Subject: categories: cubical nerve or not ?

From: Philippe Gaucher <[email-masked]>
Date: Fri, 11 Sep 1998 18:11:24 +0200

Dear all, 

If D is any category equipped with a "cohomotopy structure" (P,p0,p1,s) where 
P:D->D is a functor and p_0,p_1:P->Id_D and s:Id_D->P are natural transformations 
of functors such that p_0s=p_1s=Id_D, we get cubical sets by the following way :

we set P^0=Id_D, P^{n+1}=P(P^n) and 

d^\delta_{i,n} = P^{i-1}p_\delta P^{n-i}:P^n-->P{n-1}, i=1,...,n, \delta=0,1
s_{i,n} = P^{i-1}s P^{n+1-i}:P^n-->P^{n-1}, i=1,...,n+1

and (D(X,P^n(Y)),d^\delta_{i,n},s_{i,n})_n is a cubical set for any object X and Y 
of D.

[I am using the notations of the paper "Homotopies of small categories", Marek 
Golasinski, Fund.Math. 114 (1981) no 3, 209-217]

Now, take D = omega-Cat, I^1 the 1-cube {a -u-> b} and for P the following functor 
: if C is an omega-category and 2_n the omega-category representing C|->C_n, we set 

P(C) = Hom^l(I^1,C) (the left internal hom : take the right one if you want)

with p_0 and p_1 induced by 2_0 ==> I^1 (==> means 2 arrows) which send the point 
of 2_0 on a (resp. b) of I^1 and s : I^1 -> 2_0 which sends a and b on the point of 
2_0 and the 1-morphism of I^1 on the degenerated 1-morphism. 

So (P,p_0,p_1,s) is a cohomotopy structure on omega-Cat. 

=> (omega-Cat(X,P^n(C)),d^\delta_{i,n},s_{i,n}) for any omega-category X.

With X=2_0, omega-Cat(X,P^n(C))=omega-Cat(I^n,C), and (I think) we get the 
classical cubical nerve. We have to verify that the face and degeneracy maps are 
the same in both cases (for the underlying set, it is trivial). I am looking for a 
simple ("abstract") argument in order to aviod an explicit computation. Thank you 
for any help.

pg.