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):

HH)H H(HH) H H HH H HH\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 }
IH 0H HH H0 HI 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&& }
IH 1H HH H1 HI Ieps ϵI II I 1 H 1 I II\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}
HH ϵϵ II 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 17:49:40. See the history of this page for a list of all contributions to it.