Term for edges between graph homomorphisms?
If we define a graph to be a tuple (E, V, s: E -> V, t: E -> V), then the category Gph of graphs and graph homomorphisms is cartesian closed (in fact, a topos). For any pair of graphs G, G', there is a "hom graph" whose vertices are graph homomorphisms from G to G' and whose edges are things I've been calling "graph shifts". A graph shift S between two graph homomorphisms F, F':G -> G' assigns to each vertex g in G an edge S(g) in G' from F(g) to F'(g). Is there a more common term for a "graph shift"? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Mike Stay wrote:
Is there a more common term for a "graph shift"?
I don't know - I'm curious. But the term "transformation" would be logical. We all know about natural transformations, but we can also drop the naturality condition: given functors F, F': C -> C', a "transformation" from F to F' assigns to each object c in C a morphism from F(c) to F(c'). Since a category is a graph with extra structure, but the concept of "transformation" doesn't invoke this extra structure, we can also speak of a transformation between homomorphisms between graphs - which is just what you're calling a "graph shift". Then there's a forgetful 2-functor from [categories, functors, natural transformations] to [graphs, graph homomorphisms, transformations] But if there's already a standard term, I recommend that you use that! Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Mike, You might find the following paper relevant: R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, `Graphs of Morphisms of Graphs', Electronic Journal of Combinatorics, A1 of Volume 15(1), 2008. 1-28. However we do not seem to have given a name to the arrows of GPH(B,C) occurring in Gph(A \time B, C) \cong Gph(A, GPH(B,C)). Ronnie ----Original message----
From : metaweta@gmail.com Date : 16/03/2017 - 16:58 (GMTST) To : categories@mta.ca Subject : categories: Term for edges between graph homomorphisms?
If we define a graph to be a tuple (E, V, s: E -> V, t: E -> V), then the category Gph of graphs and graph homomorphisms is cartesian closed (in fact, a topos). For any pair of graphs G, G', there is a "hom graph" whose vertices are graph homomorphisms from G to G' and whose edges are things I've been calling "graph shifts". A graph shift S between two graph homomorphisms F, F':G -> G' assigns to each vertex g in G an edge S(g) in G' from F(g) to F'(g). Is there a more common term for a "graph shift"? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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John Baez -
Mike Stay -
RONALD BROWN