Spahn
segment object
The axioms of a segment are expressed by the commutativity of the following five diagrams (all isomorphisms being induced by the symmetric monoidal structure):
H ⊗ H ) ⊗ H → ∼ H ⊗ ( H ⊗ H ) ↓ ∨ ⊗ H ↓ H ⊗ ∨ H ⊗ H ← ∨ H ⟵ ∨ H ⊗ H \array{
H\otimes H)\otimes H&\to^\sim&H\otimes(H\otimes H)\\\downarrow^{\vee\otimes H}&&\downarrow_{H\otimes\vee}\\H\otimes H&\overset{\vee}{\leftarrow} H\overset{\vee}{\longleftarrow}&H\otimes H
}
I ⊗ H → 0 ⊗ H H ⊗ H ← H ⊗ 0 H ⊗ I ↘ ∼ ↓ ∨ ↙ ∼ H \array{I\otimes H&\rightarrow^{0\otimes H}& H\otimes H&\leftarrow^{H\otimes 0}&H\otimes I\\&\searrow_\sim&\downarrow_\vee&\swarrow_\sim&\\&&H&&
}
I ⊗ H → 1 ⊗ H H ⊗ H ← H ⊗ 1 H ⊗ I ↙ I ⊗ eps ↓ ↓ ∨ ↓ ↘ ϵ ⊗ I I ⊗ I → ∼ I → 1 H ← 1 I ← ∼ I ⊗ I \array{&&I\otimes H&\rightarrow^{1\otimes H}&H\otimes H&\leftarrow^{H\otimes 1}&H\otimes I&&\\&\swarrow^{I\otimes\eps}&\downarrow&&\downarrow_\vee&&\downarrow&\searrow^{\epsilon\otimes I}&\\I\otimes I&\rightarrow^\sim&I&\rightarrow^1&H&\leftarrow^1&I&\leftarrow^\sim&I\otimes I} H ⊗ H → ϵ ⊗ ϵ I ⊗ I I → 0 H ↓ ∨ ↓ ∼ ↓ 1 ↘ id ↓ ϵ H → ϵ I H → ϵ I \array{H\otimes H&\rightarrow^{\epsilon\otimes\epsilon}&I\otimes I&\quad&I&\rightarrow^0&H\\\downarrow^\vee&&\downarrow_\sim&\quad&\downarrow_1&\searrow^{id}&\downarrow_\epsilon\\H&\rightarrow^\epsilon&I&\quad&H&\rightarrow^\epsilon&I}
Created on November 8, 2012 at 17:38:40.
See the history of this page for a list of all contributions to it.