# Full Text: SharedProtentions

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## Page 1

Citation: Albarracin, M.; Pitliya, R.J.;
St. Clere Smithe, T.; Friedman, D.A.;
Friston, K.; Ramstead, M.J.D. Shared
Protentions in Multi-Agent Active
Inference. Entropy 2024, 26, 303.
https://doi.org/10.3390/e26040303
Academic Editor: José F. F. Mendes
Received: 21 December 2023
Revised: 14 March 2024
Accepted: 27 March 2024
Published: 29 March 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed
under
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terms
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conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
entropy
Article
Shared Protentions in Multi-Agent Active Inference
Mahault Albarracin 1,2,*
, Riddhi J. Pitliya 1,3
, Toby St. Clere Smithe 1,4
, Daniel Ari Friedman 5
, Karl Friston 1,6
and Maxwell J. D. Ramstead 1,6
1
VERSES Research Lab and Spatial Web Foundation, Los Angeles, CA 90016, USA; riddhi.jain@verses.ai (R.J.P.);
entropy@tsmithe.net (T.S.C.S.); k.friston@ucl.ac.uk (K.F.); maxwell.ramstead@verses.ai (M.J.D.R.)
2
Département d’Informatique, l’Université du Québec à Montréal, Montreal, QC H3C 3P8, Canada
3
Department of Experimental Psychology, University of Oxford, Oxford OX2 6GG, UK
4
Topos Institute, Berkeley, CA 94704, USA
5
Active Inference Institute, Davis, CA 95616, USA; daniel@activeinference.institute
6
Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London,
London WC1N 3BG, UK
*
Correspondence: albarracin.mahault@courrier.uqam.ca
Abstract: In this paper, we unite concepts from Husserlian phenomenology, the active inference
framework in theoretical biology, and category theory in mathematics to develop a comprehensive
framework for understanding social action premised on shared goals. We begin with an overview
of Husserlian phenomenology, focusing on aspects of inner time-consciousness, namely, retention,
primal impression, and protention. We then review active inference as a formal approach to modeling
agent behavior based on variational (approximate Bayesian) inference. Expanding upon Husserl’s
model of time consciousness, we consider collective goal-directed behavior, emphasizing shared
protentions among agents and their connection to the shared generative models of active inference.
This integrated framework aims to formalize shared goals in terms of shared protentions, and thereby
shed light on the emergence of group intentionality. Building on this foundation, we incorporate
mathematical tools from category theory, in particular, sheaf and topos theory, to furnish a mathe-
matical image of individual and group interactions within a stochastic environment. Specifically, we
employ morphisms between polynomial representations of individual agent models, allowing pre-
dictions not only of their own behaviors but also those of other agents and environmental responses.
Sheaf and topos theory facilitates the construction of coherent agent worldviews and provides a way
of representing consensus or shared understanding. We explore the emergence of shared protentions,
bridging the phenomenology of temporal structure, multi-agent active inference systems, and cate-
gory theory. Shared protentions are highlighted as pivotal for coordination and achieving common
objectives. We conclude by acknowledging the intricacies stemming from stochastic systems and
uncertainties in realizing shared goals.
Keywords: active inference; phenomenology; multi-agent; category theory
1. Introduction
This paper proposes to understand collective action driven by shared goals by formal-
izing core concepts from phenomenological philosophy—notably Husserl’s phenomeno-
logical descriptions of the consciousness of inner time—using mathematical tools from
category theory under the active inference approach to theoretical biology. This project falls
under the rubric of computational phenomenology [1] and pursues initial work [2,3] that
proposed an active inference version of (core aspects of) Husserl’s phenomenology. Our
specific contribution in this paper will be to extend the core aspects of Husserl’s description
of time consciousness to group action and to propose a formalization of this extension.
In detail, we unpack the notion of shared goals in a social group by appealing to the
construct of protention (or real-time, implicit anticipation) in Husserlian phenomenology.
Entropy 2024, 26, 303. https://doi.org/10.3390/e26040303
https://www.mdpi.com/journal/entropy

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We propose that individual-scale protentions can be communicated (explicitly or implicitly)
to other members of a social group, and we argue that, when properly augmented with
tools from category theory, the active inference framework allows us to model the resulting
shared protentions formally in terms of a shared generative model. To account for multiple
agents in a shared environment, we extend our model to represent the interaction of agents
having different perspectives on the social world, enabling us to model agents that predict
behavior—both their own and that of their companions—as well as the environment’s
response to their actions. We utilize sheaf-theoretic and topos-theoretic tools from category
theory to construct coherent representations of the world from the perspectives of multiple
agents, with a focus on creating “internal universes” (topoi) that represent the beliefs, per-
ceptions, and predictions of each agent. In this setting, shared protentions are an emergent
property—and possibly a necessary property—of any collective scale of self-organization,
i.e., self-organization where elements or members of an ensemble co-organize themselves.
This paper weaves together elements that might seem relevant to fairly disparate
readerships, namely, Husserlian phenomenology, active inference modeling, and cate-
gory theory. We clarify that our primary intended audience is threefold. The primary
intended audience is composed, in part, of phenomenologists who are interested in us-
ing contemporary mathematical approaches to generate formal models of the kinds of
dynamic lived experiences that are captured by phenomenological descriptions. This seg-
ment of our readership will likely intersect with proponents of the project to naturalize
phenomenology [4,5]. Our target audience also comprises active inference modelers who
have taken an interest in consciousness and phenomenological description.
We begin by reviewing aspects of Husserlian temporal phenomenology, with a partic-
ular focus on the notions of primal impression, retention, and protention. These provide
us with a conceptual foundation to think scientifically about the phenomenology of the
emergence of shared goals in a community of interacting agents. We cast these shared goals
in terms of the protentional (or future-oriented) aspects of immediate phenomenological
experience, in particular, what we call “shared protentional goals”. We then formalize
this neo-Husserlian construct with active inference, which allows for the representation
and analysis of oneself and another’s generative models and their interactions with their
environment. By using tools from category theory, namely, polynomial morphisms and
hom polynomials, we are able to design agent architectures that implement a form of
recursive cognition and prediction of other agents’ actions and environmental responses.
Finally, we propose a method for gluing together the internal universes of multiple agents
using topoi from category theory, allowing for a more robust representation and analysis
of individual and group interactions within a stochastic environment. We use these tools
to construct what we call a “consensus topos”, which represents the understanding of
the world that is shared among the agents. This consensus topos may be considered the
mathematical object representing the external world, providing a unified framework for
analyzing social action based on shared goals. Our integrative approach provides some key
first steps towards a computational phenomenology of collective action under a shared goal,
which may help us naturalize group intentionality more generally and better understand
the complex dynamics of social action.
2. From the Phenomenology of Time Consciousness to Co-Construction and Shared Goals
2.1. Overview of Husserlian Phenomenology of Inner Time-Consciousness
This section provides an examination of the intricate (but informal) descriptive study
of the conscious perception of internal time that was proposed by the originator of the
discipline of phenomenology, Husserl [1,6–8]. In philosophy, the term “phenomenology” is
used in a technical sense to denote a specific kind of philosophical discourse, namely, the
descriptive study of the dynamics, structure, and contents of the first-person, conscious
experience. (The term is also used, less formally, to denote the descriptions that result from
such an exercise and, more generally, to denote the way in which something discloses itself
to a conscious subject.) Husserl himself defined phenomenology as a systematic effort to

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offer precise descriptions of the essential or necessary properties that pertain to various
kinds of first-person experiences, by virtue of them being the kinds of experiences that they
are. Thus, we can think of phenomenology as an informal counterpart to mathematics that
is interested in the “essence” or essential properties of certain kinds of experience. Here,
we focus on Husserl’s description of inner time consciousness and interpretations thereof,
keeping in mind, of course, that Husserl scholarship is an active field of philosophical
research and that any given interpretation will be subject to dispute. In what follows, we
mainly draw from Husserl’s early lectures on time consciousness [6] and intersubjectivity [9]
and on formal approaches to Husserl [2,3].
We appeal to Husserl’s phenomenological descriptions of inner time-consciousness
over those of other phenomenologists, e.g., de Beauvoir or Merleau-Ponty, for a number
of key reasons. The first is practical: Husserl’s extensive body of work provides us with
what are arguably the most comprehensive, rigorously conducted, and rich descriptions of
first-person experience available in the phenomenological literature, which, in addition, are
perhaps the most amenable to mathematical formalization, as Husserl himself attempted
to do in a few key places. Although others in the phenomenological tradition, such as
Heidegger, have proposed descriptions of time consciousness, we would argue that they are
neither as descriptively rich as Husserl’s nor as amenable to mathematization. Second, we
chose Husserl’s phenomenological descriptions as a starting point because the recent project
of computational phenomenology has already been developed by formalizing Husserl’s
descriptions of inner time-consciousness using active inference modeling. So, we chose
Husserl’s descriptions in part to build on and make the most of previous work. We should
note that, following the tradition in naturalized phenomenology (e.g., [5]), we propose to
mainly use Husserl’s phenomenological descriptions to generate data to be explained using
generative modeling and bracket his antinaturalist philosophical commitments (see [4] for
a discussion).
Husserl argues that the consciousness of inner time is the fundamental form of con-
sciousness, acting as the background against which all other forms of conscious experience
are situated and unfold. What Husserl calls the “constitution” of objects of experience
(i.e., their disclosure to a perceiving subject) always presupposes the consciousness of time
as a background condition [6,10,11]. Crucially, much like other thinkers of the time, like
Bergson [12] and James [13], Husserl observes that the consciousness of time exhibits a
form of “temporal thickness”. In this view, the way in which objects are experienced always
evinces a kind of temporal depth. That is, any given experiential “now” carries with it a
dimension of the just-passed and the just-yet-to-come. Husserl posits that the stream of
conscious experience consists mostly of raw sensations or sensory data that arise in a basic,
unprocessed state; these are called “primal impressions.” Husserl refers to this primal
stratum of sensory experience as the “hyletic” data of consciousness (derived from the
Greek term for matter, hyle) [6]. These data are then “formatted”, so to speak, in accordance
with the cognitive principles governing the awareness of internal time.
In this context, “retention” refers to the aspect of time awareness that preserves the
previous state or path of the temporal object in a particular manner: as “living” and
contributing to the present “now” of perception. Husserl describes retention using the
metaphor of “sediment” that accumulates over time. What Husserl calls “protention”,
in contrast, refers to the element of time consciousness that looks ahead, so to speak,
and anticipates the immediate future state or path of a temporal object. Retention and
protention are both distinguished from the explicit recollection of a past event and the
explicit imagining of a possible future event.
Husserl contends that our experience of temporally extended objects consists of a
flow of anticipation (via protention) and fulfillment/frustration by new primal impres-
sions, which may or may not conform with what was predicted to happen. Our inner
time-consciousness consists of a dynamic process that anticipates what will be experienced
next based on what has just been experienced. Thus, the flow of time consciousness is com-
posed of a series of structured impressions. Primal impression, retention, and protention

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interact to shape the temporal flow of conscious experience. Retentions and protentions
create a framework of sorts that structures the ebb and flow of conscious experience,
which in turn affects how we perceive and anticipate events. The temporal thickness of
experience, in this view, consists of the protented aspects of experience interacting with
“sedimented” retentions.
2.2. Shared Protentions
Through the analysis of the phenomenological components of temporal consciousness,
we can gain a valuable understanding of how individuals develop collective objectives and
expectations and how these are incorporated into the previously mentioned mathematical
models. Indeed, missing from the above account are intersubjectivity and the sharing of goals
by agents in a shared life world. Husserl himself devoted much energy to thinking about
intersubjectivity [9]. Zooming out to the broader literature, the concept of sharing goals
or beliefs can be understood through several interrelated perspectives. Group members
may possess shared beliefs about the past as well as aligned expectations about the future
that are not explicitly expressed but that naturally coincide due to shared experiences or
comprehension [14]. We would equate these shared ways of anticipating and smoothly
coping with the world as being premised on a set of “shared protentions”.
We now make a terminological clarification. This paper concerns what we will call,
in a somewhat idiosyncratic fashion, “shared protentions”. It should be noted that, for
Husserl, in some sense, the structure of time consciousness, as an intentional relation
between primary content, retention, and protentions, is shared by all conscious subjects:
the structure of inner time-consciousness is invariant and identically the same for all
conscious subjects. That is, according to Husserl, inner time-consciousness must conform
to this structure by virtue of being the experience of the type inner time-consciousness.
The co-constitution of the lived world (Lebenswelt) has been discussed extensively by
Husserl and studied under the rubric of intersubjectivity by phenomenologists of various
sorts. Husserl distinguishes between the empirical and transcendental self or ego. For
Husserl, the self generically functions as one of two “poles” of conscious experience (the
“act-pole” and the “object-pole”). The transcendental self is, on the one hand, a kind of
abstract structure that any conscious experience contains intrinsically. The self is also a
thing that appears to this transcendental self; this is the empirical or personal self, which
also has a temporal structure and accumulates experiences, forming habits. This empirical
self is not just a static center of acts but evolves over time, integrating past acts into a
cohesive identity. In his later writings, in particular in the fifth Cartesian Meditation [15],
The Crisis of European Sciences and Transcendental Philosophy [16], and the extensive
notes On Intersubjectivity, Husserl extends his analysis of the self, considered through the
lens of intersubjectivity.
In Husserl’s account, the empirical self is co-constituted with others, where past
experiences become integrated into the self. Since these experiences almost always involve
other selves, their perspectives are thus integrated to the self. The self is thus a collective
achievement that emerges from a community of selves [17]. The self realizes its constitutive
role only within a network of intersubjective relations to other selves, implying an intrinsic
connection between individual consciousnesses. The self is co-constructed through this
intersubjective framework, where each self intentionally carries within itself the presence of
other selves, thereby forming a deep, communicative relationship that establishes the full
sense of the world. This communal interaction and the acknowledgment of each other’s
subjective experiences contribute to the continuous generation and reformation of the self,
emphasizing a dynamic, interrelated construction of identity and understanding. Our
individual temporal experiences, the immediate experience of the now, retention, and
protention, are not closed off to ourselves. They are open and connected to the temporal
experiences of others. This openness allows for a shared temporal framework, or “world-
time”, that encompasses not only our own temporal flow but also the temporal experiences
of others, making our individual sense of time inherently intersubjective [18].

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In the flow of time consciousness, the contents of experience become “formatted” by
retention and protention, as pure forms of time consciousness, as they well up. It is these
sedimented contents to which we refer—as shorthand—as “shared retentions” and “shared
protentions”, in the sense that the content is shared between conscious subjects.
Previous work on modeling Husserlian time consciousness with active inference
has associated these contents that are retained and protended by the pure protentional
and retentional structure of inner time-consciousness with the implicit knowledge that
is encoded in the parameters of a generative model, which captures the formal structure
of inference. This is to be contrasted with the kind of online, contentful inference that
corresponds to posterior state estimation in real time, i.e., with explicit predictions about
the future. Thus, we should note that protentions are not equivalent to explicit predictions
(neither in Husserl’s sense of formal structure nor in our sense of retained or protended
contents). The point we are making is that shared intentionality depends on shared
sedimented content; that is, in addition to having in common the generic form of inner
time-consciousness, shared protentional content is necessary for shared intentionality.
Assuming the plausibility of such a neo-Husserlian concept, group members can be
seen as actively and implicitly aligning their beliefs and expectations through dialogue
and interaction, thereby enhancing their ability to predict each other’s actions and inten-
tions [19], and thereby coming to perceive and act in the world in similar ways. Thus,
shared retentions and protentions might arise collectively within the group as shared styles
of appraising and engaging with the social world, extending beyond the individual agent’s
subjective experience and leading to a shared implicit comprehension that surpasses and
encompasses individual viewpoints.
In the following analysis, we propose to formally model these shared temporal struc-
tures, which involve resonant cognitive models and communication and which impact the
decision-making and behavior of agents within a social group.
3. An Overview of Active Inference
Here, we provide a brief overview of active inference, oriented towards application in
the setting of shared protentional goals (for a more complete overview of active inference,
see [20–22], and for its application to modeling phenomenological experience, also known
as “computational phenomenology”, see [23]).
Active inference is a mathematical account of the behavior of cognitive agents, model-
ing the action–perception loop in terms of variational (approximate Bayesian) inference. A
generative model is defined, encompassing beliefs about the relationship between (unob-
servable) causes s—whose temporal transitions depend upon action u—and (observable)
effects o, formalized as [24]
P(oτ, sτ, uτ|sτ−1) = P(oτ|sτ)
|
{z
}
likelihood
P(sτ|sτ−1, uτ)P(uτ)
|
{z
}
prior
This probabilistic model contains a Markov blanket in virtue of certain conditional
independencies—implicit in the above factorization—that individuate the observer from the
observed (e.g., the agent from their environment). Agents can then be read as minimizing
a variational free energy functional of (approximate Bayesian) beliefs over unobservable
causes or states Q(s), given by

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Q(sτ, uτ) = arg minQF
(1)
F = EQ[ln Q(sτ, uτ)
|
{z
}
posterior
−ln P(oτ|sτ, uτ)
|
{z
}
likelihood
−ln P(sτ, uτ)
|
{z
}
prior
]
(2)
= DKL[Q(sτ, uτ)||P(sτ, uτ|oτ)]
|
{z
}
divergence
−ln P(oτ|m)
|
{z
}
log evidence
(3)
= DKL[Q(sτ, uτ)||P(sτ, uτ)]
|
{z
}
complexity
−EQ[ln P(oτ|sτ, uτ)]
|
{z
}
accuracy
(4)
Agents interact with the environment, updating their beliefs to minimize variational
free energy. The variational free energy provides an upper bound on the log evidence
for the generative model (also known as marginal likelihood), which can be understood
in terms of optimizing Bayesian beliefs to provide a simple but accurate account of the
sensorium (i.e., minimizing complexity while maximizing accuracy). This is sometimes
referred to as self-evidencing [25].
Priors over action are based on the free energy expected following an action, such
that the most likely action an agent commits to can be expressed as a softmax function of
expected free energy:
P(u) = σ(−γ · G(u))
(5)
G(u) = EQu[ln Q(sτ+1|u) −ln Q(sτ+1|oτ+1, u) −ln P(oτ+1|c)]
(6)
= EQu[ln Q(sτ+1|oτ+1, u) −ln Q(sτ+1|u)]
|
{z
}
expected in f ormation gain
−EQu[ln P(oτ+1|c)]
|
{z
}
expected value
(7)
= DKL[Q(oτ+1|u)||P(oτ+1|c)]
|
{z
}
risk
−EQu[ln Q(oτ+1|sτ+1, u)]
|
{z
}
ambiguity
(8)
Here, G(π) is the expected free energy for policy u and γ is a precision parameter
influencing the stochasticity of action selection. Effectively, actions are selected based on
their expected value, which is the expected log likelihood of preferred observations, and
their epistemic value, which represents the expected information gain. The expected free
energy can also be rearranged in terms of risk and ambiguity, namely, the divergence
between anticipated and preferred outcomes and the imprecision of outcomes, given their
causes. A comparison of Equations (2) and (3) shows that risk is analogous to expected
complexity, while ambiguity can be associated with expected inaccuracy. In summary,
agents engage with their world by updating beliefs about the hidden or latent states,
causing observations while, at the same time, acting to solicit observations that minimize
expected free energy, namely, minimizing risk and ambiguity.
It is important to note that the active inference framework is not a metaphysical
approach or framework but rather a physics modeling framework. It only assumes the rest
of contemporary physics as a background (classical, statistical, and quantum mechanics)
and answers the question: given this background, what does it mean for something to
be reliably re-identifiable by an external observer as the thing that it is? It makes no
metaphysical claims per se (see [26–28]).
4. Active Inference and Time Consciousness
We have previously [1] framed Husserl’s descriptions of time consciousness in terms
of (Bayesian) belief updating, while further work proposed a mathematical reconstruction
of the core notions of Husserl’s phenomenology of time consciousness—retention, primal
impression, protention, and the constitution of disclosure of objects in the flow of time
consciousness—using active inference [2]. See also [29] for related work. Active inference

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foregrounds a manner in which previous experience updates an agent’s (Bayesian) beliefs
and thereby underwrites behaviors and expectations, leading to a better understanding of
the world.
Descriptions of temporal thickness from Husserl’s phenomenology are highly compat-
ible with generative modeling in active inference agents. To connect the phenomenological
ideas reviewed above with active inference, consider again how active inference agents up-
date their beliefs about the world based on the temporal flow structure. In active inference,
agents continuously update their posterior beliefs by integrating new observations with
their existing beliefs. This belief updating involves striking a balance between maintain-
ing the agent’s current beliefs and learning from new information. Retention and primal
impressions relate to the fact that, in active inference, all past knowledge contributes to
shaping beliefs about the present state of the world. In short, active inference effectively
bridges prior experiences with current expectations. In this setting, retentions are for-
malized as the encoding of new information about the world under a generative model.
Primal impressions are formalized as the new data that agents sample over time, forcing
an agent to update their beliefs about the world, leading to a better understanding of the
environment and allowing for more effective information and preference-seeking behavior.
Thus, the consciousness of inner time can be modeled as active inference, where prior
beliefs (sedimented retentions) are intermingled with ongoing sensory information (primal
impression) and contextualized by an unfolding, implicit, and future-oriented anticipation
of what will be sensed next (protention).
4.1. Mapping Husserlian Phenomenology to Active Inference Models
As illustrated in Table 1 and previously explored in [1], active inference comprehen-
sively maps to Husserlian phenomenology: observations represent the hyletic data, namely,
sensory information, that set perceptual boundaries but are not directly perceived. The
hidden states correspond to the perceptual experiences, namely, the data that are inferred
from the sensory input. The likelihood and prior transition matrices are associated with the
idea of sedimented knowledge. These matrices represent the background understanding
and expectations that scaffold perceptual encounters. The preference matrix evinces a simi-
larity to Husserl’s notions of fulfillment or frustration. It represents the expected results
or preferred observations. The initial distributions represent the prior beliefs of the agent
that are shaped by previous experiences. In particular, the habit matrix resonates with
Husserlian notions of horizon and trail set. Collectively, these aspects of the generative
model entail prior expectations and the possible course of action.
Table 1. Parameters used in the general model under the active inference framework and their
phenomenological mapping.
Parameter
Description
Phenomenological Mapping
o ∈O
Observations that capture the
sensory information received by
the agent
Represents the hyletic data, setting
perceptual boundaries but not
directly perceived
s ∈S
Hidden states that capture the
causes for the sensory information:
the latent or worldly states
Corresponds to perceptual
experiences inferred from
sensory input
P(o|s) = Cat(A)
Likelihood matrix that captures the
mapping of observations to
(sensory) states
Associated with sedimented
knowledge, representing
background understanding and
expectations
P(s|s−1, µ) = Cat(B)
Transition matrix that captures the
mapping for how states are likely
to evolve
Linked to sedimented knowledge,
shaping perceptual encounters

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Table 1. Cont.
Parameter
Description
Phenomenological Mapping
P(o+1|c) = Cat(C)
Preference matrix that captures the
preferred observations for the agent,
which drive their actions
Similar to Husserl’s notions of
fulfillment or frustration,
representing expected results
or preferences
P(s0) = Cat(D)
Initial distribution that captures the
priors over the hidden states
Represents prior beliefs shaped by
previous experiences and
current expectations
P(µ) = Cat(E)
Habit matrix that captures the prior
expectations for initial actions
Connected to Husserlian notions of
horizon and trail set, symbolizing
prior expectations
π
Policy matrix that captures the
potential policies that guide the
agent’s actions, driving the
evolution of the B matrix
Symbolizes the possible course of
action, influenced by background
information and values
4.2. Intersubjectivity and Intentionality
There is a body of related work on shared intentionality, intersubjectivity, and joint/shared
attention that is relevant in this context (see, e.g., [30–33]). Joint attention is defined as the state
in which two or more individuals focus on the same object or event. It involves mutual under-
standing and the coordination of attention, underpinned by cognitive processes that enable
individuals to recognize and follow each other’s gaze or attentional focus, thus establishing a
shared point of interest. Shared intentionality is the capacity to share psychological states,
including intentions, beliefs, and goals, with others in order to coordinate.
These concepts have been developed in a Bayesian framework, which is quite germane
to our own approach. Intersubjectivity here is seen as a consequence of agents conforming
to the dependence structures harnessed in the same (or similar) generative models and
of the capacity of the agents to consider events and their sequentiality in a similar way
and coordinate around them. For instance, ref. [34] has argued that shared intentionality
depends on agents having a shared representation of the joint goal and its context. This
is a key aspect of intersubjectivity, as the agents maintain beliefs about a shared goal that
they are trying to accomplish together. Agents continuously update their beliefs about the
joint goal, which act as a context based on observing each other’s actions. This involves
inferring the other’s intentions and goals based on their movements. Through this kind
of interactive inference, the agents align their representations and behaviors over time.
This models the intersubjective alignment and “coupling” of representations that enable
successful joint action in humans.
In a different study, ref. [35] investigated ant colony foraging behavior using an active
inference model. The study modeled ant behavior in a T-maze paradigm, simulating
how ants discover food sources and communicate these locations to the colony through
pheromone trails. This behavior is formalized via active inference, emphasizing the ants’
ability to make predictions and inferences about their environment based on sensory
inputs and prior knowledge. This study also illustrates how shared representations and
goals (e.g., locating food sources) facilitate coordinated actions within the ant colony. The
ants’ behavior, driven by hierarchical Bayesian inference, showcases a form of temporal
integration where actions are predictive of the future, based on the anticipated positions of
resources and the inferred intentions of other ants.
We draw partly from this work to justify our claim that the structure of temporality en-
ables coordination, but we take a significant step beyond merely sharing similar predictive
models. We suggest that agents do not just align their predictions of the future, but rather,
also integrate the predictive models of others into their own. This integration reflects a
deeper level of intersubjectivity, where the contents that are protended include anticipatory
representations of other agents’ future states of mind. Such integration is akin to a form of

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theory of mind, where understanding and predicting the mental states of others—including
their intentions and future actions—are essential for complex social interactions. It also
entails that the distinction between self and other gets blurrier as we move up a predictive
hierarchy, explaining the co-constitutive nature of the self.
4.3. An Active Inference Approach to Shared Protentions
Active inference often involves agents making inferences about each other’s mental
states by attributing cues to underlying causes [36]. The emergence of communication and
language in collective or federated inference provides a concrete example of how these
cues become standardized across agents [37]. Individual agents leverage their beliefs to
discern patterns of behavior- and belief-updating in others. This entails a state of mutual
predictability that can be seen as a communal or group-level reduction in (joint) free
energy [36,38]. When agents are predictable to each other, they can anticipate each other’s
actions in a complementary fashion, in a way that manifests as generalized synchrony [39].
This is similar to how language emerges in federated inference as a tool for minimizing
free energy across agents in a shared econiche [40].
As agents in a group exhibit similar behaviors, they generate observable cues in the
environment (e.g., an elephant path through a park) that guide other agents towards
the same generative models and nudge agents towards the same behavior. This has
been discussed in terms of “deontic value”, which scores the degree to which an agent’s
observation of a behavior will cause that agent to engage in that behavior [40]. Alignment
can thus be achieved by agents that share similar enough goals and exist in similar enough
environments, thereby reinforcing patterns of behavior and epistemic foraging for new
information [36,40]. Individual agents perceive the world, link observable “deontic cues”
to the latent states and policies that cause them, and observe others, using these cues to
engage in situationally appropriate ways with the world. These deontic cues might range
from basic road markings to intricate semiotics or symbols such as language [41].
Consider, for example, someone wearing a white lab coat. You and the person wearing
the lab coat share some similarities that lead you to believe that the coat means the same
thing to them as it does to you. You know that lab coats are generally worn for scientific or
medical purposes (sedimented retentions scaffolded by the cognitive niche). The person
wearing the lab coat is standing in a street near a hospital building. From all these cues,
without ever wearing a lab coat yourself or being a doctor, you can make a pretty good
guess that this individual is a doctor.
Similarly, gathering interoceptive cues to infer one’s own internal states can enable
individuals to make sense of other people’s behavior and allow them to infer the internal
states of others [42,43]. Within a multi-agent system, the environment is complex, encom-
passing both abiotic and social niches. The abiotic niche is the physical and inanimate
components of the environment, whereas the social niche comprises the interactions and
observations resulting from the activities of other agents. This differentiation underscores
that the environment is not exclusively shaped by agents but rather is shaped by an intricate,
self-sustaining interaction between living and non-living things. This suggests that agents
not only acquire knowledge about their environment to efficiently understand and navigate
it, but they also acquire knowledge about others and, implicitly, themselves in tandem.
These ideas provide some framing for the core question that will concern us presently: how
should we model such shared protentions? To model shared protentions effectively, the
use of category theory becomes a key epistemological resource. The mathematical preci-
sion and structural complexity of category theory provide a sophisticated framework for
comprehending the cohesive behaviors of agents with regard to shared objectives or future
perspectives. It explores fundamental relationships among things like the characteristics
of agents’ interactions, goals, protentions, and the organization of their environmental
resources. This theoretical framework allows for a formal and scalable understanding of
shared protentions, highlighting the interconnections and relational dynamics between
individuals in a complex setting.

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5. Category Theoretic Description of Shared Protentions in Active Inference Ensembles
We have reviewed active inference-based approaches to time consciousness and, in
particular, the construct of a shared protention. Here, we propose a category theoretic
formulation of shared protentions among an ensemble of active inference agents [44]
combining two main ideas: first, a notion of agent derived from categorical systems
theory [45], whose boundary (or Markov blanket) is described using polynomial functors;
second, the concept of sheaf, to account for agents with shared beliefs that may thus be
“glued together”.
We do not pretend to give a detailed mathematical exposition of either of these
concepts here and instead refer the interested reader to [46] on polynomial interaction
and [47] on the basic ideas of sheaf theory. For our purposes, it will be sufficient to know
some basic concepts from set theory (the notions of disjoint union and intersection) and the
basic definitions of category and functor, which we now review.
Categories capture the mathematical essence of composition, the process by which
many parts make a whole. A category X is thus determined by a collection of objects,
denoted X0, and, for each pair (a, b) of objects, a set X (a, b) of morphisms from a to b. We
denote such a morphism by f : a →b, and say it has source a and target b. Morphisms with
compatible source and target may be composed, so that f : a →b and g : b →c yield g ◦f :
a →c, and each object a is assigned an identity morphism, ida : a →a. The morphisms of
a category are required to satisfy two axioms: unitality, saying f ◦ida = f = idb ◦f; and
associativity h ◦(g ◦f ) = (h ◦g) ◦f, meaning we can simply write h ◦g ◦f for consecutive
composition. A basic example of a category is the category Set, whose objects are sets X
and whose morphisms f : X →Y are functions f (x) = y.
A functor is a morphism between categories. If we think of a category as like “a set
where there may be relationships (morphisms) between points”, then a functor is like
a function that preserves the structure of those relationships. Formally, if C and D are
categories, then a functor F : C →D is a mapping F0 : C0 →D0 along with a family of
functions Fa,b : C(a, b) →D(F0a, F0b), indexed by the objects a, b of C; one typically drops
the subscripts and infers them from the context. These mappings must satisfy the axioms
of functoriality: F(g ◦f ) = F(g) ◦F( f ) and F(ida) = idFa for all morphisms f, g and objects
a in C. Each object a in a category C induces a functor C(a, −) : C →Set, which maps each
object b to the set C(a, b) of morphisms a →b, and which sends each morphism g : b →c to
the function g ◦(−) : C(a, b) →C(a, c), which acts by post-composition, f 7→g ◦f. Beyond
these “representable” functors, both polynomial functors and sheaves are also special kinds
of functor.
5.1. Polynomial Functors
At school, we learn about polynomial functions, such as f (x) = x2 + 3x + 2; a poly-
nomial functor is to this concept precisely what a functor is to a function. Formally, one
merely changes the variables, coefficients, and exponents in the expression from numbers
to sets (this replacement may be seen to generalize polynomial functions if we note that
a number such as 3 may be seen to stand for a set {∗, ∗, ∗} of the same cardinality). In
an expression such as yA + By + C, we interpret the exponential yA as the representable
functor X 7→XA := Set(A, X), By as the product functor X 7→B × X and + as the disjoint
union of sets, so that altogether, the expression encodes the functor X 7→XA + B × X + C.
Every polynomial can be written in the form of a sum (disjoint union) of representable
functors, ∑i:I yp[i], for some indexing set I and collection of exponents {p[i]}, for example,
we can write By as ∑b:B y1, where 1 is the 1-element set {∗}. Therefore, we will henceforth
summarize the data of a polynomial p as
p = ∑
i:p(1)
yp[i]
where we now write p(1) for the indexing set.

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The mathematics of polynomial functors supplies a perhaps surprisingly rich for-
malism for describing interacting systems such as intelligent agents. We can think of a
polynomial p as describing the “interface” or “boundary” of such a system: each element
i of p(1) represents a possible shape or configuration that the system may adopt or the
possible actions that it may take, and each exponent p[i] represents the set of possible
“inputs” that it may expect (such as sense data), having adopted configuration i.
Because the type of expected sense data may depend on the configuration adopted
(just as you do not expect to “see” when you close your eyes), this generalizes the usual
notion of a Markov blanket in active inference to something more dynamic. We can thus
model an active inference agent with boundary polynomial p as predicting the activity of
its boundary p. That is to say, we collect the exponent sets p[i] together into their disjoint
union Σp := ∑i:p(1) p[i] and then understand the agent as predicting a distribution over the
whole set Σp. This amounts to predicting both its configurations i : p(1) (hence, its actions)
and, compatibly, its sense data in each p[i]. If we restrict each p[i] to being the same (so
there is no dependence of sense data on configuration), then we can recover the standard
Markov blanket: if we set p[i] = S to be the sense data and p(1) = A the actions, then
∑a:A S = A × S.
Being a category theoretic formalism, one does not just have objects (polynomials)
but also morphisms between them. These encode the data of how agents with polynomial
interfaces may interact, in particular, they encode how systems may be “nested” within
each other. Thus, a morphism φ : p →q encodes how a system with boundary p may
be nested within a system with boundary q, and consists of a pair (φ1, φ♯) of “forwards”
function φ1 : p(1) →q(1) (that encodes how p-configurations or p-actions are translated
into q-configurations) and a family of “backwards” functions φ♯
i : q[φ1(i)] →p[i] (that
encodes dually how q-sense data are translated into p-sense data). Polynomials and their
morphisms collect into a category: Poly.
Now, a morphism p →q represents simply nesting a p-system within a q-system; but
often, as here, we wish to consider how multiple agents form a coherent collective, which
means we need a way to encode multiple agents’ polynomials as a single polynomial. For
this, we can use the tensor of polynomial functors, p ⊗p′, which places the two interfaces p
and p′ “side by side”. Formally, we define p ⊗p′ as the polynomial ∑i:p(1) ∑i′:p′(1) yp[i]×p′[i′].
With this definition, we can understand a morphism p ⊗p′ →q as representing how
systems p and p′ come together to form a system with boundary q.
This is not yet enough for our purposes; we also wish to model systems that recur-
sively predict the beliefs of other agents in their environment. Behaviorally, this means
predicting how the other agents are going to act given their perceptions, which in turn
means predicting the patterns of interaction within the environment. And, formally, this
means “internalizing” these patterns into a single polynomial.
Thus, given polynomials p and q, we can define the corresponding hom polynomial
[p, q] :=
∑
φ:Poly(p.q)
y∑i:p(1) q[φ1(i)]
The set of configurations of [p, q] is the set of morphisms p →q, so to adopt a [p, q]-
configuration is to adopt a particular pattern of interaction.
Dually, the “sense-data” associated with a particular pattern of interaction φ is given
by the configurations of the “inner” system p and, for each such configuration i : p(1), the
corresponding sense data q[φ1(i)] for the outer system q in the configuration implied by i
via φ.
A prediction over [p, q] is thus a prediction over the set
Σ[p, q] =
∑
φ:Poly(p.q)∑i:p(1) q[φ1(i)]

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that is, a distribution over patterns of interaction Poly(p, q), inner configurations p(1), and
outer sense data ∑i q[φ(i)]. By way of an example, if we assume that the outer system q is
“closed” (with no further external environment), then that is to say that it has the trivial
interface q = y with only one configuration (“being”) and no non-trivial sense data. A
morphism p →y corresponds to a function p(1) →∑i p[i] (strictly speaking, a section of
the bundle ∑ip(1) p[i] →p(1)), which encodes how the environment responds with sense
data, given the p-system’s actions. Thus, a prediction over [p, y] is a prediction of the
environment’s response, along with a prediction of “how to act”.
5.2. Generative Models via Polynomials with Stochastic Feedback
In order to extend the standard formalism of active inference to this more general
setting, we need to add two more ingredients to the Poly mix: stochasticity and dynamics.
And to keep the presentation as simple as possible, we will focus on discrete models
(discrete in time and space). The key component of a generative model in discrete active
inference is a partially observable Markov decision process (POMDP), which encodes an
agent’s beliefs about the (stochastic) dynamics of its environment and the observations
(sense data) that these generate. POMDPs can be expressed very succinctly in the stochastic
Poly setting.
To incorporate stochasticity into Poly, one can adjust the morphisms of polynomials
so that they have stochastic backward components: stochastic feedback. Thus, in the resulting
category StochPoly, a morphism φ : p →q consists of a forward function φ1 : p(1) →q(1)
along with a p(1)-indexed family of stochastic backward maps φ♯
i : q[φ1(i)] ⇝p[i]. In the
discrete case, a stochastic map c : X ⇝Y is equivalent to a function c : X →DY, where
DY is the set of (finitely supported) probability distributions on Y; equivalently, the set of
conditional probability distributions c(y|x). Thus, φ♯is given by a p(1)-indexed family of
conditional probability distributions.
A discrete-time dynamical system with state space S and polynomial interface (“Markov
blanket”) p is a polynomial morphism ϑ : SyS →p. This consists of an “output” function
ϑo : S →p(1), which picks a configuration of the Markov blanket for every state in S, and
a family of “update” functions ϑu
s : p[ϑo(s)] →S, which update the state given inputs
corresponding to the current state.
Instantiating this recipe in StochPoly and letting p be a “monomial”, such as OyU, we
find that a system ρ : SyS →OyU corresponds to a POMDP with state space S, observation
space O, and action space U. That is, it is given by an output function ρo : S →O (which
encodes how the agent believes its observations are generated) and a transition matrix
ρu : S × U ⇝S, which may be written ρu(sτ+1|sτ, u).
Classical active inference then couples this generative model ρ : SyS →OyU, which
encodes the environment’s dynamics, with a “control” system α, which encodes the agent’s
interaction with the environment, i.e., how actions are generated. Classically, the coupled
agent–environment system is then assumed to be closed, that is, it is assumed to have the
trivial polynomial interface y. This means that, in general, the agent’s control system α is of
the form S∗yS∗→[p, y], since there is a canonical morphism [p, y] ⊗p →y. This system
α, along with the priors necessary to initialize the systems, is typically constituted by the
remaining data in Table 1.
5.3. Beyond Classical Active Inference: Multi-Agent Systems
Having sketched how active inference may be expressed using stochastic polynomial
dynamics, we see that there are natural loci for generalization. A first is that an agent
(or its environment) need not have a monomial interface; we can now move beyond the
classical POMDP framework. One way to make use of this is to describe agents who
believe themselves to be situated among other agents: a key step towards a mathematical
description of shared protentions. In such a context, a single agent need not believe simply
that they and their (amorphous) environment constitute the whole “closed” universe,
represented by the polynomial y. Rather, an agent who believes themselves among a

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group may believe that the universe is only closed when accounting for the behavior of the
whole group.
Thus, letting the agents’ Markov blankets be denoted by pj, we may replace the trivial
polynomial y with the hom [⊗j pj, y], so that a single agent’s control system αi obtains
the type S∗
i yS∗
i →[pi, [⊗j̸=i pj, y]]. It is possible to prove an isomorphism of polynomials
[p, [q, r]] ∼= [p ⊗q, r], and so αi equivalently has the interface [⊗j pj, y], and this type
signature holds for all agents in the group. Then, for consistency, each agent’s generative
model γi must have the type SiySi →⊗j pj, which means that each agent believes that the
dynamics of the hidden states depend on the actions of all the agents and, likewise, that
the hidden states determine the observations of all the agents.
Such agents thus predict not only their individual actions but those of their companion
agents, along with how the environment will respond to all of them. The foregoing analysis
can be repeated to arbitrary levels of nesting so that it constitutes a starting point for a
formal “theory of mind” and, indeed, a starting point for an account of agents that model
each other’s protentions.
5.4. A Sheaf-Theoretic Approach to Multi-Agent Systems
In the preceding two sections, we described how an ensemble of agents may predict
each other’s behavior by instantiating a family of polynomial generative models. However,
there is nothing in that formalism that pushes the agents’ beliefs to be in any way compati-
ble; they need not share protentions. Indeed, a true collective of agents should be a group of
agents that have “overlapping” world models that are sufficiently cohesive to promote the
development of common intentions among individuals.
In order to describe agents with such shared beliefs, we propose upgrading the
formalism using the mathematical tools of sheaf and topos theory. Sheaves are, in some
sense, the canonical structure for distributed data [47], and tools from sheaf theory allow
us to describe agents that communicate in order to reach a consensus [48].
In more detail, a sheaf over a topological space constitutes a systematic method of
keeping track of how “local” data or qualities, defined on open subsets, can be reliably
concatenated to represent a “global” situation. This attribute renders them highly valuable
in comprehending the varied and potentially contradictory convictions, perspectives, and
forecasts of individual agents within a multi-agent system. Sheaves enable the analysis
of both the diversity and agreement among various agents’ perspectives on the environ-
ment, and their ability to alter over time is crucial for adjusting and reacting to system
modifications. Sheaves formalize the concept of “shared experience” among agents, which
is essential for reaching a consensus on the structure of the external world.
Mathematically, a sheaf F is an assignment of data sets to a space X, such that the
assignment “agrees on overlaps”, meaning that, if we consider overlapping subsets U
and V of X, then F(U) and F(V) agree on the overlap U ∩V. A little more formally, if
we consider there to be a morphism U →U′ whenever U′ ⊆U, we obtain a category
O(X) whose objects are (open) subsets of X and whose morphisms are such (“opposite”)
inclusions. Then, a sheaf F is a functor O(X) →Set, such that, whenever U and V cover W
(as when W = U ∩V) so that there are morphisms ιU : U →W and ιV : V →W in O(X),
then, if u ∈U and v ∈V, there is a unique w ∈W such that F(ιU)(u) = w = F(ιV)(v).
The category of sheaves on X forms a subcategory Sh(X) of the category of functors
O(X) →Set.
Now, a space such as X is itself an object of a category of spaces Spc, whose morphisms
are the appropriate kind of functions between spaces (e.g., continuous functions between
topological spaces), and when Spc has enough structure (such as when it is a topos), there
is an equivalence between Sh(X) and the category Spc/X of bundles over X, whose objects
are morphisms πE : E →X in Spc and whose morphisms f : πE →πF are functions
f : E →F, such that πE = πF ◦f. We can use this equivalence to lift the models of the
previous section to the world of sheaves, as we now sketch (to see one direction of the

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equivalence, observe that, given a bundle πE : E →X, we can obtain a sheaf by defining
F(U) to be the pullback of πE along the inclusion U ,→X).
A bundle πE : E →X thus may itself be seen as representing a type (or collection)
of data that vary over the space X; for each x ∈X, there is a fibre Ex encoding the data
relevant to x. In this way, each polynomial Σp = ∑i:p(1) p[i] yields a “discrete” bundle
∑i:p(1) p[i] →p(1) that maps (i, x) to i. But the polynomials of the preceding sections are in
no way related to another ambient spatial structure. For instance, one might expect that the
internal state space S of an agent’s generative model is structured as a model of the agent’s
external environment, which is likely spatial. Likewise, the type of available configurations
p(1) may itself depend on where in the environment the agent finds themselves (consider
that we might suppose this S to also encode task-relevant information).
This suggests that the model’s configuration space p(1) should itself be bundled over
S, so that the polynomial p takes the form ∑i p[i] →p(1) →S as a bundle, or, better put,
as an object in Spc/S. In order for this to make sense, we need to be able to instantiate
the category Poly in Spc/S rather than Set. This is possible if Spc has enough structure
(it must be locally Cartesian closed, which it will be if it is a topos), as we have assumed.
Then, we can define a spatial generative model on the interface p over X to be a (stochastic)
system SyS →p as before, but now instantiated in Spc/S. Explicitly, this is a pair of maps
(γo, γu) making the following diagram commute (where deterministic maps compose after
stochastic ones by pushforward):
S × S
∑s∈S p[γo(s)]
∑i∈p(1) p[i]
S
p(1)
S
γo
p
⌟
γu
proj1
πp(1)
The commutativity of this diagram ensures, in particular, that the spatial structure of S
is respected. For example, if p(1) represents the observation space of the agent, then
observations may only be predicted “where they may be made”.
Now, in this spatially enhanced setting, we may recapitulate the polynomial theory-
of-mind of the preceding section and suppose that each agent j is equipped with a spatial
generative model of the form SjySj →⊗ipi. If we additionally suppose that the collection
of agents’ model spaces {Sj} covers a (perhaps larger) space X, then we can in turn ask
whether it is possible to glue these models Sj →⊗ipi together accordingly to form a “sheaf
of world models” W.
If it is possible, then we may say that the agents inhabit a shared universe, and thus,
with appropriate generative models, may be said to share protentions. Conversely, if it is
not possible, then we may ask, what is the obstruction? In this case, there must be some
disagreement between the agents. But sheaf theory supplies tools for overcoming such
disagreements [49,50], and thus to communicate to reach a consensus [48]. Even if the
disagreements are fundamental, it is usually possible to derive dynamics that will yield
as close to a sheaf as possible [51]. In future work, we hope to apply these methods to
multi-agent, active inference model in order to demonstrate this consensus-building.
A Note on Toposes
Sheaves collect into categories called toposes. A topos is a category that has both
a spatial and logical structure [52], allowing for the expression of logical propositions
and deductions within it. Topos theory, extending beyond sheaf theory, provides a more
holistic and abstract framework. Each topos is like a “categorified space” and comes with
an internal logic and language whose expressions are relative to the space that the topos
models, thereby enabling a more profound exploration of the conceptual structures within

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these spaces. In this way, each topos can be thought of as a “universe”, where the truth
of propositions may depend on where they are uttered. For example, the topos Spc/X
assumed above represents “the universe of the space X”, known as the “little topos” or
“petit topos” of X.
The tools of sheaf theory naturally extend to toposes. Thus, in the foregoing discussion,
we considered a collection of agents with internal world models {Xj}, which, in turn, induce
toposes Spc/Xj that we may consider gluing into a “shared universe” or “consensus topos”
Spc/W according to their topology or interaction pattern. Perhaps, in the end, we may
consider this shared universe to be the agents’ understanding of their actual universe,
socially constructed.
6. Closing Remarks
Our paper introduces the integration of Husserlian phenomenology, active inference
in theoretical biology, and category theory. We have formalized collective action and
shared goals using mathematical tools of increasing generalization. We were able to an-
chor this formalism in phenomenology by delving into Husserl’s phenomenology of inner
time-consciousness, emphasizing retention, primal impression, and protention. We then
proposed that these concepts could be connected in the formation of shared goals in social
groups. With a short overview of active inference, we cast the action–perception loop of
cognitive agents as variational inference, furnishing an isomorphic construct to time con-
sciousness. Building on this introduction, we were able to review the relationship between
Husserl’s time consciousness and active inference established in a previous paper, showing
how past experiences and expectations influence present behavior and understanding. We
then proceeded to leverage a category theory to model shared protentions among active
inference agents, using concepts like polynomial functors and sheaves. These sophisticated
tools were necessary to account for the complexity of shared protentions, leveraging ex-
isting tools of category theory. Our paper achieves a conceptual and mathematical image
of the interconnection among agents, enabling them to coordinate in large groups across
spatiotemporal scales. It dissolves the boundaries between externalist and internalist per-
spectives by demonstrating the intrinsic connections of perceptions extended in time. This
formalization elucidates how agents co-construct their world and interconnect through this
process, offering a novel approach to understanding collective action and shared goals. As
is the case with most work in computational neuroscience and psychology, this paper is
only one step in a broader, multi-step, iterative process where we transition from abstract
theory- and model-building (as in this paper) to experimental validation. The proposal
in this paper represents a point of departure for future work on the phenomenology of
shared intentionality, indeed, it is formulated quite broadly and does not provide a specific
testable model yet. Moving towards such a specific model making testable hypotheses will
be left for future follow-up work.
Author Contributions: Conceptualization, M.A., R.J.P., D.A.F. and M.J.D.R.; Formal analysis, T.S.C.S.;
Writing—original draft, M.A.; Writing—review & editing, M.A., T.S.C.S., D.A.F., K.F. and M.J.D.R.;
Supervision, M.J.D.R. All authors have read and agreed to the published version of the manuscript.
Funding: We wish to acknowledge the Wellcome center grant 088130/Z/09/Z for funding this
manuscript in open access.
Institutional Review Board Statement: Not applicable.
Data Availability Statement: No new data were created or analyzed in this study. Data sharing is
not applicable to this article.
Conflicts of Interest: The authors declare no conflicts of interest.
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*Extraction method: pymupdf*