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Colcombet and Petri\c{s}an argued that automata may be usefully considered from a functorial perspective, introducing a general notion of "$\mathcal{V}$-automaton" based on functors into $\mathcal{V}$. This enables them to recover different standard notions of automata by choosing $\mathcal{V}$ appropriately, and they further analyzed the determinization for \textbf{Rel}-automata using the Kleisli adjunction between \textbf{Set} and \textbf{Rel}. In this paper, we revisit Colcombet and Petri\c{s}an's analysis from a fibrational perspective, building on Melli\`es and Zeilberger's recent alternative but related definition of categorical automata as functors $p : \mathcal{Q} \to \mathcal{C}$ satisfying the finitary fiber and unique lifting of factorizations property. In doing so, we improve the understanding of determinization in three regards: Firstly, we carefully describe the universal property of determinization in terms of forward-backward simulations. Secondly, we generalize the determinization procedure for \textbf{Rel} automata using a local adjunction between \textbf{SpanSet} and \textbf{Rel}, which provides us with a canonical forward simulation. Finally we also propose an alterative determinization based on the multiset relative adjunction which retains paths, and we leverage this to provide a canonical forward-backward simulation. |
