% !TEX root =  ../main.tex
\section{Further Discussion and Exemplification of RCK}\label{app:disc&ex}

\noindent\textbf{A Paradigm Shift.} Our work provides a \textit{novel conceptual foundation} for interpreting causality. 
Each subject infers causal knowledge from personal experience and can only see the world through its own perspective. Although the subjectiveness of causality has already been partially explored by \citet{richens2024robust} and, very recently, \citet{bookelias2025causalai}, we move further by making causality not only \textit{subjective} but also \textit{relational}. Consequently, asking for a unique \enquote{true} causal description of a system is an ill-posed question, as the very same notion of causality is inherently relative in our framework. However, it is important to highlight that the relativity of causality is different from the relativism (in its philosophical meaning) of causality: we do not undermine the meaning of things, we question its description as a monolithic object. In our setting, global causal traits of the system emerge only when there is local agreement throughout the network–the global sections of the sheaf of causal knowledge. This parallels how certain physical quantities in physics remain invariant across all reference frames, while other quantities remain inherently relative. But still, the relative treatment better fits the notion of physical reality we are able to describe.  Consider a sudden drop in a company’s stock price. Different agents (the subjects) in the financial system (the network sheaf) propose distinct explanations (their CK): equity analysts blame shifting market sentiment; management highlights an unfavorable earnings report; institutional investors worry about looming regulatory risks; retail shareholders point to negative social media coverage fueling panic selling. However, viewed within the broader network, the interplay of these perspectives (the RCK) discloses a nuanced mix of market sentiment, regulatory uncertainty, company-specific fundamentals, and social influence—no single narrative fully explains the drop. A unified description emerges only when there is local agreement across the entire network (the global sections of the sheaf of causal knowledge). See Figure \ref{fig:explanation} for a simple but comprehensive description of our framework.

\begin{figure*}
    \centering
\includegraphics[width=.9\linewidth]{figures/explanation.png}
    \caption{In the top figure, we depict the classical, non-relative approach to causality: a subject interacts with a \textcolor{system}{system} that is assumed to be completely describable by an underlying \enquote{true} SCM $\mathsf{M}$. We refer to the (convex) space of all the interventional and observational probability measures entailed by $\mathsf{M}$ as the \enquote{true} \textcolor{ck}{Causal Knowledge $\mathsf{CK}(\mathsf{M})$}. In this classical setting, the \textcolor{system}{system} and $\mathsf{M}$ are thus in a bijection, the other subjects are assumed to be part of the \textcolor{system}{system}, and the observer subject interacts with the \textcolor{system}{system} following (or inferring) \textcolor{ck}{$\mathsf{CK}(\mathsf{M})$}. The \textit{Relativity of Causal Knowledge} fundamentally challenges this paradigm by making causality subjective and relational, i.e., by breaking the bijection between the \textcolor{system}{system} and a \enquote{true} SCM, and isolating the subjects from the \textcolor{system}{system} while allowing them to interact with each other. The novel, core mathematical object implementing our relative paradigm is the \textit{Network (Co)Sheaf of Causal Knowledge}, which can be informally thought of as a graph on whose nodes and edges are attached certain convex spaces, called \textit{stalks}, that can interact through certain mappings, called \textit{restriction and extension maps}. In our relative setting, as we show in the bottom left figure, the \textcolor{system}{system} can be pictured as a spherical planet, and the subjects as satellites orbiting around it. Each subject can only observe the planet from certain angles--its \textit{perspective}--, positioning it on a specific orbit. The orbits represent then the subjective \textcolor{ck}{Causal Knowledge} of the subjects--\textcolor{ck}{$\mathsf{CK}(\mathsf{M}^{\rho})$} and \textcolor{ck}{$\mathsf{CK}(\mathsf{M}^{\sigma})$}--, i.e., the (convex) spaces of all the interventional and observational probability measures entailed by the subjective SCMs that the subjects use to describe the underlying \textcolor{system}{system}--$\mathsf{M}^{\rho}$ and $\mathsf{M}^{\sigma}$. In sheaf jargon, each subject is a node, and each space \textcolor{ck}{$\mathsf{CK}(\mathsf{M}^{\cdot})$} is a \textcolor{ck}{\textit{node stalk}}. As such, a point \textcolor{cochain}{$\chi^{\rho}\in \mathsf{CK}(\mathsf{M}^{\rho})$} on the orbit of a subject $\rho$ is a \textit{specific probability measure}. In sheaf jargon, a collection of points, one per each subject's orbit --\textcolor{cochain}{$\{\chi^{\rho}, \chi^{\sigma}\}$}-- is a \textcolor{cochain}{\textit{0-cochain}}. Two subjects $\rho$ and $\sigma$ can communicate if their SCMs $\mathsf{M}^{\rho}$ and $\mathsf{M}^{\sigma}$ admit \textit{a shared abstraction}, i.e., a backbone space \textcolor{backbone}{$\tau$}. In this case, $\rho$ and $\sigma$ can map their 0-cochain values $\chi^{\rho}$ and $\chi^{\sigma}$ to more abstract  representations in the backbone space \textcolor{backbone}{$\alpha_{\mathcal{X}}^{\rho \triangleleft \tau}\left(\chi^\rho\right)$} and \textcolor{backbone}{$\alpha_{\mathcal{X}}^{\sigma \triangleleft \tau}\left(\chi^\sigma\right)$} through surjective mappings $\alpha_{\mathcal{X}}^{\rho \triangleleft \tau}$ and $\alpha_{\mathcal{X}}^{\sigma \triangleleft \tau}$, respectively. \enquote{More abstract} here means a coarse-grained but interventionally consistent representation, telling us essentially the same story about (subjective) cause-and-effect, but at different levels of detail.
    In sheaf jargon, $\rho$ and $\sigma$ are connected by an \textcolor{backbone}{edge $\tau$}, the backbone space is an \textcolor{backbone}{\textit{edge stalk}}, and the mappings are the \textit{restriction maps}.  Therefore, a collection of more abstract representations \textcolor{backbone}{$\{\alpha_{\mathcal{X}}^{\rho \triangleleft \tau}\left(\chi^\rho\right), \alpha_{\mathcal{X}}^{\sigma \triangleleft \tau}\left(\chi^\sigma\right)\}$} is a \textcolor{backbone}{\textit{1-cochain}}. It is now clear that, in our relative setting, global traits of the underlying system only emerge by studying network-level \textit{invariants}. Among them, \enquote{local agreement} of the subjects is particularly important. In sheaf jargon, local agreement refers to \textit{global sections}, i.e., \textcolor{cochain}{0-cochains} whose values are mapped, per each edge, to the same more abstract value-- \textcolor{backbone}{$\alpha_{\mathcal{X}}^{\rho \triangleleft \tau}\left(\chi^\rho\right)= \alpha_{\mathcal{X}}^{\sigma \triangleleft \tau}\left(\chi^\sigma\right)$}. As we show in the bottom right figure, a probability measure \textcolor{cochain}{$\chi^{\rho}$} $\in$ \textcolor{ck}{$\mathsf{CK}(\mathsf{M}^{\rho})$} in the CK of a subject $\rho$ can be mapped to a (usually less informative) probability measure \textcolor{cochain}{$\chi^{\rho,\sigma}$} $\in$ \textcolor{ck}{$\mathsf{CK}(\mathsf{M}^{\rho})$} of another subject $\sigma$, connected to $\rho$ through an edge \textcolor{backbone}{$\tau$}, by first applying a restriction map $\alpha_{\mathcal{X}}^{\rho \triangleleft \tau}$ and then an  extension map $\beta_{\mathcal{X}}^{\sigma \triangleleft \tau}$, i.e., \textcolor{cochain}{$\chi^{\rho,\sigma}=\beta_{\mathcal{X}}^{\sigma \triangleleft \tau}(\alpha_{\mathcal{X}}^{\rho \triangleleft \tau}(\chi^{\rho}))$}. The \textcolor{ck}{\textit{Relative Causal Knowledge} $\mathsf{CK}^{\rho,\sigma}$} of a subject $\rho$ from the perspective of $\sigma$ is then the subspace \textcolor{ck}{$\mathsf{CK}^{\rho,\sigma}$} $\subseteq$ 
 \textcolor{ck}{$\mathsf{CK}(\mathsf{M}^{\sigma})$} being the image of $\beta_{\mathcal{X}}^{\sigma \triangleleft \tau}(\alpha_{\mathcal{X}}^{\rho \triangleleft \tau}(\cdot))$. In the general case, RCK is definable for any pair of subjects for which there exists a connecting path in the underlying network, not only for subjects directly connected by an edge (see \Cref{def:rel_caus_know}).}
    \label{fig:explanation}
\end{figure*}

\noindent\textbf{The Role of Causal Abstractions.} A subject cannot simply share all of its CK with some other subject because that would lead to an inherently inconsistent notion of perspective and, thus, of relativity. Think about physics: if we move from one reference frame to another, we don't just use the same measurements, but we transform them to make them consistent in the new frame. The notion of perspective is crucial in our framework because it is useful to model a variety of possible elements in a network of subjects: privacy or fairness constraints, maximizing mutual information, or \enquote{simply} modeling the impossibility of a human being to analyze a system if not through their own eyes. In practice, this translates into the uselessness of communicating CK whose distributions have different support on different random variables w.r.t. to the causal knowledge of the receiver subject, which would not know how to use it (see the toy example below). Therefore, the abstractions are a convenient yet rigorous way to model a \enquote{shared discourse space} among subjects to enable communication. In this sense, an abstraction is a backbone space: if the subjects operate on different random variables, abstraction and interventional consistency are arguably the best ways to provide them with a rigorous communication medium.

\noindent\textbf{A Clarification on the Meaning of Cochains.} A network comprises three subjects $\sigma$, $\rho$, and $\gamma$, each of them injecting into the network three soft-intervened probability measures from their own CK. These measures represent their current causal representation of the world. As stated in the main body and in Figure \ref{fig:explanation}, the collection of these three probability measures forms a 0-cochain, i.e., a valuation of the node stalks of the network sheaf. In other words, the value of the 0-cochain at each node is just an object (a probability measure) of the node stalk (the node's CK). Via the restriction maps, the 0-cochain entails a 1-cochain on the edges, i.e., the collection of the abstracted probability measures. Although the subjects share a common CA, the abstracted probability measures might disagree on the edge stalk due to their different understanding of the world at that moment. Then, when a subject, say $\rho$, embeds onto its own node stalk, the abstract probability measure of subjects $\sigma$ and $\gamma$ from the edge stalks, it individuates different probability measures belonging to its CK. The latter represents the understanding of the world of $\sigma$ and $\gamma$ expressed in terms of the causal variables of subject $\rho$. Subject $\rho$ can combine its understanding with those of $\sigma$ and $\gamma$ to accomplish its task, leveraging the closure over convex combinations of the CK (\Cref{th:convex_comb_prob_meas}). A global section can then be seen as a 0-cochain whose values, for each pair of nodes connected by an edge, are mapped to the same 1-cochain value by the restriction maps. 



\noindent\textbf{A Toy Example.} Suppose there is some underlying system and consider two subjects $\rho$ and $\sigma$.  The former observes some random variables $X, Z$, and $Y$, and knows that these random variables are related by a structural causal model (SCM) with DAG $X \rightarrow Z \rightarrow Y$. In particular, $\rho$ knows the joint distribution of these variables and also their joint distribution under every possible intervention--that is, $\rho$'s CK. This SCM is compatible with a more abstract SCM whose random variables are $U$ and $M$, and that has DAG $U \rightarrow M$. \enquote{More abstract} here means that there exist surjective mappings porting $X, Z$ and $Y$ into $U$ and $M$. The more abstract SCM is interventionally consistent, i.e., if we pick a variable in the initial SCM, set it to a specific value (\textit{an intervention}), and do the corresponding intervention in the more abstract model, they both tell us essentially the same story about cause-and-effect, but at different levels of detail. Now, suppose $\sigma$ observes variables $P, W, T, L$ and knows they are related by an SCM with DAG $P \rightarrow W \rightarrow T \rightarrow L$. This second SCM is also compatible with the same more abstract SCM with DAG $U \rightarrow M$. Additionally, $\sigma$ can access $\rho$'s CK only through this shared abstract representation and vice versa. If $\sigma$ and $\rho$ are connected by an edge $\tau$, then the Relative CK (RCK) of $\rho$ from the perspective of $\sigma$ is the result of first applying the restriction map to port the CK of $\rho$ on $\tau$, and then porting that more abstract realization of the CK of $\rho$ to $\sigma$ by applying the extension map. In the general case, RCK is definable for any pair of subjects for which there exists a connecting path in the underlying network, not only for subjects directly connected by an edge. Moreover, if more than one possible path connects two subjects, the RCK is path-dependent (see \Cref{def:rel_caus_know}).

\noindent\textbf{Is RCK Needed?} As a little mental exercise to highlight the need for our relative treatment of causality, let us forget for a moment our framework and suppose that there exists a protocol for communicating the entire CK from $\rho$ to $\sigma$ and vice versa. In this case, $\sigma$ receives a certain state (\textit{a cochain value}) of the CK over $\{X, Y, Z\}$ from $\rho$ but, in order to utilize this information, $\sigma$ needs to translate this information onto $\{P, W, T, L\}$. As such, any protocol needs a translation step. The need for this translation step is an informal necessary condition for proving the need of a relative description of causality, beyond our personal philosophical perspective. Consequently, any protocol that uses causal abstraction as the translation step would be a specific instance of our framework. This statement holds true even in more nuanced corner cases: consider again $\rho$ and $\sigma$ but now suppose that both share the same causal model over the same set of random variables, say $\{X, Y, Z\}$. Consider again the protocol above. This time $\sigma$ receives from $\rho$ a certain state of its own causal model according to the perspective of $\rho$ in that context. Then, $\sigma$ can directly combine the $\rho$ 's perspective with its own and accomplish the assigned task. This situation is allowed within our proposed framework. In particular, it is equivalent to set \emph{(i)} the causal abstraction as a causal model over three random variables $\{U, V, W\}$ entailing rotated versions of the probability measures of both $\rho$ and $\sigma$, \emph{(ii)} the restriction maps as rotation matrices, and \emph{(iii)} the extension maps as the transpose of such rotation matrices. The trivial example is when the restrictions and extensions are rotations with a null angle, i.e., the identity.

\noindent\textbf{A Practical and Brief Example in Decentralized Agentic AI.}
In the context of AI, think about the latent spaces of two  autonomous AI agents: they cannot communicate their internal current causal representations (the current 0-cochain) without aligning the latent spaces, i.e., without mapping one latent space into the other. Assume that the latent space of the first agent is a realization of a convex space of probability measures over $\mathbb{R}^d$ and its current causal representation is a point in this space (its current 0-cochain value), and that the latent space of the second agent is a realization of a convex space of probability measures over $\mathbb{R}^n$, with $n \neq d$, and its current causal representation is a point in this space (its current 0-cochain value). The subjects clearly cannot compare or exchange their current causal representations without any additional alignment step.
Consider, however, the simplest scenario: linear alignment, where both restriction and extension maps are represented as matrices.
In this case, the alignment matrix can be viewed as the product of a restriction matrix and an extension matrix. Notably, this factorization aligns closely with the approach commonly employed in low-rank adaptation methods (e.g., LoRA \cite{hu2022lora}). Please notice that this is the case even if the agents communicate in natural language, as they would need to be able to explain to each other how to relate their internal causal representations. At this point, it should be clear as well that the internal current causal representation of one agent can be entirely transmitted to the other agent without any transformation only if the two latent spaces coincide. As a consequence, please notice that the restriction and extension maps are not only useful to allow subjects to communicate, but they can also encode how they should communicate (the perspective). This is why we say that they can be designed to enforce privacy, fairness, or whatever other criterion of interest.



