Like suggested by someone else in this forum, here I propose creating an article, but will wait for agreement or disagreement before creating it.
The article would be called
category of simple graphs with embeddings
and would
be partly modelled on, and aiming for consistency with, the article category of simple graphs
treat and compare at least three wide subcategories of category of simple graphs, namely
(weak.emb) countable simple graphs with weak graph-embeddings
(strong.emb) countable simple graphs with all strong graph-embeddings
(isom.emb) countable simple graphs with all isometric graph-embeddings
Part of the motivation for this:
Apart from the personal motivation of giving structure to my n-th attempt to get the manuscript into a satisfying form, this comparison would perhaps also be mildy interesting from a pure categorical point of view, since
Part of my motivation for creating left cancellative categories is our interest in category (isom.emb).
Do you agree that such an article could fit into the nlab?
Incidentally, I know that wide subcategories are a concept to which sometimes a certain four-letter-word is applied. Nevertheless, it seems to me that
^{ 1 } Actually, it seems that we can prove something considerably stronger, namely that this class of graphs is not closed under elementary equivalence. (synonyms: the class is not elementarily closed$=$ the class is not $\Sigma\Delta$-elementary) Moreover, what we are mostly interested in is the non-elementarily-closedness (and hence non-elementarity) of various subclasses of the class of all vertex-reconstructible graphs. (Proving that the latter class is not elementarily closed does not need https://arxiv.org/abs/1606.02926) For example, it seems we can prove that the class of all vertex-reconstructible locally-finite forests is not elementarily closed. For proving that, the methods of https://arxiv.org/abs/1606.02926 appear essential.
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