# Ag.algebraic geometry – Simplicial set represented by a set (unordered)

Let $$X$$ to be a set (finished if you want) and form the whole simplicial $$F ^ { bullet} (X)$$ with
$$F ^ {n} (X) = mathrm {Hom} _ { mathrm {set}} ([n], X)$$
where the right side indicates arbitrary maps of sets (of course
it would not be logical to say the order while preserving $$X$$ do not come
with a command).

I wonder a description of $$F ^ { bullet} (X)$$. For example if $$X = {0,1 }$$ then there are 2 0-simplices, can also call them $$[0] and [1]$$ and 2 1-simplices $$[0, 1]$$ and $$[1,0]$$ glued together to form a copy of $$S ^ 1$$.

edit: As Goodwillie pointed out in his answer, this is not the end of the story, there are many more non-degenerate simplices of higher dimension.

Is there a similar description when $$X = {0, 1, 2 }$$?

A closely related question is whether there is a deputy right to
forgetful functor of the simplex category $$Delta$$ (ordered finished
sets), for example, finite sets (unordered) – and if so, what is it?

Example where such simplicial sets appear: given a map of the topological spaces $$f: X to Y$$ we can always form a
simplicial object $$mathcal {S} ^ { bullet} (f)$$ with
$$mathcal {S} ^ {n} = prod nolimits_ {X} ^ {n} = underlay {X times_ {Y} cdots times_ {Y} X} _ {n text {times}}$$
with facial maps and degeneracy given by projections and diagonals
respectively. Taking connected components gives a simplicial set.

When $$Y$$ is the union $$bigcup_ {i = 1} ^ {N} H_ {i}$$ of the coordinate
hyperplans in $$mathbb {C} ^ {N}$$ and $$f: X = coprod_ {i = 1} ^ {N} H_ {i} to bigcup_ {i = 1} ^ {N} H_ {i} = Y$$ is the obvious map, I believe the simplicial
together we get is $$F ^ { bullet} ( {1, points, n })$$.