Eckmann-Hilton for triple loop space
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Jamie Vicary
May 5, 2018, 6:06:15 PM5/5/18
to Homotopy Type Theory
Dear all,
The formalization of the Eckmann-Hilton argument is given in Chapter
2.1 of the HoTT book. Just to fix notation, let me recall the
statement: for any type A, and for any paths a,b:id_A->id_A, there is
a path sigma_{a,b}:a.b->b.a which reverses the order of composition.
If A is itself a loop space, it is then expected that
sigma_{a,b}.sigma_{b,a} = id. (For this to be interesting, I suppose
the sigma_{a,b} would have to be chosen as a natural family, rather
than arbitrarily.) My question is whether this standard observation
has been formalized in homotopy type theory, and if so, where.
Best wishes,
Jamie
The formalization of the Eckmann-Hilton argument is given in Chapter
2.1 of the HoTT book. Just to fix notation, let me recall the
statement: for any type A, and for any paths a,b:id_A->id_A, there is
a path sigma_{a,b}:a.b->b.a which reverses the order of composition.
If A is itself a loop space, it is then expected that
sigma_{a,b}.sigma_{b,a} = id. (For this to be interesting, I suppose
the sigma_{a,b} would have to be chosen as a natural family, rather
than arbitrarily.) My question is whether this standard observation
has been formalized in homotopy type theory, and if so, where.
Best wishes,
Jamie
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