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Consider the category $BiGraph$ of bipartite graphs (vertices are named "places" and "transitions" like in Petri nets) and continous maps. A subgraph is named open iff it is place-bordered, and a map is named continous iff it is continous with repect to the induced topologies.

Formally: Consider a bipartite graph with vertex set $X$ and adjacency relation $ad$, and an induced subgraph spanned by a vertex set $U \subset X$. Then the subgraph is open iff for all $p,t\in X:$

$$t \in U \ and \ (p,t) \in ad \implies p \in U.$$ (Transitions from $U$ are adjacent to places only in $U$)

My question: Does the category $BiGraph$ have products and coproducts?

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    $\begingroup$ Kבבב ב רקע חח נהנה ה ההגה חחדח. בדג ב בדבר דדדדססב בדבש סבדססבסססס $\endgroup$
    – Tomer Schlank
    Commented Mar 10, 2016 at 16:43
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    $\begingroup$ @Tomer Schlank Did you use the wrong font? $\endgroup$
    – Jo Wehler
    Commented Mar 10, 2016 at 19:12
  • $\begingroup$ There are some imprecisions in Section 3.2 that make it difficult for me to understand. For instance there is $p \in X$ and $(p,t) \in ad$ for $t\in U$, but there is no quantifier on $t$. I assume it is $\exists t$? To define uniquely a topology, your definition also assume that any $p$ is taken iff there is some $t$ that forces it to belong to $U$, or do you assume that a $p$ can belong to an open without being linked to any of the $t$ in $U$? $\endgroup$
    – logicute
    Commented Mar 11, 2016 at 0:27
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    $\begingroup$ Wow, my phone decided to answer the question while in my pocket. But didnt do a very good job. It seems AI has a way to go. $\endgroup$
    – Tomer Schlank
    Commented Mar 11, 2016 at 5:27
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    $\begingroup$ @TomerSchlank, if you read your comment backwards at 33rpm, it's a message from the devil answering the question and proving the Riemann hypothesis. $\endgroup$
    – Manuel Bärenz
    Commented Sep 7, 2016 at 11:48

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Yes - the product is the categorical product of graphs, which is also bipartite (it's easy to see that if $G, H$ are bipartite, then $G\times H$ cannot contain odd circles), and the coproduct is the disjoint union.

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    $\begingroup$ I am not quite convinced: The categorical product considers (strong) morphisms. But continous maps in the sense above allow to contract edges. I assume continous maps correspond to weak morphisms of graphs. Continous maps are continous with respect to the net topology of the graphs. $\endgroup$
    – Jo Wehler
    Commented Mar 10, 2016 at 13:52
  • $\begingroup$ @JoWehler Maybe you could describe a bit more your problem, since 'place-bordered subgraph' may not make a lot of sense for most graph -theorists, who have never heard of Petri nets. $\endgroup$
    – logicute
    Commented Mar 10, 2016 at 14:00
  • $\begingroup$ @Dominic In Petri net theory all graphs are bipartite with two sorts of vertices, named place or transition. A subgraph is place-bordered when all vertices of the subgraph with an edge to a vertex of the complement are places. With respect to the net topology the place-bordered subgraphs are the open subsets by definition. Note that the closed subsets, the transition-bordered subgraphs, form a topology too. Notably singletons with a place (transition) are open (closed). - If necessary, I can explain more about the context, see also Chapter 3 of arxiv.org/abs/cs/0608038 $\endgroup$
    – Jo Wehler
    Commented Mar 10, 2016 at 14:12

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