As planned, I’m going to try to say a few words on the cotangent complex in the following posts. There is a bit of background I’d like to go through first, starting with the theory of simplicial sets. The reason is that the cotangent complex is most naturally thought of using the model structure on simplicial rings, on which it is a non-abelian version of a derived functor.
1.1. The simplex category
Definition 1 Let
be the category of finite (nonempty) ordered sets and order-preserving morphisms. The object
will denote the set
with the usual ordering. Thus
is equivalent to the subcategory consisting of the
. This is called the simplex category.
There is a functor from to the category
of topological spaces. Given
, we send it to the standard topological
-simplex
that consists of points
such that each
and
. Given a morphism
of ordered sets, we define
by sending
Here the empty sum is to be regarded as zero.
1.2. Simplicial sets
Definition 2 A simplicial set
is a contravariant functor from
to the category of sets. In other words, it is a presheaf on the simplex category. A morphism of simplicial sets is a natural transformation of functors. The class of simplicial sets thus becomes a category
. Asimplicial object in a category
is a contravariant functor
.
We have just seen that the category is equivalent to the subcategory consisting of the
. As a result, a simplicial set
is given by specifying sets
for each
, together with maps
for each map in
. The set
is called the set of
-simplices of
.
Example 1 Let
be a topological space. Then we define its singular simplicial set
as follows. We let
. Using the functoriality of
discussed above, it is clear that there are maps
for each
.
Example 2 Given
, we define the standard
-simplex
via
Given a category
, we know that there is a way of generating presheaves on
. For each
, we consider the presheaf
defined as
; the presheaves obtained are the representable presheaves. The standard simplices are a special case of that.
Proposition 3 Let
be a simplicial set. Then there is a natural bijection
In other words, mapping from a standard -simplex into
is equivalent to giving an
-simplex of
. Proof: Immediate from Yoneda’s lemma.
1.3. Generalities on presheaves
We are interested in describing functors on the category of simplicial sets. It will be convenient to describe them first on the standard -simplices
. In general, this will be sufficient to characterize the functor. We are going to see that the values on the standard
-simplices will be enough, in many cases, to determine a functor. We shall discuss this in a general context of presheaves on any small category, though. Let
be any small category. We shall, most often, take
to be
. Let
be the category of presheaves on
.
Proposition 4 Any presheaf on
is canonically the colimit of representable presheaves.
Proof: Let be a presheaf on
. For each
, we let
be the representable presheaf defined, as above, by
. Now form the category
whose objects are morphisms of presheaves
, such that the morphisms between
and
are given by commutative diagrams
Note that these commutative diagrams depend on nothing fancy: a morphism is just a map
, in view of Yoneda’s lemma. There is a functor
sending
to
. The image of this functor consists of representable presheaves (clear) and, by definition of the category
, there is a map
that commutes with the diagrams. So there is induced a map
This is a map from a colimit of representable presheaves to . The claim is that it is an isomorphism.
But by the Yoneda lemma, if and
, then there is a map
in
such that the identity in
is sent to
in
. It follows that we can hit any element in any part of the presheaf
by a representable presheaf. Thus the map (1) is surjective.
Now let be a fixed object. We want to show that the map
is injective. Note that we can calculate direct limits in “piecewise.”
Suppose two elements and
are mapped to the same element of
. Then
correspond to maps
given by elements
, and
correspond to maps
. The fact that they map to the same thing in
means that
, where the star denotes pulling back. Call
.
We are now going to show that are identified in the colimit. To see this, we construct diagrams
and
The first comes from the map and the map
given by
; the map
, given by
, is just the object
. The second diagram is similar. The first shows that the object
of the colimit is identified with the identity of
by
in the diagram (where
is an element of
by the map
given by
). Similarly
is identified with this in the colimit, so
are identified. It follows that (1) is also injective.
The fact that this colimit is “canonical” follows from the fact that if is a morphism of presheaves, there is a functor between the categories
associated to each of them. To clarify, if
is a presheaf, then we have described a category
and a functor
such that
is the colimit of
. This association is functorial; if
is a morphism of presheaves, then there is a functor
that fits into an obvious commutative diagram.
Corollary 5 Any simplicial set is canonically a colimit of standard
-simplices.
Proof: This follows from the previous result with .
Warning: Just because every element of is a colimit of representable presheaves does not mean that every element of
is representable, even if
is cocomplete. For instance, the empty presheaf (which assigns to each element of the category
) is not representable. I learned this from Todd Trimble’s answer here. The problem is that the Yoneda embedding does not commute with colimits.
May 3, 2011 at 11:24 pm
[…] that last time, we introduced the notion of a simplicial set. As these were presheaves on the category of finite […]