Question on simplicial homotopies

Question on simplicial homotopies

[Michael Barr, Prof.]

Most of you know the combinatorial definition of simplicial homotopy in terms of families h^i: X_n --> Y_{n+1} satisfying a number of identities, including d^0h^0 = f_n and d^{n+1}h^n = g_n, to define a homotopy from f to  g.  We (John Kennison, Bob Raphael, and I) have been using a notion we call reduced homotopy which consists in a series of maps r^i: X_n --> Y_n satisfying certain identities.  It turns out that the r^i = d^{i+1}h^i = d^ih^i (for all except the extreme values of i, 0 and n+1 when only the one or the other is defined) carry all the information and can have technical advantages in certain cases.  What I would like to know is whether they appear anywhere in the literature that can be referred to.


Michael


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