in trade execution, actively updating the
exchange rate between cryptocurrencies. Metaspecification is introduced as a class of formal
methods to address the slate of risks arising from vulnerabilities in incorrect and incomplete
contract specification rather than incorrect code. One such project, ConCert, proposes a smart
contract verification tool for contract axiomatization and metaspecification in the interactive
theorem prover Coq. Metaspecification addresses properties not specifically encoded in the
contract such as consistency, completeness, and economic viability. In a practical example,
metaspecification analysis identified a standard economic primitive (bonding curves which relate
the price and supply of an asset) to be lacking in Dexter2, a Tezos-based AMM program. The
ConCert project presents a framework for smart contract metaspecification as a Coq-based
formal verification tool for third-generation blockchains (e.g. Tezos, Cardano, Polkadot, Solana).
3 Higher Categorical Cryptoeconomics
Identifying and mobilizing algebraic structure found in formal systems is a first-line approach to
categorical instantiation portable to computational substrates and higher formulations. Ongoing
progress in mathematical research frontiers articulates how algebraic structures may be
represented through matrices and linear transformations (e.g. Banach spaces, C*-algebras,
representation theory, KK-theory, index theory, and tempiric representations [Pisier, 2024]) and
may have interpretations in categorical cryptoeconomics. A key advance is Lawvere theory
which provides a way to describe algebraic theories using category theory (the objects represent
types or data structures and the morphisms represent functions or operations between types)
[Kock, 2023]. An application is categorical deep learning, finding that all studied examples of
geometrical deep learning neural networks are instances of the same kind of universal algebraic
structure which can be represented as a 2-category (objects are vectors spaces and morphisms are
parametric maps) [Gavranovic et al., 2024]. Similarly, categorical cryptoeconomics could benefit
from a systemic investigation of algebraic structures. Certain archetypal well-formedness
constructions in blockchains have been discovered such as zero-knowledge proofs. A categorical
approach might populate the formalism space more quickly as blockchains are defined on
manifolds and topologies with the formal structure indicating the optimal formulations for
security, efficiency, and interoperability in mining, contracts, and typed data structures.
A progression of higher categorical cryptoeconomic ideas are introduced: HoTT (homotopy
type theory) blockchains, Petri net computad ledgers, coregulator DAOs (decentralized
autonomous organizations), and cohomology ZKPs (zero-knowledge proofs) (Table 3). The
progression is first nodes that themselves are equipped with additional structure (e.g. simplicial
sets, fibrations, and 2-Segal spaces), second, nodes switched as gradients, third, time-modulated
node and path propagation, and finally, physics-agnostic node and path multiplexing.
Table 3. Categorical Blockchain Solutions to Open Problems.
Higher Categorical Blockchains Topic Advance
1 HoTT, HIT, K-theory Blockchains Interoperability Nodes are simplicial sets, fibrations, Segal spaces,
paths are higher homotopies
2
Computad Petri net Ledgers
Scaling
Nodes are gradient switched
3
Coregulator DAOs
Complexity
Nodes & paths allow time
-
modulated propagation
4
Galois cohomology relativistic ZKPs
Security
Nodes & paths allow physics
-
agnostic branching
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HoTT Blockchains and Higher HoTT (K-Theory) Blockchains. Homotopy type theory
(HoTT) is a research frontier in theoretical computer science and category theory. HoTT is a
standard primitive linking formal programming’s type theory (allowable formats) with category
theory’s homotopy (same shape analysis) [IAS, 2013]. The key insight is that objects with the
same shape have the same logical structure, and therefore can be mobilized into other
translations, duality views, and formal systems on this basis. Type theory and homotopy
mutually extend each other. HoTT extends set theory in that the homotopy level captures the
higher information in the type system (e.g. groupoids (groups with more structure) have higher
information (more structure) than sets). In order to reason effectively about type objects, it is
necessary to know their homotopy level [Awodey, 2010].
Homotopy (same shape) is a higher topological term generally meaning a 2-category or
functor relationship in that there are paths (mappings) between paths in homotopical systems,
possibly up to an arbitrary level. In homotopical systems (vs basic topological systems), the
paths are paths between paths, homotopies between paths, and higher homotopies. This means
that three higher categorical terms are somewhat synonyms: homotopy, infinity-category, and
infinity-groupoid; all connoting various levels of paths between paths, homotopies between
paths, and higher homotopies up to some level. The HoTT primitive is actualized in the
realization that types are really infinity groupoids (having the structure of higher homotopies of
homotopies between homotopies (morphisms between morphisms) which are equivalences.
Hence, HoTT’s unique features (e.g. ability to represent homotopical structures and higher-order
types) provide a more effective and nuanced representation of formal theories [Rodin, 2018].
A further representation of the higher homotopies or infinity-groupoid idea in type theory
is Higher Inductive Types (HIT). Inductive types are used to build new theories while deductive
types are used to test existing theories. Inductive types construct complex data types from
simpler ones. Higher inductive types define not only points but paths, surfaces, and higher-
dimensional topological structures (e.g. points (elements), paths (equalities between elements),
surfaces (equalities between paths), and higher-dimensional structures). The result is that HIT is
a tool for defining complex topological spaces within a type theory in a computable setting.
HoTT Blockchains. In the categorical cryptoeconomic context, HoTT blockchains means
immediately computationally implementable forms (type theory) together with the higher
homotopical exploration of arbitrarily high paths between paths (e.g. a structure like a Merkle
root). HoTT blockchains thus provide a rigorous mathematical framework for the analysis and
design of blockchain systems, addressing fundamental challenges in security, scalability, and
formal verification. HoTT blockchains use the categorical structures inherent in homotopy type
theory, particularly infinity-groupoids, to model the complex relationships within distributed
ledgers. This topological perspective allows for a more nuanced representation of blockchain
components, including transactions, smart contracts, and network topology, within a cohesive
categorical framework. The incorporation of higher inductive types enables the encoding of
intricate algebraic and geometric structures directly into the blockchain's type system, facilitating
more sophisticated reasoning about system properties.
The formal power of the HoTT Blockchains concept is visible in multiple areas: it
enables rigorous security proofs for cryptographic systems, supports advanced smart contracts
with formal correctness, and offers novel scalability solutions like sharding mechanisms and
layer-2 protocols. Additionally, HoTT's abstract categorical foundations aid blockchain
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interoperability and automated theorem proving, enhancing verification and trust in diverse
blockchain systems. HoTT blockchain examples already exist. First is the blockchain
cohomology project in which the blockchain sheaves and Poincaré consensus protocol manifolds
are subject to type-level verification; homology is used to implement the topological framework
within a type system [Meldman, 2018]. Second is the UTxOs blockchain algebra project which
represents the system of equations as four type equations (input, output, transaction, and
validator (code-script check of data-key pairs) [Gabbay, 2021].
Higher HoTT (K-Theory) Blockchains. Higher HoTT (K-Theory) Blockchains are the notion of
each transaction being its own 2-Segal space, simplicial space, or vector bundles (K-theory).
Drawing on the notion of higher inductive types defining not only point transactions but the
higher structure of paths, surfaces, and higher-dimensional topological structures, mobilizing the
notion of field in physics. The implication is rich transaction ecologies that give way to novel
structure such as measures of qualitative well-being.
Categorical Networks and Computad Petri net Ledgers. There are many innovations in
blockchain Petri nets and higher Petri nets including sigma nets, kinetic nets, and colored petri
nets (tokens with a data value). However, despite allowing one or all token dynamics treatment,
do not allow treating as gradient (as in gradient descent loss functions in deep learning) that
might be solved with higher blockchain Petri nets.
Higher-token Petri net computads are just one element towards a broader homotopy type
theory of longevity and biosystems more broadly. A homotopy type theory continues into higher-
dimensional formulations: space-valued and time-valued paths lift to paths between paths and
higher homotopies, creating an overall abstraction apparatus, one of whose proximate
consequences could be a cognitive interface to digital twin systems
Categorical Networks. Categorical network science or higher-order networks is the categorical
formulation of networks to capture relationships beyond the linear interaction of two nodes [Bick
et al., 2023]. Geometric and topological primitives (building blocks) are used such as persistent
homology (multiscalar features in topological data analysis), simplicial complexes (basic shapes
glued together), and hypergraphs (general graphs).
Higher Petri Nets. Various forms of higher Petri nets (Petri nets with additional categorical
structure) have been developed. Whole-grain nets are a geometric-algebraic dual interpretation of
Petri nets amenable to both non-fungible token (NFT) and fungible token modeling [Kock,
2023]. The geometric interpretation treats NFTs by explicitly mapping sets of tokens (with
individual bookkeeping). The algebraic interpretation treats fungible tokens using multi-sets (as
collective state transitions). Sigma-nets offer further refinement selectivity as to which non-
fungible (individual) and fungible (collective) tokens are fired [Baez et al., 2021]. Petri net
computads (inspired by categorical deep learning) and higher tokens (extended from categorical
networks) may be combined in pathway-based systems to study life sciences problems.
Petri Net Computads. The notion of Petri net computads encapsulates the space-valued computad
formulation of probabilistic molecular programming for longevity. Computational probabilistic
categories are needed to model probabilistic outcomes in longevity such as genetic mutation
impact, intervention efficacy, and population-scale programs.
Page 11
Higher Tokens. Higher tokens are the concept of a 2-category (maps between maps) extension of
Petri net token systems with blockchain primitives such as UTXO (unspent transaction output)
and ZKP (zero-knowledge proofs). The higher-dimensional 2-category of tokens may help to
reduce the complexity in whole-grain sigma-net formulations, and to enhance functionality,
offering gradient-based control and smart routing in Petri net digital twin systems. Higher-token
Petri nets could provide a Level 2 system (like the Lightning Network) for smart contract
longevity DAOs (decentralized autonomous organizations) and other DeSci (decentralized
science) applications. Higher tokens are a 2-category for longevity but connote the additional
structure of an infinity category concretized in a homotopy theory.
HoTT Petri Nets. Higher-token Petri net computads are just one element towards a broader
homotopy type theory of longevity and biosystems more broadly. A homotopy type theory
continues into higher-dimensional formulations: space-valued and time-valued paths lift to paths
between paths and higher homotopies, creating an overall abstraction apparatus, one of whose
proximate consequences could be a cognitive interface to digital twin systems. The computad
Petri net is proposed for the pathway-based modeling of longevity (in the computational digital
health twin modeling of the quantitative progress in aging analysis (twenty biomarkers, twelve
hallmarks, and eight top interventions) [Swan et al., 2025].
Coregulator DAOs. Coregulator DAOs are the proposal for a memory evolutive system (a
system retaining memory yet evolving in novel ways) interpreted in the blockchain DAO
context. A DAO (decentralized autonomous organization) is a blockchain-based dApp
(decentralized application) that is essentially an AI operated by resource-using tokens. Although
higher Petri nets allow greater functionality in certain ways, for sophisticated interoperable
landscapes of cross-chain processes, what might be helpful is the autonomous operations of the
coregulator DAO. A coregulator is a method for mapping landscapes to a colimit (master node of
objects and morphisms), making predictions about the next state, taking action, and comparing
action to prediction [Ehresmann & Vanbremeersch, 2007]. Similar to a reinforcement learning
agent updating its action policy per action-taking reward feedback, the coregulator DAO updates
its predictive action-taking per forward-looking anticipative predictions and backward-looking
memory. Further, the coregulator structure allows the morphisms to be modulated with
information and delayed or hastened in propagation. Coregulator DAOs is the idea of
implementing the sophistication of the coregulator in an automated smart contract architecture. A
Hebbian rather than Markovian learning model, MES have been proposed as a mathematical
model for longevity [Ehresmann & Vanbremeersch, 2007], personalized medicine [Simeonov &
Ehresmann, 2017], AI safety [Ehresmann & Vanbremeersch, 2023], genetic algorithms
[Mitavskiy et al., 2013], and cognition [Andreatta et al., 2023] and might also be applied to
spiking neural networks [Izhikevich, 2006].
Galois Cohomology Relativistic ZKPs. Galois cohomology relativistic ZKPs is the notion of
using cohomology (assigning algebraic invariants to a topological space) to create a physics-
agnostic proof structure in branching spacetime. Galois theory is concerned with polynomial
roots. Polynomial formulations such as Galois fields are used in cryptography (e.g. in the
Advanced Encryption Standard (AES)) as they have well-defined algebraic structure which is
readily composable into cryptographic algorithms. Galois field-based encryption has been
Page 12
proposed to provide additional layers for security and privacy in next-generation blockchains
[Lesavich & Lesavich, 2017]. Galois cohomology further pairs the idea of polynomials with
topological invariants to define fields with higher local duality [Galet, 2024]. Such advanced
treatments of fields could be helpful as various blockchain proof structures are implemented in
quantum and relativistic domains, extending existing formulations of relativistic ZKP [Shi,
2024]. Blockchains instantiate event histories but more foundationally create the atoms of
domain-specific time and space, the flexible generative capacity to create formal digital systems.
4 Risks and Limitations
The global computational infrastructure is heightening into a complex juggernaut which may be
beyond the comprehension of any one or even teams of humans. There is substantial concern
about the role of generative AI in science, business, government, and daily life. However, the
enduring complexity, fecundity, and survivability of biology on Earth is perhaps a demonstration
proof of resistance and adaption to change [Lane, 2015]. It is possible that AI agents will
continue to be the interface to the growing complexity of the computational infrastructure.
Regarding categorical approaches, the flexibility of the loosely-coupled yet formal systems of
category theory seems like a useful “hammer” to apply to almost any kind of “nail,” however,
such an attitude may result in unwarranted overapplication (or “categoricitis” [Buliga, 2019]).
5 Conclusion
The use of categorical methods may grow as the programmable computational infrastructure
continues to develop with AI, deep learning, quantum computing, blockchains, robotics, and
autonomous vehicle networks. One factor is the available tools for non-mathematicians to apply
category theory, e.g. CatColab (https://topos.institute/work/catcolab/) and Categorica [Gorard,
2024] (https://github.com/JonathanGorard/Categorica). This work provides a thematic summary
of twenty categorical cryptoeconomic projects and introduces four progressively higher
categorical cryptoeconomic primitives: HoTT (homotopy type theory) blockchains, Petri net
computad ledgers, coregulator DAOs (decentralized autonomous organizations) and cohomology
ZKPs (zero-knowledge proofs). The envisioned technical progression is one from higher
algebras (representation theory, operator algebras, KK-theory) to Lawvere theory, to inductive
HoTT, to computable formats. Blockchains comprise just one instance in the smart network
technologies 2-category which also includes AI, quantum, robotics, and autonomous vehicles.
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