This post is an exposition of the material in the paper “Homotopy is not concrete” by P. Freyd, of whose existence I learned from this MO discussion.
A category 
followed by the product functor 
Nonetheless, not every category is concrete, and the following example shows that a very natural one is not:
Theorem 1 (Freyd) The homotopy category
of pointed spaces is not concrete.
In other words, a homotopy type is somehow too complex to be encoded simply as a set with appropriate structure.
The idea of the proof is essentially the following. In a category of structured sets, a given object can only have so many subobjects, because a set has only so many subsets. But there are categories where an object may have an enormous collection of subobjects, because the definition of a subobject is purely arrow-theoretic. So a category where objects can have lots of subobjects is probably not concrete. (more…)
, for 
an algebraically closed field. In other words, it was to determine whether a finitely generated projective module over the ring ![{k[x_1, \dots, x_n]}](http://s0.wp.com/latex.php?latex=%7Bk%5Bx_1%2C+%5Cdots%2C+x_n%5D%7D&bg=ffffff&fg=000000&s=0 "{k[x_1, \dots, x_n]}")
is necessarily free. The topological analog, whether (topological) vector bundles on 
are trivial is easy because 
. At least, any loop space was necessarily an algebra over that monad, and conversely any algebra over that monad was homotopy equivalent to a loop space. There is an alternative and compelling idea of Segal which gives a condition somewhat easier to check.
: the construction of the space 
. It is known that any topological group 
, and conversely (though perhaps it is not as well known) that any loop space is homotopy equivalent to a topological group. (This can be proved using the 
-idea. Given a topological group 
should and runs with that.
is a point.
induced by the 
inclusions ![{[1] \rightarrow [n]}](http://s0.wp.com/latex.php?latex=%7B%5B1%5D+%5Crightarrow+%5Bn%5D%7D&bg=ffffff&fg=000000&s=0 "{[1] \rightarrow [n]}")
(sending 
and 
to consecutive elements) is an isomorphism.
is such a simplicial set, then we take 
as the underlying set of the monoid, and the map 
comes from the boundary map 
; the identity element comes from the map 
. So monoids can be described as simplicial sets satisfying certain properties (just as commutative monoids can).
together with a multiplication map 
and a distinguished unit element 
, satisfying certain identities. Let us say that we are interested in a homotopical version of this idea, especially a version of the idea of an abelian group. Then, we could try to work in the category of topological abelian groups, but this is somewhat uninteresting from the point of view of homotopy theory: every topological abelian is weakly homotopy equivalent to a product of Eilenberg-MacLane spaces. Alternatively, we could demand that one has a topological space together with a multiplication law which is commutative up to homotopy; however, as we’ve seen, this isn’t enough structure to perform a construction such as the classifying space.
of finite sets and partially defined maps as follows. The objects of 
in 
and a map 
. The composition of two partially defined maps is just the ordinary composition, defined wherever it makes sense.
to 
. Given a partially defined map 
between 
, we get a map 
sending a tuple 
to the following tuple:
. The functors 
have the following properties:
.
is an isomorphism, where the maps are induced by the maps 
defined only on 
.
be the maps listed above; they are very simple, being defined only at one point (that is, 
such that 
and such that for each 
, and one can phrase this as an equivalence of categories between such functors and abelian monoids. 
, when is 
for some 
?
is that admits a homotopy associative multiplication map
with coefficients in a field is a graded Hopf algebra. There are strong structure theorems for Hopf algebras, though. For instance, in the finite-dimensional case and in characteristic zero, they are tensor products of exterior algebras, by a theorem of Milnor and Moore. Moreover, for a double loop space 
, the H space structure is homotopy commutative.
) category theory is the ability to do this in a much more general context.
be a map with 
. Associated to this, one can form a CW complex 
; that is, we attach a 
-cell to 
via the map 
. This CW complex has one cell in dimension 
determines a generator 
of 
and the map 
determines a generator 
of 
; there are no other elements in cohomology other than the unit. Consequently, we have
as above such that 
is the Hopf invariant of 
determines only on the homotopy class of 
is, by definition, the map such that the mapping cone 
; it follows that the Hopf fibration has Hopf invariant one.
summand in the homotopy groups of spheres comes from.)
be a compact manifold, 
vector bundles over 
to compute the index of 
, which we saw last time was a well-defined number
. (Henceforth, we shall identify the tangent bundle 
with the cotangent bundle 
, by choice of a Riemannian metric; the specific metric is not really important since 
-theory is a homotopy invariant.) In fact, we have that 
for 
the unit ball bundle and 
the unit sphere bundle. But we have seen that to give an element of 
of a finite complex, the Stiefel-Whitney classes of any real vector bundle vanish.
cannot be distinguished using Stiefel-Whitney classes from the trivial bundle. The argument relies on the 
