Eric Forgy Shape Dependence in Commutative Diagrams

Are these diagrams different?

Loop ( $L = 1$ )

\bullet\righttoleftarrow 1_\bullet

Degenerate Loop ( $L = 2$ )

\bullet\underset{1_\bullet}{\stackrel{1_\bullet}{\leftrightarrows}}\bullet

Degenerate Loop ( $L = 3$ )

\array{ \bullet & \stackrel{1_\bullet}{\to} & \bullet \\ {} & \mathllap{\scriptsize{1_\bullet}}{\nwarrow} & \darr\scriptsize{1_\bullet} \\ {} & {} & \bullet }

Degenerate Triangle

\array{ \bullet & \stackrel{1_\bullet}{\to} & \bullet \\ {} & \mathllap{\scriptsize{1_\bullet}}{\searrow} & \darr\scriptsize{1_\bullet} \\ {} & {} & \bullet }

What if I think that the first three are the same but the last one is different? (But I'm really not sure what $1_\bullet$ is supposed to be.) —Toby

Eric: Hi Toby. Now that I think I understand what’s going on, I’d be interested in hearing how the second and third are the same.

What if I change the question? If instead of diagrams, what if they were categories? Would they be the same category? When I started this page, I was wrongly thinking diagrams were categories.

Revised on April 4, 2010 at 11:46:12 by Eric Forgy