[ALGTOP-L] RE : From simplicial to cosimplicial objects

Joyal, André joyal.andre at uqam.ca
Wed Oct 8 12:18:22 EDT 2008


The duality observed by Klein and discussed by Rognes is a special case
of the Stone duality between the category of finite posets and
and the category of finite distributive lattices:

http://en.wikipedia.org/wiki/Stone_duality

I will be more explicit.
In the category of lattices, the lattice maps are required to preserve the smallest element 0 and 
the largest element 1, in addition to preserving binary infima and suprema.
If X is a finite poset, then the poset X^*=hom(X,[1])  is a finite distributive lattice
(where hom is the poset of poset maps).
If D is a finite distributive lattice, then D^*=Hom(D,[1]) is a finite poset 
(where Hom is the poset of distributive lattice maps)
The canonical maps X-->X^** and D-->D^** are isomorphisms.

An example of lattice is an interval, where an interval is defined to be
an ordered set with first and last elements 0 and 1.
A morphisms of intervals f:J-->K is an order preserving maps such that f(0)=0 and f(1)=1.
The Stone duality restricts to a duality between the category of finite ordinals
and the category of finite intervals.  
If n is a finite ordinal, then the poset n^*=hom(n,[1])=n+1  is a finite interval
(where hom is the poset of order preserving maps).
If J is a finite interval, then J^*=Hom(J,[1]) is a finite ordinal
(where Hom is the poset of morphisms of intervals)
The canonical maps n-->n^** and J-->J^** are isomorphisms.

An interval J is said to be strict if its first and last elements are distinct.
The Stone duality restricts to a duality between the category Delta of finite non-empty ordinals
and the category of finite strict intervals.  
If [n] is a finite non-empty ordinal, then the poset [n]^*=hom([n],[1])=[n+1]  is a finite strict interval
(where hom is the poset of order preserving maps).
If [n+1] is a finite strict interval, then [n+1]^*=Hom([n+1],[1])=[n] is a finite non-empty ordinal
(where Hom is the poset of morphisms between intervals)

A simplicial set is normally defined to be a contravariant set valued functor on the category Delta. 
It follows from the duality that it can be defined to be a covariant set valued functor on the category of finite strict intervals. 
This other description can be used to "explain" some of the nice properties of the geometric realisation functor.
See the book of MacLane and Moerdijk:

http://cs.ioc.ee/yik/lib/1/MacLane1.html

The simplicial interval Delta[1] is a strict interval in the topos of simplicial sets SS.
And the topos SS is freely generated by it.

André Joyal

-------- Message d'origine--------
De: algtop-l-bounces+joyal.andre=uqam.ca at lists.lehigh.edu de la part de John R. Klein
Date: mar. 07/10/2008 11:11
À: algtop
Objet : [ALGTOP-L] From simplicial to cosimplicial objects
 
Suppose D is the category of finite ordered sets and D^o is its opposite.

It has occurred to me that there is an embedding D --> D^o  which maps

[n] ---> [n+1]

(where [n] = {0 < 1 < 2 ... < n}), where the
i-th degeneracy

s_i : [n] ---> [n-1]         (0 \le i  \le n-1)

maps to the (i+1)-st face

d_{i+1} : [n] --> [n+1]

the ith face d_i:[n] --> [n+1] maps to s_i: [n+2] --> [n+1]


My question: is this known? (and is it correct?)

(If it's correct, then it gives a way of producing cosimplicial
objects from simplicial ones.)

jk








-- 
John R. Klein, Professor
Department of Mathematics
Wayne State University
Room 1213 FAB, 656 W. Kirby
Detroit, MI 48202
voice:  (313) 577-3174
fax: (313) 577-7596

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