# Product and coproduct for bipartite graphs

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?

• Kבבב ב רקע חח נהנה ה ההגה חחדח. בדג ב בדבר דדדדססב בדבש סבדססבסססס Commented Mar 10, 2016 at 16:43
• @Tomer Schlank Did you use the wrong font? Commented Mar 10, 2016 at 19:12
• 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 $U$? Commented Mar 11, 2016 at 0:27
• 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. Commented Mar 11, 2016 at 5:27
• @TomerSchlank, if you read your comment backwards at 33rpm, it's a message from the devil answering the question and proving the Riemann hypothesis. Commented Sep 7, 2016 at 11:48

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.