Categorical Constructions in Graphs Laura Scull Fort Lewis College, Durango, CO FMCS May 2019
Category of Graphs Products Exponentials Homotopy Outline The Category of Graphs Products of Graphs Exponential Object Homotopy
Category of Graphs Products Exponentials Homotopy The Category of Graphs Throughout this talk, Gph is the category where: • Objects are graphs G with: • A finite set of vertices V ( G ) • A finite set of edges E ( G ) which are unordered pairs of vertices { v w } • We have at most one edge between any two vertices; loops are allowed.
Category of Graphs Products Exponentials Homotopy The Category of Graphs Throughout this talk, Gph is the category where: • Objects are graphs G with: • A finite set of vertices V ( G ) • A finite set of edges E ( G ) which are unordered pairs of vertices { v w } • We have at most one edge between any two vertices; loops are allowed. • Morphisms f : G → H maps vertices to vertices and respect adjacency: • A set map V ( G ) → V ( H ) • If v w ∈ E ( G ), then f ( v ) f ( w ) ∈ E ( H )
Category of Graphs Products Exponentials Homotopy The Category of Graphs In Gph, maps respect adjacency: 3 4 f(1) f(4) f f(2),f(3) 1 2
Category of Graphs Products Exponentials Homotopy Graph Morphsims Let P n denote the graph: · · · n 1 2 3 • Map P n → G : − → 1 2 3 4
Category of Graphs Products Exponentials Homotopy Graph Morphsims Let P n denote the graph: · · · n 1 2 3 • Map P n → G : f (2) f (1) f (3) − → 1 2 3 4 f (4) • a list of vertices v 1 v 2 v 3 . . . v n such that v n v n +1 ∈ E ( G ) • a walk in G
Category of Graphs Products Exponentials Homotopy Graph Morphisms Let K n denote the graph with n vertices, all connected to each other (but not themselves): 3 4 1 2 • Map K n → G : 3 4 − → 1 2
Category of Graphs Products Exponentials Homotopy Graph Morphisms Let K n denote the graph with n vertices, all connected to each other (but not themselves): • Map K n → G : 3 4 f (3) f (1) f (4) − → 1 2 f (2) • a set of vertices { v 1 v 2 v 3 . . . v n } which are all connected pairwise • a clique in G
Category of Graphs Products Exponentials Homotopy Graph Morphisms Let K 2 denote the graph with 2 vertices: 1 2 • Map G → K 2 : − → 1 2
Category of Graphs Products Exponentials Homotopy Graph Morphisms Let K 2 denote the graph with 2 vertices: 1 2 • Map G → K 2 : − → 1 2 • NOT POSSIBLE! • each vertex gets assigned a label, either 1 or 2, and is not connected to anyone of the same label • a bipartition of G
Category of Graphs Products Exponentials Homotopy Graph Morphisms Let K n denote the graph with n vertices, all connected to each other (but not themselves). • Map G → K n : 3 4 − → 1 2
Category of Graphs Products Exponentials Homotopy Graph Morphisms Let K n denote the graph with n vertices, all connected to each other (but not themselves). • Map G → K n : 3 4 − → 1 2 • each vertex gets assigned a label (colour), and is not connected to anyone of the same label • a (proper) colouring of G
Category of Graphs Products Exponentials Homotopy Homotopy for Graphs? Continuous deformation does not make sense for graphs. Previous approach: create a simplicial space associated with a graph, apply homotopy to it. • Babson and Kosolov • Dochtermann To develop homotopy internal to Gph: A homotopy is a map X × I → Y ? A homotopy is a map X → Y I ?
� � � � Category of Graphs Products Exponentials Homotopy Graph Products A (categorical) product X × Y : Y � X 2 X 1 X 1 × X 2 Excercise: What is the categorical graph product?
Category of Graphs Products Exponentials Homotopy Graph Products Definition The (categorical) product graph G × H (also called the tensor or Kronecker product) is defined by: • A vertex is a pair ( v , w ) where v ∈ V ( G ) and w ∈ V ( H ). • There is an edge ( v 1 , w 1 ) ( v 2 , w 2 ) ∈ E ( G × H ) whenever v 1 v 2 ∈ E ( G ) and w 1 w 2 ∈ E ( H ).
Category of Graphs Products Exponentials Homotopy Graph Products Example Let G be given by a single looped vertex x , and H = K 2 : x 0 1 G H Products: ( x , x ) G × G
Category of Graphs Products Exponentials Homotopy Graph Products Example Let G be given by a single looped vertex x , and H = K 2 : x 0 1 G H Products: ( x , x ) G × G
Category of Graphs Products Exponentials Homotopy Graph Products Example Let G be given by a single looped vertex x , and H = K 2 : x 0 1 G H Products: ( x , 0) ( x , 1) ( x , x ) G × G G × H
Category of Graphs Products Exponentials Homotopy Graph Products Example Let G be given by a single looped vertex x , and H = K 2 : x 0 1 G H Products: ( x , 0) ( x , 1) ( x , x ) G × G G × H
Category of Graphs Products Exponentials Homotopy Graph Products Example Let G be given by a single looped vertex x , and H = K 2 : x 0 1 G H Products: (0 , 0) (0 , 1) ( x , 0) ( x , 1) ( x , x ) (1 , 0) (1 , 1) G × G G × H H × H
Category of Graphs Products Exponentials Homotopy Graph Products Example Let G be given by a single looped vertex x , and H = K 2 : x 0 1 G H Products: (0 , 0) (0 , 1) ( x , 0) ( x , 1) ( x , x ) (1 , 0) (1 , 1) G × G G × H H × H
Category of Graphs Products Exponentials Homotopy Graph Products 1 2 A B 4 3 G H
Category of Graphs Products Exponentials Homotopy Graph Products A 4 B 4 A 3 B 3 1 2 A 2 B 2 A 1 B 1 4 3 A B G × H G H
Category of Graphs Products Exponentials Homotopy Graph Products A 4 B 4 A 3 B 3 1 2 A 2 B 2 A 1 B 1 A B 4 3 G × H G H
Category of Graphs Products Exponentials Homotopy Graph Products This is the categorical product: A 4 B 4 A 3 B 3 1 2 A 2 B 2 A 1 B 1 4 3 A B G × H G H
Category of Graphs Products Exponentials Homotopy π 0 ( G ) • For spaces, π 0 ( X ) is the set of connected components. • For graphs, we can make a similar definition. • If we exclude isolated vertices, this is a homotopy invariant. • BUT: it is not product preserving. (0 , 0) (0 , 1) 0 1 (1 , 0) (1 , 1) H H × H
Category of Graphs Products Exponentials Homotopy π 0 ( G ) Joint with FLC Senior Capstone Class 2018 Definition Let G be a graph with no isolated vertices. Define π 0 ( G ) as the graph with • vertices are equivalence classes of vertices in G • v ∼ w if there is an even length walk from v to w • there is an edge connecting [ v ] to [ w ] when there is an odd length walk from v to w
Category of Graphs Products Exponentials Homotopy Example of π 0 ( G ) G =
Category of Graphs Products Exponentials Homotopy Example of π 0 ( G ) π 0 ( G ) = G =
Category of Graphs Products Exponentials Homotopy Example of π 0 ( G ) 1 2 4 3 H
Category of Graphs Products Exponentials Homotopy Example of π 0 ( G ) [2] = [4] 1 2 π 0 ( G ) = 4 3 [1] = [3] H
Category of Graphs Products Exponentials Homotopy π 0 ( G ) Example a 1 a 2 a 3 G = b 1 b 2 b 3 b 4
Category of Graphs Products Exponentials Homotopy π 0 ( G ) Example a 1 a 2 a 3 G = b 1 b 2 b 3 b 4 A π 0 ( G ) = B
Category of Graphs Products Exponentials Homotopy π 0 ( G ) Results: • π 0 defines a functor from Gph to Gph • π 0 is a homotopy invariant • π 0 ( G × H ) ≃ π 0 ( G ) × π 0 ( H )
Category of Graphs Products Exponentials Homotopy Examples ( b , a ) ( b , b ) b G × G G a ( a , a ) ( a , b ) a b G
Category of Graphs Products Exponentials Homotopy Examples ( b , a ) ( b , b ) b G × G π 0 ( G × G ) G π 0 ( G ) a ( a , a ) ( a , b ) π 0 ( G ) a b G
Category of Graphs Products Exponentials Homotopy Examples 1 A B 3 2 G H
Category of Graphs Products Exponentials Homotopy Examples 1 A B 3 2 G H [ A ] [ B ] [1] = [2] = [3] π 0 ( G ) π 0 ( H )
Category of Graphs Products Exponentials Homotopy Examples 1 A B 3 2 G H A 3 B 3 A 2 B 2 A 1 B 1 G × H
Category of Graphs Products Exponentials Homotopy Examples A 3 B 3 A 2 B 2 A 1 B 1 π 0 ( G × H ) G × H
Category of Graphs Products Exponentials Homotopy Examples 1 2 A B 4 3 G H
Category of Graphs Products Exponentials Homotopy Examples 1 2 A B 4 3 G H [ A ] [ B ] [1] = [3] [2] = [4] π 0 ( G ) π 0 ( H )
Category of Graphs Products Exponentials Homotopy Examples 1 2 4 3 A B G H A 4 B 4 A 3 B 3 A 2 B 2 A 1 B 1 G × H
Category of Graphs Products Exponentials Homotopy Examples A 4 B 4 A 3 B 3 A 2 B 2 A 1 B 1 π 0 ( G × H ) G × H
Category of Graphs Products Exponentials Homotopy Factoring Graphs We know that the following two products are the same: × = × = To what extent can we find other multiple factorizations?
Category of Graphs Products Exponentials Homotopy Exponential Objects in Graphs Recall we had another way of approaching homotopy: • G → H I where H I is the exponential object Excercise: What do exponential objects look like in graphs?
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