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\title{The Relativity of Causal Knowledge}
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\author[1,2]{\href{mailto:<gabriele.dacunto@uniroma1.it>?Subject=About "The Relativity of Causal Knowledge"}{Gabriele D'Acunto}{}}
\author[3]{\href{mailto:<cbattiloro@hsph.harvard.edu>?Subject=About "The Relativity of Causal Knowledge"}{Claudio Battiloro}{}}
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\affil[1]{%
    Information Engineering, Electronics and Telecommunications Dept.\\
    Sapienza University\\
    Rome, Italy
    % \texttt{gabriele.dacunto@uniroma1.it}
}
\affil[2]{%
    National Inter-University Consortium for Telecommunications (CNIT)\\
    Parma, Italy
}
\affil[3]{%
    Biostatistics Dept.\\
    Harvard University\\
    Cambridge, MA, USA
    % \texttt{cbattiloro@hsph.harvard.edu}\\ 
}


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\begin{document}

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\begin{abstract}
Recent advances in \emph{artificial intelligence} reveal the limits of purely predictive systems and call for a shift toward causal \emph{and} collaborative reasoning.
%% IDEA
Drawing inspiration from the revolution of Grothendieck in mathematics, 
we introduce the \emph{relativity of causal knowledge}, which posits structural causal models (SCMs) are inherently imperfect, subjective representations embedded within networks of relationships.
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By leveraging category theory, we arrange SCMs into a functor category and show that their observational and interventional probability measures naturally form convex structures.
This result allows us to encode non-intervened SCMs with convex spaces of probability measures.
Next, using sheaf theory, we construct the \emph{network sheaf and cosheaf of causal knowledge}. 
These structures enable the transfer of causal knowledge across the network while incorporating interventional consistency and the perspective of the subjects, ultimately leading to the formal, mathematical definition of \emph{relative causal knowledge}. 
\end{abstract}

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\begin{contributions} % will be removed in pdf for initial submission 
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    The Authors equally contributed to the original idea, the thorough conceptual development of the framework, and the paper writing.
    GD developed the category-theoretic framework, and established and proved the related theoretical results.
    GD and CB codeveloped the sheaf-theoretic characterization of RCK.
\end{contributions}

\begin{acknowledgements} % will be removed in pdf for initial submission,
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The work of Gabriele D'Acunto was supported by \emph{(i)} the European Union under the Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU, partnership on `` Telecommunications of the Future'' (PE00000001 - program `` RESTART''), 
and \emph{(ii)} the SNS JU project 6G-GOALS \cite{strinati2024goal} under the EU's Horizon program Grant Agreement No 101139232.

The work of Claudio Battiloro was supported by the National Institutes of Health (NIH) Grant 1R01ES037156-01.

The Authors thank Dr. Hans Riess from Duke University and Dr. Fabio Massimo Zennaro from University of Bergen for valuable comments on a preliminary version of the work.
\end{acknowledgements}

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\title{The Relativity of Causal Knowledge\\(Supplementary Material)}
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\appendix
\input{appendix/background}
\input{appendix/examples}
\input{appendix/invertibility}
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