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For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and transitive binary relation) on $S$.

Now, fix a nonempty universe $\Omega$, and let $\mathsf{Dig}$ and $\mathsf{Pre}$ be the usual categories of $\Omega$-small digraphs and $\Omega$-small presets. Do we know of any maximal full subcategories of $\mathsf{Dig}$ and $\mathsf{Pre}$ that happen to be isomorphic to each other? If so, I wound much appreciate some references.

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  • $\begingroup$ Could you tell me what it means to be full? $\endgroup$
    – Joel David Hamkins
    Commented Jan 24, 2015 at 21:05
  • $\begingroup$ Yes, sorry. I had assumed it was standard terminology. Given a cat $\sf C$, it's said that $\sf D$ is a full subcat of $\sf C$ if, well, $\sf D$ is a subcat of $\sf C$ and $\hom_{\sf D}(X,Y) = \hom_{\sf C}(X,Y)$ for all objects $X,Y \in {\rm Ob}(\sf D)$. $\endgroup$
    – Salvo Tringali
    Commented Jan 24, 2015 at 21:20

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