These are lecture notes giving a detailed introduction to classical homotopy theory, starting with the concept of homotopy in topological spaces and motivating from this the “abstract homotopy theory” in general model categories.
For background on basic topology see at Introduction to Topology.
For application to homological algebra see at Introduction to Homological Algebra.
For application to stable homotopy theory see at Introduction to Stable homotopy theory.
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
While the field of algebraic topology clearly originates in topology, it is not actually interested in topological spaces regarded up to topological isomorphism, namely homeomorphism (“point-set topology”), but only in topological spaces regarded up to weak homotopy equivalence – hence it is interested only in the “weak homotopy types” of topological spaces. This is so notably because ordinary cohomology groups are invariants of the (weak) homotopy type of topological spaces but do not detect their homeomorphism class.
The category of topological spaces obtained by forcing weak homotopy equivalences to become isomorphisms is the “classical homotopy category” Ho(Top). This homotopy category however has forgotten a little too much information: homotopy theory really wants the weak homotopy equivalences not to become plain isomorphisms, but to become actual homotopy equivalences. The structure that reflects this is called a model category structure (short for “category of models for homotopy types”). For classical homotopy theory this is accordingly called the classical model structure on topological spaces. This we review here.
This section recalls relevant concepts from actual topology (“point-set topology”) and highlights facts that motivate the axiomatics of model categories below. We prove two technical lemmas (lemma and lemma ) that serve to establish the abstract homotopy theory of topological spaces further below.
Literature (Hirschhorn 15)
Throughout, let Top denote the category whose objects are topological spaces and whose morphisms are continuous functions between them. Its isomorphisms are the homeomorphisms.
(Further below we restrict attention to the full subcategory of compactly generated topological spaces.)
To begin with, we recall some basics on universal constructions in Top: limits and colimits of diagrams of topological spaces; exponential objects.
Generally, recall:
A diagram in a category is a small category and a functor
A cone over this diagram is an object equipped with morphisms for all , such that all these triangles commute:
Dually, a co-cone under the diagram is equipped with morphisms such that all these triangles commute:
A limit over the diagram is a universal cone, denoted , that is: a cone such that every other cone uniquely factors through it , making all the resulting triangles commute.
Dually, a colimit over the diagram is a universal co-cone, denoted .
We now discuss limits and colimits in Top. The key for understanding these is the fact that there are initial and final topologies:
Let be a set of topological spaces, and let be a bare set. Then
For a set of functions out of , the initial topology is the topology on with the minimum collection of open subsets such that all are continuous.
For a set of functions into , the final topology is the topology on with the maximum collection of open subsets such that all are continuous.
For a single topological space, and a subset of its underlying set, the initial topology , def. , is the subspace topology, making
a topological subspace inclusion.
Conversely, for an epimorphism, the final topology on is the quotient topology.
Let be a small category and let be an -diagram in Top (a functor from to ), with components denoted , where and a topology on . Then:
The limit of exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology, def. , for the functions which are the limiting cone components:
Hence
The colimit of exists and is the topological space whose underlying set is the colimit in Set of the underlying diagram of sets, and whose topology is the final topology, def. for the component maps of the colimiting cocone
Hence
(e.g. Bourbaki 71, section I.4)
The required universal property of (def. ) is immediate: for
any cone over the diagram, then by construction there is a unique function of underlying sets making the required diagrams commute, and so all that is required is that this unique function is always continuous. But this is precisely what the initial topology ensures.
The case of the colimit is formally dual.
The limit over the empty diagram in is the point with its unique topology.
For a set of topological spaces, their coproduct is their disjoint union.
In particular:
For , the -indexed coproduct of the point, is the set itself equipped with the final topology, hence is the discrete topological space on .
For a set of topological spaces, their product is the Cartesian product of the underlying sets equipped with the product topology, also called the Tychonoff product.
In the case that is a finite set, such as for binary product spaces , then a sub-basis for the product topology is given by the Cartesian products of the open subsets of (a basis for) each factor space.
The equalizer of two continuous functions in is the equalizer of the underlying functions of sets
(hence the largest subset of on which both functions coincide) and equipped with the subspace topology, example .
The coequalizer of two continuous functions in is the coequalizer of the underlying functions of sets
(hence the quotient set by the equivalence relation generated by for all ) and equipped with the quotient topology, example .
For
two continuous functions out of the same domain, the colimit under this diagram is also called the pushout, denoted
(Here is also called the pushout of , or the cobase change of along .)
This is equivalently the coequalizer of the two morphisms from to the coproduct of with (example ):
If is an inclusion, one also writes and calls this the attaching space.
By example the pushout/attaching space is the quotient topological space
of the disjoint union of and subject to the equivalence relation which identifies a point in with a point in if they have the same pre-image in .
(graphics from Aguilar-Gitler-Prieto 02)
Notice that the defining universal property of this colimit means that completing the span
to a commuting square
is equivalent to finding a morphism
For a topological subspace inclusion, example , the pushout
is the quotient space or cofiber, denoted .
An important special case of example :
For write
for the standard topological n-disk (equipped with its subspace topology as a subset of Cartesian space);
Notice that and that .
Let
be the canonical inclusion of the standard (n-1)-sphere as the boundary of the standard n-disk (both regarded as topological spaces with their subspace topology as subspaces of the Cartesian space ).
Then the colimit in Top under the diagram
i.e. the pushout of along itself, is the n-sphere :
(graphics from Ueno-Shiga-Morita 95)
Another kind of colimit that will play a role for certain technical constructions is transfinite composition. First recall
A partial order is a set equipped with a relation such that for all elements
1) (reflexivity) ;
2) (transitivity) if and then ;
3) (antisymmetry) if and then .
This we may and will equivalently think of as a category with objects the elements of and a unique morphism precisely if . In particular an order-preserving function between partially ordered sets is equivalently a functor between their corresponding categories.
A bottom element in a partial order is one such that for all a. A top element is one for which .
A partial order is a total order if in addition
4) (totality) either or .
A total order is a well order if in addition
5) (well-foundedness) every non-empty subset has a least element.
An ordinal is the equivalence class of a well-order.
The successor of an ordinal is the class of the well-order with a top element freely adjoined.
A limit ordinal is one that is not a successor.
The finite ordinals are labeled by , corresponding to the well-orders . Here is the successor of . The first non-empty limit ordinal is .
Let be a category, and let be a class of its morphisms.
For an ordinal (regarded as a category), an -indexed transfinite sequence of elements in is a diagram
such that
takes all successor morphisms in to elements in
is continuous in that for every nonzero limit ordinal , restricted to the full-subdiagram is a colimiting cocone in for restricted to .
The corresponding transfinite composition is the induced morphism
into the colimit of the diagram, schematically:
We now turn to the discussion of mapping spaces/exponential objects.
For a topological space and a locally compact topological space (in that for every point, every neighbourhood contains a compact neighbourhood), the mapping space
is the topological space
whose underlying set is the set of continuous functions ,
whose open subsets are unions of finitary intersections of the following subbase elements of standard open subsets:
the standard open subset for
a compact topological space subset
an open subset
is the subset of all those continuous functions that fit into a commuting diagram of the form
Accordingly this is called the compact-open topology on the set of functions.
The construction extends to a functor
For a topological space and a locally compact topological space (in that for each point, each open neighbourhood contains a compact neighbourhood), the topological mapping space from def. is an exponential object, i.e. the functor is right adjoint to the product functor : there is a natural bijection
between continuous functions out of any product topological space of with any and continuous functions from into the mapping space.
A proof is spelled out here (or see e.g. Aguilar-Gitler-Prieto 02, prop. 1.3.1).
In the context of prop. it is often assumed that is also a Hausdorff topological space. But this is not necessary. What assuming Hausdorffness only achieves is that all alternative definitions of “locally compact” become equivalent to the one that is needed for the proposition: for every point, every open neighbourhood contains a compact neighbourhood.
Proposition fails in general if is not locally compact. Therefore the plain category Top of all topological spaces is not a Cartesian closed category.
This is no problem for the construction of the homotopy theory of topological spaces as such, but it becomes a technical nuisance for various constructions that one would like to perform within that homotopy theory. For instance on general pointed topological spaces the smash product is in general not associative.
On the other hand, without changing any of the following discussion one may just pass to a more convenient category of topological spaces such as notably the full subcategory of compactly generated topological spaces (def. ) which is Cartesian closed. This we turn to below.
The fundamental concept of homotopy theory is clearly that of homotopy. In the context of topological spaces this is about contiunous deformations of continuous functions parameterized by the standard closed interval:
Write
for the standard topological interval, a compact connected topological subspace of the real line.
Equipped with the canonical inclusion of its two endpoints
this is the standard interval object in Top.
For , the product topological space , example , is called the standard cylinder object over . The endpoint inclusions of the interval make it factor the codiagonal on
For two continuous functions between topological spaces , then a left homotopy
is a continuous function
out of the standard cylinder object over , def. , such that this fits into a commuting diagram of the form
(graphics grabbed from J. Tauber here)
Let be a topological space and let be two of its points, regarded as functions from the point to . Then a left homotopy, def. , between these two functions is a commuting diagram of the form
This is simply a continuous path in whose endpoints are and .
For instance:
Let
be the continuous function from the standard interval to itself that is constant on the value 0. Then there is a left homotopy, def. , from the identity function
given by
A key application of the concept of left homotopy is to the definition of homotopy groups:
For a topological space, then its set of connected components, also called the 0-th homotopy set, is the set of left homotopy-equivalence classes (def. ) of points , hence the set of path-connected components of (example ). By composition this extends to a functor
For , and for any point, the th homotopy group of at is the group
whose underlying set is the set of left homotopy-equivalence classes of maps that take the boundary of to and where the left homotopies are constrained to be constant on the boundary;
whose group product operation takes and to with
where the first map is a homeomorphism from the unit -cube to the -cube with one side twice the unit length (e.g. ).
By composition, this construction extends to a functor
from pointed topological spaces to graded groups.
Notice that often one writes the value of this functor on a morphism as .
At this point we don’t go further into the abstract reason why def. yields group structure above degree 0, which is that positive dimension spheres are H-cogroup objects. But this is important, for instance in the proof of the Brown representability theorem. See the section Brown representability theorem in Part S.
A continuous function is called a homotopy equivalence if there exists a continuous function the other way around, , and left homotopies, def. , from the two composites to the identity:
and
If here is constant along , is said to exhibit as a deformation retract of .
For a topological space and its standard cylinder object of def. , the projection and the inclusion are homotopy equivalences, def. , and in fact are homotopy inverses to each other:
The composition
is immediately the identity on (i.e. homotopic to the identity by a trivial homotopy), while the composite
is homotopic to the identity on by a homotopy that is pointwise in that of example .
A continuous function is called a weak homotopy equivalence if its image under all the homotopy group functors of def. is an isomorphism, hence if
and for all and all
Every homotopy equivalence, def. , is a weak homotopy equivalence, def. .
In particular a deformation retraction, def. , is a weak homotopy equivalence.
First observe that for all Top the inclusion maps
into the standard cylinder object, def. , are weak homotopy equivalences: by postcomposition with the contracting homotopy of the interval from example all homotopy groups of have representatives that factor through this inclusion.
Then given a general homotopy equivalence, apply the homotopy groups functor to the corresponding homotopy diagrams (where for the moment we notationally suppress the choice of basepoint for readability) to get two commuting diagrams
By the previous observation, the vertical morphisms here are isomorphisms, and hence these diagrams exhibit as the inverse of , hence both as isomorphisms.
The converse of prop. is not true generally: not every weak homotopy equivalence between topological spaces is a homotopy equivalence. (For an example with full details spelled out see for instance Fritsch, Piccinini: “Cellular Structures in Topology”, p. 289-290).
However, as we will discuss below, it turns out that
every weak homotopy equivalence between CW-complexes is a homotopy equivalence (Whitehead's theorem, cor. );
every topological space is connected by a weak homotopy equivalence to a CW-complex (CW approximation, remark ).
For , the projection from the cylinder object of , def. , is a weak homotopy equivalence, def. . This means that the factorization
of the codiagonal in def. , which in general is far from being a monomorphism, may be thought of as factoring it through a monomorphism after replacing , up to weak homotopy equivalence, by .
In fact, further below (prop. ) we see that has better properties than the generic monomorphism has, in particular better homotopy invariant properties: it has the left lifting property against all Serre fibrations that are also weak homotopy equivalences.
Of course the concept of left homotopy in def. is accompanied by a concept of right homotopy. This we turn to now.
For a topological space, its standard topological path space object is the topological path space, hence the mapping space , prop. , out of the standard interval of def. .
The endpoint inclusion into the standard interval, def. , makes the path space of def. factor the diagonal on through the inclusion of constant paths and the endpoint evaluation of paths:
This is the formal dual to example . As in that example, below we will see (prop. ) that this factorization has good properties, in that
is a Serre fibration.
So while in general the diagonal is far from being an epimorphism or even just a Serre fibration, the factorization through the path space object may be thought of as replacing , up to weak homotopy equivalence, by its path space, such as to turn its diagonal into a Serre fibration after all.
For two continuous functions between topological spaces , then a right homotopy is a continuous function
into the path space object of , def. , such that this fits into a commuting diagram of the form
We consider topological spaces that are built consecutively by attaching basic cells.
Write
for the set of canonical boundary inclusion maps of the standard n-disks, example . This going to be called the set of standard topological generating cofibrations.
For and for , an -cell attachment to is the pushout (“attaching space”, example ) of a generating cofibration, def.
along some continuous function .
A continuous function is called a topological relative cell complex if it is exhibited by a (possibly infinite) sequence of cell attachments to , in that it is a transfinite composition (def. ) of pushouts (example )
of coproducts (example ) of generating cofibrations (def. ).
A topological space is a cell complex if is a relative cell complex.
A relative cell complex is called a finite relative cell complex if it is obtained from a finite number of cell attachments.
A (relative) cell complex is called a (relative) CW-complex if the above transfinite composition is countable
and if is obtained from by attaching cells precisely only of dimension .
Strictly speaking a relative cell complex, def. , is a function , together with its cell structure, hence together with the information of the pushout diagrams and the transfinite composition of the pushout maps that exhibit it.
In many applications, however, all that matters is that there is some (relative) cell decomposition, and then one tends to speak loosely and mean by a (relative) cell complex only a (relative) topological space that admits some cell decomposition.
The following lemma , together with lemma below are the only two statements of the entire development here that involve the concrete particular nature of topological spaces (“point-set topology”), everything beyond that is general abstract homotopy theory.
Assuming the axiom of choice and the law of excluded middle, every compact subspace of a topological cell complex, def. , intersects the interior of a finite number of cells.
(e.g. Hirschhorn 15, section 3.1)
So let be a topological cell complex and a compact subspace. Define a subset
by choosing one point in the interior of the intersection with of each cell of that intersects .
It is now sufficient to show that has no accumulation point. Because, by the compactness of , every non-finite subset of does have an accumulation point, and hence the lack of such shows that is a finite set and hence that intersects the interior of finitely many cells of .
To that end, let be any point. If is a 0-cell in , write . Otherwise write for the unique cell of that contains in its interior. By construction, there is exactly one point of in the interior of . Hence there is an open neighbourhood containing no further points of beyond possibly itself, if happens to be that single point of in .
It is now sufficient to show that may be enlarged to an open subset of containing no point of , except for possibly itself, for that means that is not an accumulation point of .
To that end, let be the ordinal that labels the stage of the transfinite composition in the cell complex-presentation of at which the cell above appears. Let be the ordinal of the full cell complex. Then define the set
and regard this as a partially ordered set by declaring a partial ordering via
This is set up such that every element of with the maximum value is an extension that we are after.
Observe then that for a chain in (a subset on which the relation restricts to a total order), it has an upper bound in given by the union . Therefore Zorn's lemma applies, saying that contains a maximal element .
Hence it is now sufficient to show that . We argue this by showing that assuming leads to a contradiction.
So assume . Then to construct an element of that is larger than , consider for each cell at stage its attaching map and the corresponding preimage open set . Enlarging all these preimages to open subsets of (such that their image back in does not contain ), then . This is a contradiction. Hence , and we are done.
It is immediate and useful to generalize the concept of topological cell complexes as follows.
For any category and for any sub-class of its morphisms, a relative -cell complex is a morphism in which is a transfinite composition (def. ) of pushouts of coproducts of morphisms in .
Write
for the set of inclusions of the topological n-disks, def. , into their cylinder objects, def. , along (for definiteness) the left endpoint inclusion.
These inclusions are similar to the standard topological generating cofibrations of def. , but in contrast to these they are “acyclic” (meaning: trivial on homotopy classes of maps from “cycles” given by n-spheres) in that they are weak homotopy equivalences (by prop. ).
Accordingly, is to be called the set of standard topological generating acyclic cofibrations.
For a CW-complex (def. ), then its inclusion into its standard cylinder (def. ) is a -relative cell complex (def. , def. ).
First erect a cylinder over all 0-cells
Assume then that the cylinder over all -cells of has been erected using attachment from . Then the union of any -cell of with the cylinder over its boundary is homeomorphic to and is like the cylinder over the cell “with end and interior removed”. Hence via attaching along the cylinder over is erected.
The maps in def. are finite relative cell complexes, def. . In other words, the elements of are -relative cell complexes.
There is a homeomorphism
such that the map on the right is the inclusion of one hemisphere into the boundary n-sphere of . This inclusion is the result of attaching two cells:
Every -relative cell complex (def. , def. ) is a weak homotopy equivalence, def. .
Let be a -relative cell complex.
First observe that with the elements of being homotopy equivalences for all (by example ), each of the stages in the relative cell complex is also a homotopy equivalence. We make this fully explicit:
By definition, such a stage is a pushout of the form
Then the fact that the projections are strict left inverses to the inclusions gives a commuting square of the form
and so the universal property of the colimit (pushout) gives a factorization of the identity morphism on the right through
which exhibits as a strict left inverse to . Hence it is now sufficient to show that this is also a homotopy right inverse.
To that end, let
be the left homotopy that exhibits as a homotopy right inverse to by example . For each consider the commuting square
Regarded as a cocone under the span in the top left, the universal property of the colimit (pushout) gives a continuous function
for each . For this construction reduces to the provious one in that is the composite which we need to homotope to the identity; while is the identity. Since is clearly also continuous in it constitutes a continuous function
which exhibits the required left homotopy.
So far this shows that each stage in the transfinite composition defining is a homotopy equivalence, hence, by prop. , a weak homotopy equivalence.
This means that all morphisms in the following diagram (notationally suppressing basepoints and showing only the finite stages)
are isomorphisms.
Moreover, lemma gives that every representative and every null homotopy of elements in already exists at some finite stage . This means that also the universally induced morphism
is an isomorphism. Hence the composite is an isomorphism.
Given a relative -cell complex , def. , it is typically interesting to study the extension problem along , i.e. to ask which topological spaces are such that every continuous function has an extension along
If such extensions exists, it means that is sufficiently “spread out” with respect to the maps in . More generally one considers this extension problem fiberwise, i.e. with both and (hence also ) equipped with a map to some base space :
Given a category and a sub-class of its morphisms, then a morphism in is said to have the right lifting property against the morphisms in if every commuting diagram in of the form
with , has a lift , in that it may be completed to a commuting diagram of the form
We will also say that is a -injective morphism if it satisfies the right lifting property against .
A continuous function is called a Serre fibration if it is a -injective morphism; i.e. if it has the right lifting property, def. , against all topological generating acylic cofibrations, def. ; hence if for every commuting diagram of continuous functions of the form
has a lift , in that it may be completed to a commuting diagram of the form
Def. says, in view of the definition of left homotopy, that a Serre fibration is a map with the property that given a left homotopy, def. , between two functions into its codomain, and given a lift of one the two functions through , then also the homotopy between the two lifts. Therefore the condition on a Serre fibration is also called the homotopy lifting property for maps whose domain is an n-disk.
More generally one may ask functions to have such homotopy lifting property for functions with arbitrary domain. These are called Hurewicz fibrations.
The precise shape of and in def. turns out not to actually matter much for the nature of Serre fibrations. We will eventually find below (prop. ) that what actually matters here is only that the inclusions are relative cell complexes (lemma ) and weak homotopy equivalences (prop. ) and that all of these may be generated from them in a suitable way.
But for simple special cases this is readily seen directly, too. Notably we could replace the n-disks in def. with any homeomorphic topological space. A choice important in the comparison to the classical model structure on simplicial sets (below) is to instead take the topological n-simplices . Hence a Serre fibration is equivalently characterized as having lifts in all diagrams of the form
Other deformations of the -disks are useful in computations, too. For instance there is a homeomorphism from the -disk to its “cylinder with interior and end removed”, formally:
and hence is a Serre fibration equivalently also if it admits lifts in all diagrams of the form
The following is a general fact about closure of morphisms defined by lifting properties which we prove in generality below as prop. .
A Serre fibration, def. has the right lifting property against all retracts (see remark ) of -relative cell complexes (def. , def. ).
The following statement is foreshadowing the long exact sequences of homotopy groups (below) induced by any fiber sequence, the full version of which we come to below (example ) after having developed more of the abstract homotopy theory.
Let be a Serre fibration, def. , let be any point and write
for the fiber inclusion over that point. Then for every choice of lift of the point through , the induced sequence of homotopy groups
is exact, in that the kernel of is canonically identified with the image of :
It is clear that the image of is in the kernel of (every sphere in becomes constant on , hence contractible, when sent forward to ).
For the converse, let be represented by some . Assume that is in the kernel of . This means equivalently that fits into a commuting diagram of the form
where is the contracting homotopy witnessing that .
Now since is a lift of , there exists a left homotopy
as follows:
(for instance: regard as embedded in such that is identified with the basepoint on the boundary of and set ).
The pasting of the top two squares that have appeared this way is equivalent to the following commuting square
Because is a Serre fibration and by lemma and prop. , this has a lift
Notice that is a basepoint preserving left homotopy from to some . Being homotopic, they represent the same element of :
But the new representative has the special property that its image in is not just trivializable, but trivialized: combining with the previous diagram shows that it sits in the following commuting diagram
The commutativity of the outer square says that is constant, hence that is entirely contained in the fiber . Said more abstractly, the universal property of fibers gives that factors through , hence that is in the image of .
The following lemma , together with lemma above, are the only two statements of the entire development here that crucially involve the concrete particular nature of topological spaces (“point-set topology”), everything beyond that is general abstract homotopy theory.
The continuous functions with the right lifting property, def. against the set of topological generating cofibrations, def. , are precisely those which are both weak homotopy equivalences, def. as well as Serre fibrations, def. .
We break this up into three sub-statements:
A) -injective morphisms are in particular weak homotopy equivalences
Let have the right lifting property against
We check that the lifts in these diagrams exhibit as being an isomorphism on all homotopy groups, def. :
For the existence of these lifts says that every point of is in the image of , hence that is surjective. Let then be a map that hits two connected components, then the existence of the lift says that if they have the same image in then they were already the same connected component in . Hence is also injective and hence is a bijection.
Similarly, for , if represents an element in that becomes trivial in , then the existence of the lift says that it already represented the trivial element itself. Hence has trivial kernel and so is injective.
Finally, to see that is also surjective, hence bijective, observe that every elements in is equivalently represented by a commuting diagram of the form
and so here the lift gives a representative of a preimage in .
B) -injective morphisms are in particular Serre fibrations
By an immediate closure property of lifting problems (we spell this out in generality as prop. , cor. below) an -injective morphism has the right lifting property against all relative cell complexes, and hence, by lemma , it is also a -injective morphism, hence a Serre fibration.
C) Acyclic Serre fibrations are in particular -injective morphisms
Let be a Serre fibration that induces isomorphisms on homotopy groups. In degree 0 this means that is an isomorphism on connected components, and this means that there is a lift in every commuting square of the form
(this is being surjective) and in every commuting square of the form
(this is being injective). Hence we are reduced to showing that for every diagram of the form
has a lift.
To that end, pick a basepoint on and write and for its images in and , respectively
Then the diagram above expresses that and hence by assumption on it follows that , which in turn mean that there is making the upper triangle of our lifting problem commute:
It is now sufficient to show that any such may be deformed to a which keeps making this upper triangle commute but also makes the remaining lower triangle commute.
To that end, notice that by the commutativity of the original square, we already have at least this commuting square:
This induces the universal map from the pushout of its cospan in the top left, which is the n-sphere (see this example):
This universal morphism represents an element of the th homotopy group:
By assumption that is a weak homotopy equivalence, there is a with
hence on representatives there is a lift up to homotopy
Morever, we may always find of the form for some . (“Paste to the reverse of .”)
Consider then the map
and observe that this represents the trivial class:
This means equivalently that there is a homotopy
fixing the boundary of the -disk.
Hence if we denote homotopy by double arrows, then we have now achieved the following situation
and it now suffices to show that may be lifted to a homotopy of just , fixing the boundary, for then the resulting homotopic is the desired lift.
To that end, notice that the condition that fixes the boundary of the -disk means equivalently that it extends to a morphism
out of the pushout that identifies in the cylinder over all points lying over the boundary. Hence we are reduced to finding a lift in
But inspection of the left map reveals that it is homeomorphic again to , and hence the lift does indeed exist.
In the above we discussed three classes of continuous functions between topological spaces
and we saw first aspects of their interplay via lifting properties.
A fundamental insight due to (Quillen 67) is that in fact all constructions in homotopy theory are elegantly expressible via just the abstract interplay of these classes of morphisms. This was distilled in (Quillen 67) into a small set of axioms called a model category structure (because it serves to make all objects be models for homotopy types.)
This abstract homotopy theory is the royal road for handling any flavor of homotopy theory, in particular the stable homotopy theory that we are after in Part 1. Here we discuss the basic constructions and facts in abstract homotopy theory, then below we conclude this Introduction to Homotopy Theory by showing that topological spaces equipped with the above system of classes continuous functions is indeed an example of abstract homotopy theory in this sense.
Literature (Dwyer-Spalinski 95)
A category with weak equivalences is
such that
contains all the isomorphisms of ;
is closed under two-out-of-three: in every commuting diagram in of the form
if two of the three morphisms are in , then so is the third.
It turns out that a category with weak equivalences, def. , already determines a homotopy theory: the one given given by universally forcing weak equivalences to become actual homotopy equivalences. This may be made precise and is called the simplicial localization of a category with weak equivalences (Dwyer-Kan 80a, Dwyer-Kan 80b, Dwyer-Kan 80c). However, without further auxiliary structure, these simplicial localizations are in general intractable. The further axioms of a model category serve the sole purpose of making the universal homotopy theory induced by a category with weak equivalences be tractable:
A model category is
such that
the class makes into a category with weak equivalences, def. ;
The pairs and are both weak factorization systems, def. .
One says:
elements in are weak equivalences,
elements in are cofibrations,
elements in are fibrations,
elements in are acyclic cofibrations,
elements in are acyclic fibrations.
The form of def. is due to (Joyal, def. E.1.2). It implies various other conditions that (Quillen 67) demands explicitly, see prop. and prop. below.
We now dicuss the concept of weak factorization systems appearing in def. .
Let be any category. Given a diagram in of the form
then an extension of the morphism along the morphism is a completion to a commuting diagram of the form
Dually, given a diagram of the form
then a lift of through is a completion to a commuting diagram of the form
Combining these cases: given a commuting square
then a lifting in the diagram is a completion to a commuting diagram of the form
Given a sub-class of morphisms , then
dually:
A weak factorization system (WFS) on a category is a pair of classes of morphisms of such that
Every morphism of may be factored as the composition of a morphism in followed by one in
The classes are closed under having the lifting property, def. , against each other:
is precisely the class of morphisms having the left lifting property against every morphisms in ;
is precisely the class of morphisms having the right lifting property against every morphisms in .
For a category, a functorial factorization of the morphisms in is a functor
which is a section of the composition functor .
In def. we are using the following standard notation, see at simplex category and at nerve of a category:
Write and for the ordinal numbers, regarded as posets and hence as categories. The arrow category is equivalently the functor category , while has as objects pairs of composable morphisms in . There are three injective functors , where omits the index in its image. By precomposition, this induces functors . Here
sends a pair of composable morphisms to their composition;
sends a pair of composable morphisms to the first morphisms;
sends a pair of composable morphisms to the second morphisms.
A weak factorization system, def. , is called a functorial weak factorization system if the factorization of morphisms may be chosen to be a functorial factorization , def. , i.e. such that lands in and in .
Not all weak factorization systems are functorial, def. , although most (including those produced by the small object argument (prop. below), with due care) are.
Let be a category and let be a class of morphisms. Write and , respectively, for the sub-classes of -projective morphisms and of -injective morphisms, def. . Then:
Both classes contain the class of isomorphisms of .
Both classes are closed under composition in .
is also closed under transfinite composition.
Both classes are closed under forming retracts in the arrow category (see remark ).
is closed under forming pushouts of morphisms in (“cobase change”).
is closed under forming pullback of morphisms in (“base change”).
is closed under forming coproducts in .
is closed under forming products in .
We go through each item in turn.
containing isomorphisms
Given a commuting square
with the left morphism an isomorphism, then a lift is given by using the inverse of this isomorphism . Hence in particular there is a lift when and so . The other case is formally dual.
closure under composition
Given a commuting square of the form
consider its pasting decomposition as
Now the bottom commuting square has a lift, by assumption. This yields another pasting decomposition
and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that has the right lifting property against and is hence in . The case of composing two morphisms in is formally dual. From this the closure of under transfinite composition follows since the latter is given by colimits of sequential composition and successive lifts against the underlying sequence as above constitutes a cocone, whence the extension of the lift to the colimit follows by its universal property (cf. eg. Hirschhorn (2002), Lem. 10.3.1).
closure under retracts
Let be the retract of an , i.e. let there be a commuting diagram of the form.
Then for
a commuting square, it is equivalent to its pasting composite with that retract diagram
Here the pasting composite of the two squares on the right has a lift, by assumption:
By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence has the left lifting property against all and hence is in . The other case is formally dual.
closure under pushout and pullback
Let and and let
be a pullback diagram in . We need to show that has the right lifting property with respect to all . So let
be a commuting square. We need to construct a diagonal lift of that square. To that end, first consider the pasting composite with the pullback square from above to obtain the commuting diagram
By the right lifting property of , there is a diagonal lift of the total outer diagram
By the universal property of the pullback this gives rise to the lift in
In order for to qualify as the intended lift of the total diagram, it remains to show that
commutes. To do so we notice that we obtain two cones with tip :
one is given by the morphisms
with universal morphism into the pullback being
the other by
with universal morphism into the pullback being
The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is unique this implies the required identity of morphisms.
The other case is formally dual.
closure under (co-)products
Let be a set of elements of . Since colimits in the presheaf category are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects induced via its universal property by the set of morphisms :
Now let
be a commuting square. This is in particular a cocone under the coproduct of objects, hence by the universal property of the coproduct, this is equivalent to a set of commuting diagrams
By assumption, each of these has a lift . The collection of these lifts
is now itself a compatible cocone, and so once more by the universal property of the coproduct, this is equivalent to a lift in the original square
This shows that the coproduct of the has the left lifting property against all and is hence in . The other case is formally dual.
An immediate consequence of prop. is this:
Let be a category with all small colimits, and let be a sub-class of its morphisms. Then every -injective morphism, def. , has the right lifting property, def. , against all -relative cell complexes, def. and their retracts, remark .
By a retract of a morphism in some category we mean a retract of as an object in the arrow category , hence a morphism such that in there is a factorization of the identity on through
This means equivalently that in there is a commuting diagram of the form
In every category the class of isomorphisms is preserved under retracts in the sense of remark .
For
a retract diagram and an isomorphism, the inverse to is given by the composite
More generally:
Given a model category in the sense of def. , then its class of weak equivalences is closed under forming retracts (in the arrow category, see remark ).
Let
be a commuting diagram in the given model category, with a weak equivalence. We need to show that then also .
First consider the case that .
In this case, factor as a cofibration followed by an acyclic fibration. Since and by two-out-of-three (def. ) this is even a factorization through an acyclic cofibration followed by an acyclic fibration. Hence we obtain a commuting diagram of the following form:
where is uniquely defined and where is any lift of the top middle vertical acyclic cofibration against . This now exhibits as a retract of an acyclic fibration. These are closed under retract by prop. .
Now consider the general case. Factor as an acyclic cofibration followed by a fibration and form the pushout in the top left square of the following diagram
where the other three squares are induced by the universal property of the pushout, as is the identification of the middle horizontal composite as the identity on . Since acyclic cofibrations are closed under forming pushouts by prop. , the top middle vertical morphism is now an acyclic fibration, and hence by assumption and by two-out-of-three so is the middle bottom vertical morphism.
Thus the previous case now gives that the bottom left vertical morphism is a weak equivalence, and hence the total left vertical composite is.
Consider a composite morphism
If has the left lifting property against , then is a retract of .
If has the right lifting property against , then is a retract of .
We discuss the first statement, the second is formally dual.
Write the factorization of as a commuting square of the form
By the assumed lifting property of against there exists a diagonal filler making a commuting diagram of the form
By rearranging this diagram a little, it is equivalent to
Completing this to the right, this yields a diagram exhibiting the required retract according to remark :
Small object argument
Given a set of morphisms in some category , a natural question is how to factor any given morphism through a relative -cell complex, def. , followed by a -injective morphism, def.
A first approximation to such a factorization turns out to be given simply by forming by attaching all possible -cells to . Namely let
be the set of all ways to find a -cell attachment in , and consider the pushout of the coproduct of morphisms in over all these:
This gets already close to producing the intended factorization:
First of all the resulting map is a -relative cell complex, by construction.
Second, by the fact that the coproduct is over all commuting squres to , the morphism itself makes a commuting diagram
and hence the universal property of the colimit means that is indeed factored through that -cell complex ; we may suggestively arrange that factorizing diagram like so:
This shows that, finally, the colimiting co-cone map – the one that now appears diagonally – almost exhibits the desired right lifting of against the . The failure of that to hold on the nose is only the fact that a horizontal map in the middle of the above diagram is missing: the diagonal map obtained above lifts not all commuting diagrams of into , but only those where the top morphism factors through .
The idea of the small object argument now is to fix this only remaining problem by iterating the construction: next factor in the same way into
and so forth. Since relative -cell complexes are closed under composition, at stage the resulting is still a -cell complex, getting bigger and bigger. But accordingly, the failure of the accompanying to be a -injective morphism becomes smaller and smaller, for it now lifts against all diagrams where factors through , which intuitively is less and less of a condition as the grow larger and larger.
The concept of small object is just what makes this intuition precise and finishes the small object argument. For the present purpose we just need the following simple version:
For a category and a sub-set of its morphisms, say that these have small domains if there is an ordinal (def. ) such that for every and for every -relative cell complex given by a transfinite composition (def. )
every morphism factors through a stage of order :
The above discussion proves the following:
(small object argument)
Let be a locally small category with all small colimits. If a set of morphisms has all small domains in the sense of def. , then every morphism in factors through a -relative cell complex, def. , followed by a -injective morphism, def.
We discuss how the concept of homotopy is abstractly realized in model categories, def. .
Let be a model category, def. , and an object.
where is a weak equivalence and is a fibration.
where is a weak equivalence. and is a cofibration.
For every object in a model category, a cylinder object and a path space object according to def. exist: the factorization axioms guarantee that there exists
a factorization of the codiagonal as
a factorization of the diagonal as
The cylinder and path space objects obtained this way are actually better than required by def. : in addition to being just a weak equivalence, for these this is actually an acyclic fibration, and dually in addition to being a weak equivalence, for these it is actually an acyclic cofibration.
Some authors call cylinder/path-space objects with this extra property “very good” cylinder/path-space objects, respectively.
One may also consider dropping a condition in def. : what mainly matters is the weak equivalence, hence some authors take cylinder/path-space objects to be defined as in def. but without the condition that is a cofibration and without the condition that is a fibration. Such authors would then refer to the concept in def. as “good” cylinder/path-space objects.
The terminology in def. follows the original (Quillen 67, I.1 def. 4). With the induced concept of left/right homotopy below in def. , this admits a quick derivation of the key facts in the following, as we spell out below.
Let be a model category. If is cofibrant, then for every cylinder object of , def. , not only is a cofibration, but each
is an acyclic cofibration separately.
Dually, if is fibrant, then for every path space object of , def. , not only is a fibration, but each
is an acyclic fibration separately.
We discuss the case of the path space object. The other case is formally dual.
First, that the component maps are weak equivalences follows generally: by definition they have a right inverse and so this follows by two-out-of-three (def. ).
But if is fibrant, then also the two projection maps out of the product are fibrations, because they are both pullbacks of the fibration
hence is the composite of two fibrations, and hence itself a fibration, by prop. .
Path space objects are very non-unique as objects up to isomorphism:
If is a fibrant object in a model category, def. , and for and two path space objects for , def. , then the fiber product is another path space object for : the pullback square
gives that the induced projection is again a fibration. Moreover, using lemma and two-out-of-three (def. ) gives that is a weak equivalence.
For the case of the canonical topological path space objects of def , with then this new path space object is , the mapping space out of the standard interval of length 2 instead of length 1.
Let be two parallel morphisms in a model category.
Let be two parallel morphisms in a model category.
Let be cofibrant. If there is a left homotopy then there is also a right homotopy (def. ) with respect to any chosen path space object.
Let be fibrant. If there is a right homotopy then there is also a left homotopy with respect to any chosen cylinder object.
In particular if is cofibrant and is fibrant, then by going back and forth it follows that every left homotopy is exhibited by every cylinder object, and every right homotopy is exhibited by every path space object.
We discuss the first case, the second is formally dual. Let be the given left homotopy. Lemma implies that we have a lift in the following commuting diagram
where on the right we have the chosen path space object. Now the composite is a right homotopy as required:
For a cofibrant object in a model category and a fibrant object, the relations of left homotopy and of right homotopy (def. ) on the hom set coincide and are both equivalence relations.
That both relations coincide under the (co-)fibrancy assumption follows directly from lemma .
The symmetry and reflexivity of the relation is obvious.
That right homotopy (hence also left homotopy) with domain is a transitive relation follows from using example to compose path space objects.
We discuss the construction that takes a model category, def. , and then universally forces all its weak equivalences into actual isomorphisms.
Let be a model category, def. . Write for the category whose
objects are those objects of which are both fibrant and cofibrant;
morphisms are the homotopy classes of morphisms of , hence the equivalence classes of morphism under the equivalence relation of prop. ;
and whose composition operation is given on representatives by composition in .
This is, up to equivalence of categories, the homotopy category of the model category .
Def. is well defined, in that composition of morphisms between fibrant-cofibrant objects in indeed passes to homotopy classes.
Fix any morphism between fibrant-cofibrant objects. Then for precomposition
to be well defined, we need that with also . But by prop we may take the homotopy to be exhibited by a right homotopy , for which case the statement is evident from this diagram:
For postcomposition we may choose to exhibit homotopy by left homotopy and argue dually.
We now spell out that def. indeed satisfies the universal property that defines the localization of a category with weak equivalences at its weak equivalences.
(Whitehead theorem in model categories)
Let be a model category. A weak equivalence between two objects which are both fibrant and cofibrant is a homotopy equivalence.
By the factorization axioms in the model category and by two-out-of-three (def. ), every weak equivalence factors through an object as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with and both fibrant and cofibrant, so is , and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.
So let be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is formally dual. Then in fact it has a genuine right inverse given by a lift in the diagram
To see that is also a left inverse up to left homotopy, let be any cylinder object on (def. ), hence a factorization of the codiagonal on as a cofibration followed by a an acyclic fibration
and consider the commuting square
which commutes due to being a genuine right inverse of . By construction, this commuting square now admits a lift , and that constitutes a left homotopy .
Given a model category , consider a choice for each object of
a factorization of the initial morphism, such that when is already cofibrant then ;
a factorization of the terminal morphism, such that when is already fibrant then .
Write then
for the functor to the homotopy category, def. , which sends an object to the object and sends a morphism to the homotopy class of the result of first lifting in
and then lifting (here: extending) in
First of all, the object is indeed both fibrant and cofibrant (as well as related by a zig-zag of weak equivalences to ):
Now to see that the image on morphisms is well defined. First observe that any two choices of the first lift in the definition are left homotopic to each other, exhibited by lifting in
Hence also the composites are left homotopic to each other, and since their domain is cofibrant, then by lemma they are also right homotopic by a right homotopy . This implies finally, by lifting in
that also and are right homotopic, hence that indeed represents a well-defined homotopy class.
Finally to see that the assignment is indeed functorial, observe that the commutativity of the lifting diagrams for and imply that also the following diagram commutes
Now from the pasting composite
one sees that is a lift of and hence the same argument as above gives that it is homotopic to the chosen .
For the following, recall the concept of natural isomorphism between functors: for two functors, then a natural transformation is for each object a morphism in , such that for each morphism in the following is a commuting square:
Such is called a natural isomorphism if its are isomorphisms for all objects .
For a category with weak equivalences, its localization at the weak equivalences is, if it exists,
such that
sends weak equivalences to isomorphisms;
is universal with this property, in that:
for any functor out of into any category , such that takes weak equivalences to isomorphisms, it factors through up to a natural isomorphism
and this factorization is unique up to unique isomorphism, in that for and two such factorizations, then there is a unique natural isomorphism making the evident diagram of natural isomorphisms commute.
For a model category, the functor in def. (for any choice of and ) exhibits as indeed being the localization of the underlying category with weak equivalences at its weak equivalences, in the sense of def. :
First, to see that that indeed takes weak equivalences to isomorphisms: By two-out-of-three (def. ) applied to the commuting diagrams shown in the proof of lemma , the morphism is a weak equivalence if is:
With this the “Whitehead theorem for model categories”, lemma , implies that represents an isomorphism in .
Now let be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as
uniquely up to unique natural isomorphism. Now by construction of and in def. , is the identity on the full subcategory of fibrant-cofibrant objects. It follows that if exists at all, it must satisfy for all with and both fibrant and cofibrant that
(hence in particular ).
But by def. that already fixes on all of , up to unique natural isomorphism. Hence it only remains to check that with this definition of there exists any natural isomorphism filling the diagram above.
To that end, apply to the above commuting diagram to obtain
Here now all horizontal morphisms are isomorphisms, by assumption on . It follows that defining makes the required natural isomorphism:
Due to theorem we may suppress the choices of cofibrant and fibrant replacement in def. and just speak of the localization functor
up to natural isomorphism.
In general, the localization of a category with weak equivalences (def. ) may invert more morphisms than just those in . However, if the category admits the structure of a model category , then its localization precisely only inverts the weak equivalences:
Let be a model category (def. ) and let be its localization functor (def. , theorem ). Then a morphism in is a weak equivalence precisely if is an isomorphism in .
(e.g. Goerss-Jardine 96, II, prop 1.14)
While the construction of the homotopy category in def. combines the restriction to good (fibrant/cofibrant) objects with the passage to homotopy classes of morphisms, it is often useful to consider intermediate stages:
Given a model category , write
for the system of full subcategory inclusions of:
the category of fibrant-cofibrant objects ,
all regarded a categories with weak equivalences (def. ), via the weak equivalences inherited from , which we write , and .
Of course the subcategories in def. inherit more structure than just that of categories with weak equivalences from . and each inherit “half” of the factorization axioms. One says that has the structure of a “fibration category” called a “Brown-category of fibrant objects”, while has the structure of a “cofibration category”.
We discuss properties of these categories of (co-)fibrant objects below in Homotopy fiber sequences.
The proof of theorem immediately implies the following:
For a model category, the restriction of the localization functor from def. (using remark ) to any of the sub-categories with weak equivalences of def.
exhibits equivalently as the localization also of these subcategories with weak equivalences, at their weak equivalences. In particular there are equivalences of categories
The following says that for computing the hom-sets in the homotopy category, even a mixed variant of the above will do; it is sufficient that the domain is cofibrant and the codomain is fibrant:
For with cofibrant and fibrant, and for fibrant/cofibrant replacement functors as in def. , the morphism
(on homotopy classes of morphisms, well defined by prop. ) is a natural bijection.
We may factor the morphism in question as the composite
This shows that it is sufficient to see that for cofibrant and fibrant, then
is an isomorphism, and dually that
is an isomorphism. We discuss this for the former; the second is formally dual:
First, that is surjective is the lifting property in
which says that any morphism comes from a morphism under postcomposition with .
Second, that is injective is the lifting property in
which says that if two morphisms become homotopic after postcomposition with , then they were already homotopic before.
We record the following fact which will be used in part 1.1 (here):
Let be a model category (def. ). Then every commuting square in its homotopy category (def. ) is, up to isomorphism of squares, in the image of the localization functor of a commuting square in (i.e.: not just commuting up to homotopy).
Let
be a commuting square in the homotopy category. Writing the same symbols for fibrant-cofibrant objects in and for morphisms in representing these, then this means that in there is a left homotopy of the form
Consider the factorization of the top square here through the mapping cylinder of
This exhibits the composite as an alternative representative of in , and as an alternative representative for , and the commuting square
as an alternative representative of the given commuting square in .
For and two categories with weak equivalences, def. , then a functor is called a homotopical functor if it sends weak equivalences to weak equivalences.
Given a homotopical functor (def. ) between categories with weak equivalences whose homotopy categories and exist (def. ), then its (“total”) derived functor is the functor between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. ):
While many functors of interest between model categories are not homotopical in the sense of def. , many become homotopical after restriction to the full subcategories of fibrant objects or of cofibrant objects, def. . By corollary this is just as good for the purpose of homotopy theory.
Therefore one considers the following generalization of def. :
(left and right derived functors)
Consider a functor out of a model category (def. ) into a category with weak equivalences (def. ).
If the restriction of to the full subcategory of fibrant object becomes a homotopical functor (def. ), then the derived functor of that restriction, according to def. , is called the right derived functor of and denoted by :
If the restriction of to the full subcategory of cofibrant object becomes a homotopical functor (def. ), then the derived functor of that restriction, according to def. , is called the left derived functor of and denoted by :
The key fact that makes def. practically relevant is the following:
Let be a model category with full subcategories of fibrant objects and of cofibrant objects respectively (def. ). Let be a category with weak equivalences.
A functor out of the category of fibrant objects
is a homotopical functor, def. , already if it sends acylic fibrations to weak equivalences.
A functor out of the category of cofibrant objects
is a homotopical functor, def. , already if it sends acylic cofibrations to weak equivalences.
The following proof refers to the factorization lemma, whose full statement and proof we postpone to further below (lemma ).
We discuss the case of a functor on a category of fibrant objects , def. . The other case is formally dual.
Let be a weak equivalence in . Choose a path space object (def. ) and consider the diagram
where the square is a pullback and on the top left is our notation for the universal cone object. (Below we discuss this in more detail, it is the mapping cocone of , def. ).
Here:
is an acyclic fibration because it is the pullback of .
is a weak equivalence, because the factorization lemma states that the composite vertical morphism factors through a weak equivalence, hence if is a weak equivalence, then is by two-out-of-three (def. ).
Now apply the functor to this diagram and use the assumption that it sends acyclic fibrations to weak equivalences to obtain
But the factorization lemma , in addition says that the vertical composite is a fibration, hence an acyclic fibration by the above. Therefore also is a weak equivalence. Now the claim that also is a weak equivalence follows with applying two-out-of-three (def. ) twice.
Let be model categories and consider a functor. Then:
If preserves cofibrant objects and acyclic cofibrations between these, then its left derived functor (def. ) exists, fitting into a diagram
If preserves fibrant objects and acyclic fibrants between these, then its right derived functor (def. ) exists, fitting into a diagram
Let be a functor between two model categories (def. ).
If preserves fibrant objects and weak equivalences between fibrant objects, then the total right derived functor (def. ) in
is given, up to isomorphism, on any object by applying to a fibrant replacement of and then forming a cofibrant replacement of the result:
If preserves cofibrant objects and weak equivalences between cofibrant objects, then the total left derived functor (def. ) in
is given, up to isomorphism, on any object by appying to a cofibrant replacement of and then forming a fibrant replacement of the result:
We discuss the first case, the second is formally dual. By the proof of theorem we have
But since is a homotopical functor on fibrant objects, the cofibrant replacement morphism is a weak equivalence in , hence becomes an isomorphism under . Therefore
Now since is assumed to preserve fibrant objects, is fibrant in , and hence acts on it (only) by cofibrant replacement.
In practice it turns out to be useful to arrange for the assumptions in corollary to be satisfied by pairs of adjoint functors. Recall that this is a pair of functors and going back and forth between two categories
such that there is a natural bijection between hom-sets with on the left and those with on the right:
for all objects and . This being natural means that is a natural transformation, hence that for all morphisms and the following is a commuting square:
We write to indicate an adjunction and call the left adjoint and the right adjoint of the adjoint pair.
The archetypical example of a pair of adjoint functors is that consisting of forming Cartesian products and forming mapping spaces , as in the category of compactly generated topological spaces of def. .
If is any morphism, then the image is called its adjunct, and conversely. The fact that adjuncts are in bijection is also expressed by the notation
For an object , the adjunct of the identity on is called the adjunction unit .
For an object , the adjunct of the identity on is called the adjunction counit .
Adjunction units and counits turn out to encode the adjuncts of all other morphisms by the formulas
.
Let be model categories. A pair of adjoint functors between them
is called a Quillen adjunction (and , are called left/right Quillen functors, respectively) if the following equivalent conditions are satisfied
preserves cofibrations and preserves fibrations;
preserves acyclic cofibrations and preserves acyclic fibrations;
preserves cofibrations and acylic cofibrations;
preserves fibrations and acyclic fibrations.
First observe that
(i) A left adjoint between model categories preserves acyclic cofibrations precisely if its right adjoint preserves fibrations.
(ii) A left adjoint between model categories preserves cofibrations precisely if its right adjoint preserves acyclic fibrations.
We discuss statement (i), statement (ii) is formally dual. So let be an acyclic cofibration in and a fibration in . Then for every commuting diagram as on the left of the following, its -adjunct is a commuting diagram as on the right here:
If preserves acyclic cofibrations, then the diagram on the right has a lift, and so the -adjunct of that lift is a lift of the left diagram. This shows that has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if preserves fibrations, the same argument run from right to left gives that preserves acyclic fibrations.
Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.
Let be a Quillen adjunction, def. .
For a fibrant object and a path space object (def. ), then is a path space object for .
For a cofibrant object and a cylinder object (def. ), then is a cylinder object for .
Consider the second case, the first is formally dual.
First observe that because is left adjoint and hence preserves colimits, hence in particular coproducts.
Hence
is a cofibration.
Second, with cofibrant is an acyclic cofibration (lemma ), and so then is
Therefore by two-out-of-three (def. ) preserves the weak equivalence .
For a Quillen adjunction, def. , then also the corresponding left and right derived functors, def. , via cor. , form a pair of adjoint functors
By def. and lemma it is sufficient to see that for with cofibrant and fibrant, then there is a natural bijection
Since by the adjunction isomorphism for such a natural bijection exists before passing to homotopy classes , it is sufficient to see that this respects homotopy classes. To that end, use from lemma that with a cylinder object for , def. , then is a cylinder object for . This implies that left homotopies
given by
are in bijection to left homotopies
given by
For two model categories, a Quillen adjunction (def.)
is called a Quillen equivalence, to be denoted
if the following equivalent conditions hold.
The right derived functor of (via prop. , corollary ) is an equivalence of categories
The left derived functor of (via prop. , corollary ) is an equivalence of categories
For every cofibrant object , the derived adjunction unit, hence the composite
(of the adjunction unit with any fibrant replacement as in def. ) is a weak equivalence;
and for every fibrant object , the derived adjunction counit, hence the composite
(of the adjunction counit with any cofibrant replacement as in def. ) is a weak equivalence.
For every cofibrant object and every fibrant object , a morphism is a weak equivalence precisely if its adjunct morphism is:
That follows from prop. (if in an adjoint pair one is an equivalence, then so is the other).
To see the equivalence , notice (prop.) that a pair of adjoint functors is an equivalence of categories precisely if both the adjunction unit and the adjunction counit are natural isomorphisms. Hence it is sufficient to show that the morphisms called derived adjunction unit and derived adjunction counit above indeed represent the adjunction (co-)unit of in the homotopy category. We show this now for the adjunction unit, the case of the adjunction counit is formally dual.
To that end, first observe that for , then the defining commuting square for the left derived functor from def.
(using fibrant and fibrant/cofibrant replacement functors , from def. with their universal property from theorem , corollary ) gives that
where the second isomorphism holds because the left Quillen functor sends the acyclic cofibration to a weak equivalence.
The adjunction unit of on is the image of the identity under
By the above and the proof of prop. , that adjunction isomorphism is equivalently that of under the isomorphism
of lemma . Hence the derived adjunction unit is the -adjunct of
which indeed (by the formula for adjuncts) is
To see that :
Consider the weak equivalence . Its -adjunct is
by assumption 4) this is again a weak equivalence, which is the requirement for the derived unit in 3). Dually for derived counit.
To see :
Consider any a weak equivalence for cofibrant , firbant . Its adjunct sits in a commuting diagram
where is any lift constructed as in def. .
This exhibits the bottom left morphism as the derived adjunction unit, hence a weak equivalence by assumption. But since was a weak equivalence, so is (by two-out-of-three). Thereby also and , are weak equivalences by Ken Brown's lemma and the assumed fibrancy of . Therefore by two-out-of-three (def. ) also the adjunct is a weak equivalence.
In certain situations the conditions on a Quillen equivalence simplify. For instance:
If in a Quillen adjunction (def. ) the right adjoint “creates weak equivalences” (in that a morphism in is a weak equivalence precisly if is) then is a Quillen equivalence (def. ) precisely already if for all cofibrant objects the plain adjunction unit
is a weak equivalence.
By prop. , generally, is a Quillen equivalence precisely if
for every cofibrant object , the derived adjunction unit
is a weak equivalence;
for every fibrant object , the derived adjunction counit
is a weak equivalence.
Consider the first condition: Since preserves the weak equivalence , then by two-out-of-three (def. ) the composite in the first item is a weak equivalence precisely if is.
Hence it is now sufficient to show that in this case the second condition above is automatic.
Since also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image
under is.
Moreover, assuming, by the above, that on the cofibrant object is a weak equivalence, then by two-out-of-three this composite is a weak equivalence precisely if the further composite with is
By the formula for adjuncts, this composite is the -adjunct of the original composite, which is just
But is a weak equivalence by definition of cofibrant replacement.
We now discuss how the category Top of topological spaces satisfies the axioms of abstract homotopy theory (model category) theory, def. .
Say that a continuous function, hence a morphism in Top, is
a classical weak equivalence if it is a weak homotopy equivalence, def. ;
a classical fibration if it is a Serre fibration, def. ;
a classical cofibration if it is a retract (rem. ) of a relative cell complex, def. .
and hence
a acyclic classical cofibration if it is a classical cofibration as well as a classical weak equivalence;
a acyclic classical fibration if it is a classical fibration as well as a classical weak equivalence.
Write
for the classes of these morphisms, respectively.
We first prove now that the classes of morphisms in def. satisfy the conditions for a model category structure, def. (after some lemmas, this is theorem below). Then we discuss the resulting classical homotopy category (below) and then a few variant model structures whose proof follows immediately along the line of the proof of :
The model structure on compactly generated topological spaces and ;
The model structure on topologically enriched functors and .
The classical weak equivalences, def. , satify two-out-of-three (def. ).
Since isomorphisms (of homotopy groups) satisfy 2-out-of-3, this property is directly inherited via the very definition of weak homotopy equivalence, def. .
Every morphism in Top factors as a classical cofibration followed by an acyclic classical fibration, def. :
By lemma the set of topological generating cofibrations, def. , has small domains, in the sense of def. (the n-spheres are compact). Hence by the small object argument, prop. , factors as an -relative cell complex, def. , hence just a plain relative cell complex, def. , followed by an -injective morphisms, def. :
By lemma the map is both a weak homotopy equivalence as well as a Serre fibration.
Every morphism in Top factors as an acyclic classical cofibration followed by a fibration, def. :
By lemma the set of topological generating acyclic cofibrations, def. , has small domains, in the sense of def. (the n-disks are compact). Hence by the small object argument, prop. , factors as an -relative cell complex, def. , followed by a -injective morphisms, def. :
By definition this makes a Serre fibration, hence a fibration.
By lemma a relative -cell complex is in particular a relative -cell complex. Hence is a classical cofibration. By lemma it is also a weak homotopy equivalence, hence a clasical weak equivalence.
Every commuting square in Top with the left morphism a classical cofibration and the right morphism a fibration, def.
admits a lift as soon as one of the two is also a classical weak equivalence.
A) If the fibration is also a weak equivalence, then lemma says that it has the right lifting property against the generating cofibrations , and cor. implies the claim.
B) If the cofibration on the left is also a weak equivalence, consider any factorization into a relative -cell complex, def. , def. , followed by a fibration,
as in the proof of lemma . By lemma the morphism is a weak homotopy equivalence, and so by two-out-of-three (prop. ) the factorizing fibration is actually an acyclic fibration. By case A), this acyclic fibration has the right lifting property against the cofibration itself, and so the retract argument, lemma gives that is a retract of a relative -cell complex. With this, finally cor. implies that has the right lifting property against .
Finally:
The systems and from def. are weak factorization systems.
Since we have already seen the factorization property (lemma , lemma ) and the lifting properties (lemma ), it only remains to see that the given left/right classes exhaust the class of morphisms with the given lifting property.
For the classical fibrations this is by definition, for the the classical acyclic fibrations this is by lemma .
The remaining statement for and follows from a general argument (here) for cofibrantly generated model categories (def. ), which we spell out:
So let be in , we need to show that then is a retract (remark ) of a relative cell complex. To that end, apply the small object argument as in lemma to factor as
It follows that has the left lifting property against , and hence by the retract argument (lemma ) it is a retract of . This proves the claim for .
The analogous argument for , using the small object argument for , shows that every is a retract of a -cell complex. By lemma and lemma a -cell complex is both an -cell complex and a weak homotopy equivalence. Retracts of the former are cofibrations by definition, and retracts of the latter are still weak homotopy equivalences by lemma . Hence such is an acyclic cofibration.
In conclusion, prop. and prop. say that:
The classes of morphisms in of def. ,
define a model category structure (def. ) , the classical model structure on topological spaces or Serre-Quillen model structure .
In particular
every object in is fibrant;
the cofibrant objects in are the retracts of cell complexes.
Hence in particular the following classical statement is an immediate corollary:
(Whitehead theorem)
Every weak homotopy equivalence (def. ) between topological spaces that are homeomorphic to a retract of a cell complex, in particular to a CW-complex (def. ), is a homotopy equivalence (def. ).
In proving theorem we have in fact shown a bit more that stated. Looking back, all the structure of is entirely induced by the set (def. ) of generating cofibrations and the set (def. ) of generating acyclic cofibrations (whence the terminology). This phenomenon will keep recurring and will keep being useful as we construct further model categories, such as the classical model structure on pointed topological spaces (def. ), the projective model structure on topological functors (thm. ), and finally various model structures on spectra which we turn to in the section on stable homotopy theory.
Therefore we make this situation explicit:
A model category (def. ) is called cofibrantly generated if there exists two subsets
of its class of morphisms, such that
the (acyclic) cofibrations of are precisely the retracts, of -relative cell complexes (-relative cell complexes), def. .
For a cofibrantly generated model category, def. , with generating (acylic) cofibrations (), then its classes of weak equivalences, fibrations and cofibrations are equivalently expressed as injective or projective morphisms (def. ) this way:
;
;
;
It is clear from the definition that , so that the closure property of prop. gives an inclusion
For the converse inclusion, let . By the small object argument, prop. , there is a factorization . Hence by assumption and by the retract argument lemma , is a retract of an -relative cell complex, hence is in .
This proves the first statement. Together with the closure properties of prop. , this implies the second claim.
The proof of the third and fourth item is directly analogous, just with replaced for .
With the classical model structure on topological spaces in hand, we now have good control over the classical homotopy category:
The Serre-Quillen classical homotopy category is the homotopy category, def. , of the classical model structure on topological spaces from theorem : we write
From just theorem , the definition (def. ) gives that
is the category whose objects are retracts of cell complexes (def. ) and whose morphisms are homotopy classes of continuous functions. But in fact more is true:
Theorem in itself implies that every topological space is weakly equivalent to a retract of a cell complex, def. . But by the existence of CW approximations, this cell complex may even be taken to be a CW complex.
(Better yet, there is Quillen equivalence to the classical model structure on simplicial sets which implies a functorial CW approximation given by forming the geometric realization of the singular simplicial complex of .)
Hence the Serre-Quillen classical homotopy category is also equivalently the category of just the CW-complexes whith homotopy classes of continuous functions between them
It follows that the universal property of the homotopy category (theorem )
implies that there is a bijection, up to natural isomorphism, between
functors out of which agree on homotopy-equivalent maps;
functors out of all of which send weak homotopy equivalences to isomorphisms.
This statement in particular serves to show that two different axiomatizations of generalized (Eilenberg-Steenrod) cohomology theories are equivalent to each other. See at Introduction to Stable homotopy theory – S the section generalized cohomology functors (this prop.)
Beware that, by remark , what is not equivalent to is the category
obtained from all topological spaces with morphisms the homotopy classes of continuous functions. This category is “too large”, the correct homotopy category is just the genuine full subcategory
Beware also the ambiguity of terminology: “classical homotopy category” some literature refers to instead of . However, here we never have any use for and will not mention it again.
Let be a CW-complex, def. . Then the standard topological cylinder of def.
(obtained by forming the product space with the standard topological interval ) is indeed a cylinder object in the abstract sense of def. .
We describe the proof informally. It is immediate how to turn this into a formal proof, but the notation becomes tedious. (One place where it is spelled out completely is Ottina 14, prop. 2.9.)
So let be a presentation of as a CW-complex. Proceed by induction on the cell dimension.
First observe that the cylinder over is a cell complex: First itself is a disjoint union of points. Adding a second copy for every point (i.e. attaching along ) yields , then attaching an inteval between any two corresponding points (along ) yields .
So assume that for it has been shown that has the structure of a CW-complex of dimension . Then for each cell of , attach it twice to , once at , and once at .
The result is with a hollow cylinder erected over each of its -cells. Now fill these hollow cylinders (along ) to obtain .
This completes the induction, hence the proof of the CW-structure on .
The construction also manifestly exhibits the inclusion as a relative cell complex.
Finally, it is clear (prop. ) that is a weak homotopy equivalence.
Conversely:
Let be any topological space. Then the standard topological path space object (def. )
(obtained by forming the mapping space, def. , with the standard topological interval ) is indeed a path space object in the abstract sense of def. .
To see that is a weak homotopy equivalence it is sufficient, by prop. , to exhibit a homotopy equivalence. Let the homotopy inverse be . Then the composite
is already equal to the identity. The other we round, the rescaling of paths provides the required homotopy
To see that is a fibration, we need to show that every commuting square of the form
has a lift.
Now first use the adjunction from prop. to rewrite this equivalently as the following commuting square:
This square is equivalently (example ) a morphism out of the pushout
By the same reasoning, a lift in the original diagram is now equivalently a lifting in
Inspection of the component maps shows that the left vertical morphism here is the inclusion into the square times of three of its faces times . This is homeomorphic to the inclusion (as in remark ). Therefore a lift in this square exsists, and hence a lift in the original square exists.
A pointed object is of course an object equipped with a point , and a morphism of pointed objects is a morphism that takes to . Trivial as this is in itself, it is good to record some basic facts, which we do here.
Passing to pointed objects is also the first step in linearizing classical homotopy theory to stable homotopy theory. In particular, every category of pointed objects has a zero object, hence has zero morphisms. And crucially, if the original category had Cartesian products, then its pointed objects canonically inherit a non-cartesian tensor product: the smash product. These ingredients will be key below in the section on stable homotopy theory.
Let be a category and let be an object.
The slice category is the category whose
objects are morphisms in ;
morphisms are commuting triangles in .
Dually, the coslice category is the category whose
objects are morphisms in ;
morphisms are commuting triangles in .
There are the canonical forgetful functors
given by forgetting the morphisms to/from .
We here focus on this class of examples:
For a category with terminal object , the coslice category (def. ) is the corresponding category of pointed objects: its
objects are morphisms in of the form (hence an object equipped with a choice of point; i.e. a pointed object);
morphisms are commuting triangles of the form
(hence morphisms in which preserve the chosen points).
In a category of pointed objects , def. , the terminal object coincides with the initial object, both are given by itself, pointed in the unique way.
In this situation one says that is a zero object and that is a pointed category.
It follows that also all hom-sets of are canonically pointed sets, pointed by the zero morphism
Let be a category with terminal object and finite colimits. Then the forgetful functor from its category of pointed objects, def. , has a left adjoint
given by forming the disjoint union (coproduct) with a base point (“adjoining a base point”).
Let be a category with all limits and colimits. Then also the category of pointed objects , def. , has all limits and colimits.
Moreover:
the limits are the limits of the underlying diagrams in , with the base point of the limit induced by its universal property in ;
the colimits are the limits in of the diagrams with the basepoint adjoined.
It is immediate to check the relevant universal property. For details see at slice category – limits and colimits.
Given two pointed objects and , then:
their product in is simply ;
their coproduct in has to be computed using the second clause in prop. : since the point has to be adjoined to the diagram, it is given not by the coproduct in , but by the pushout in of the form:
This is called the wedge sum operation on pointed objects.
Generally for a set in
For a CW-complex, def. then for every the quotient (example ) of its -skeleton by its -skeleton is the wedge sum, def. , of -spheres, one for each -cell of :
For a category of pointed objects with finite limits and finite colimits, the smash product is the functor
given by
hence by the pushout in
In terms of the wedge sum from def. , this may be written concisely as
For a general category in def. , the smash product need not be associative, namely it fails to be associative if the functor does not preserve the quotients involved in the definition.
In particular this may happen for Top.
A sufficient condition for to preserve quotients is that it is a left adjoint functor. This is the case in the smaller subcategory of compactly generated topological spaces, we come to this in prop. below.
These two operations are going to be ubiquituous in stable homotopy theory:
symbol | name | category theory |
---|---|---|
wedge sum | coproduct in | |
smash product | tensor product in |
Let be pointed topological spaces. Then
denotes the standard interval object from def. , with a djoint basepoint adjoined, def. . Now for any pointed topological space, then
is the reduced cylinder over : the result of forming the ordinary cyclinder over as in def. , and then identifying the interval over the basepoint of with the point.
(Generally, any construction in properly adapted to pointed objects is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)
Just like the ordinary cylinder receives a canonical injection from the coproduct formed in , so the reduced cyclinder receives a canonical injection from the coproduct formed in , which is the wedge sum from example :
For pointed topological spaces with a locally compact topological space, then the pointed mapping space is the topological subspace of the mapping space of def.
on those maps which preserve the basepoints, and pointed by the map constant on the basepoint of .
In particular, the standard topological pointed path space object on some pointed (the pointed variant of def. ) is the pointed mapping space .
The pointed consequence of prop. then gives that there is a natural bijection
between basepoint-preserving continuous functions out of a smash product, def. , with pointed continuous functions of one variable into the pointed mapping space.
Given a morphism in a category of pointed objects , def. , with finite limits and colimits,
In the situation of example , both the pullback as well as the pushout are equivalently computed in . For the pullback this is the first clause of prop. . The second clause says that for computing the pushout in , first the point is to be adjoined to the diagram, and then the colimit over the larger diagram
be computed. But one readily checks that in this special case this does not affect the result. (The technical jargon is that the inclusion of the smaller diagram into the larger one in this case happens to be a final functor.)
Let be a model category and let be an object. Then both the slice category as well as the coslice category , def. , carry model structures themselves – the model structure on a (co-)slice category, where a morphism is a weak equivalence, fibration or cofibration iff its image under the forgetful functor is so in .
In particular the category of pointed objects, def. , in a model category becomes itself a model category this way.
The corresponding homotopy category of a model category, def. , we call the pointed homotopy category .
This is immediate:
By prop. the (co-)slice category has all limits and colimits. By definition of the weak equivalences in the (co-)slice, they satisfy two-out-of-three, def. , because the do in .
Similarly, the factorization and lifting is all induced by : Consider the coslice category , the case of the slice category is formally dual; then if
commutes in , and a factorization of exists in , it uniquely makes this diagram commute
Similarly, if
is a commuting diagram in , hence a commuting diagram in as shown, with all objects equipped with compatible morphisms from , then inspection shows that any lift in the diagram necessarily respects the maps from , too.
For any model category, with its pointed model structure according to prop. , then the corresponding homotopy category (def. ) is, by remark , canonically enriched in pointed sets, in that its hom-functor is of the form
Write for the classical model structure on pointed topological spaces, obtained from the classical model structure on topological spaces (theorem ) via the induced coslice model structure of prop. .
Its homotopy category, def. ,
we call the classical pointed homotopy category.
The fibrant objects in the pointed model structure , prop. , are those that are fibrant as objects of . But the cofibrant objects in are now those for which the basepoint inclusion is a cofibration in .
For from def. , then the corresponding cofibrant pointed topological spaces are tyically referred to as spaces with non-degenerate basepoints or . Notice that the point itself is cofibrant in , so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.
While the existence of the model structure on is immediate, via prop. , for the discussion of topologically enriched functors (below) it is useful to record that this, too, is a cofibrantly generated model category (def. ), as follows:
Write
and
respectively, for the sets of morphisms obtained from the classical generating cofibrations, def. , and the classical generating acyclic cofibrations, def. , under adjoining of basepoints (def. ).
The sets and in def. exhibit the classical model structure on pointed topological spaces of def. as a cofibrantly generated model category, def. .
(This is also a special case of a general statement about cofibrant generation of coslice model structures, see this proposition.)
Due to the fact that in a basepoint is freely adjoined, lemma goes through verbatim for the pointed case, with replaced by , as do the other two lemmas above that depend on point-set topology, lemma and lemma . With this, the rest of the proof follows by the same general abstract reasoning as above in the proof of theorem .
The category Top has the technical inconvenience that mapping spaces (def. ) satisfying the exponential property (prop. ) exist in general only for a locally compact topological space, but fail to exist more generally. In other words: Top is not cartesian closed. But cartesian closure is necessary for some purposes of homotopy theory, for instance it ensures that
the smash product (def. ) on pointed topological spaces is associative (prop. below);
there is a concept of topologically enriched functors with values in topological spaces, to which we turn below;
geometric realization of simplicial sets preserves products.
The first two of these are crucial for the development of stable homotopy theory in the next section, the third is a great convenience in computations.
Now, since the homotopy theory of topological spaces only cares about the CW approximation to any topological space (remark ), it is plausible to ask for a full subcategory of Top which still contains all CW-complexes, still has all limits and colimits, still supports a model category structure constructed in the same way as above, but which in addition is cartesian closed, and preferably such that the model structure interacts well with the cartesian closure.
Such a full subcategory exists, the category of compactly generated topological spaces. This we briefly describe now.
Literature (Strickland 09)
Let be a topological space.
A subset is called compactly closed (or -closed) if for every continuous function out of a compact Hausdorff space , then the preimage is a closed subset of .
The space is called compactly generated if its closed subsets exhaust (hence coincide with) the -closed subsets.
Write
for the full subcategory of Top on the compactly generated topological spaces.
Write
for the functor which sends any topological space to the topological space with the same underlying set , but with open subsets the collection of all -open subsets with respect to .
We need to show that for a -closed subset, then the preimage is closed subset.
Let be any continuous function out of a compact Hausdorff space . Since is -closed by assumption, we have that is closed in . This means that is -closed in . But by the assumption that is compactly generated, it follows that is already closed.
For there is a natural bijection
This means equivalently that the functor (def. ) together with the inclusion from def. forms an pair of adjoint functors
This in turn means equivalently that is a coreflective subcategory with coreflector . In particular is idemotent in that there are natural homeomorphisms
Hence colimits in exist and are computed as in Top. Also limits in exist, these are obtained by computing the limit in Top and then applying the functor to the result.
The following is a slight variant of def. , appropriate for the context of .
For (def. ) the compactly generated mapping space is the compactly generated topological space whose underlying set is the set of continuous functions , and for which a subbase for its topology has elements , for any open subset and a continuous function out of a compact Hausdorff space given by
If is (compactly generated and) a Hausdorff space, then the topology on the compactly generated mapping space in def. agrees with the compact-open topology of def. . Beware that it is common to say “compact-open topology” also for the topology of the compactly generated mapping space when is not Hausdorff. In that case, however, the two definitions in general disagree.
The category of def. is cartesian closed:
for every then the operation of forming the Cartesian product in (which by cor. is applied to the usual product topological space) together with the operation of forming the compactly generated mapping space (def. ) forms a pair of adjoint functors
For proof see for instance (Strickland 09, prop. 2.12).
For , the operation of forming the pointed mapping space (example ) inside the compactly generated mapping space of def.
is left adjoint to the smash product operation on pointed compactly generated topological spaces.
For a small category and a diagram, then the compactly generated mapping space construction from def. preserves limits in its covariant argument and sends colimits in its contravariant argument to limits:
and
The first statement is an immediate implication of being a right adjoint, according to cor. .
For the second statement, we use that by def. a compactly generated topological space is uniquely determined if one knows all continuous functions out of compact Hausdorff spaces into it. Hence it is sufficient to show that there is a natural isomorphism
for any compact Hausdorff space.
With this, the statement follows by cor. and using that ordinary hom-sets take colimits in the first argument and limits in the second argument to limits:
Moreover, compact generation fixes the associativity of the smash product (remark ):
On pointed (def. ) compactly generated topological spaces (def. ) the smash product (def. )
is associative and the 0-sphere is a tensor unit for it.
Since is a left adjoint by prop. , it presevers colimits and in particular quotient space projections. Therefore with then
The analogous reasoning applies to yield also .
Corollary together with prop. says that under the smash product the category of pointed compactly generated topological spaces is a closed symmetric monoidal category with tensor unit the 0-sphere.
Notice that by prop. also unpointed compactly generated spaces under Cartesian product form a closed symmetric monoidal category, hence a cartesian closed category
The fact that is still closed symmetric monoidal but no longer Cartesian exhibits as being “more linear” than . The “full linearization” of is the closed symmteric monoidal category of structured spectra under smash product of spectra which we discuss in section 1.
Due to the idempotency (cor. ) it is useful to know plenty of conditions under which a given topological space is already compactly generated, for then applying to it does not change it and one may continue working as in .
Every CW-complex is compactly generated.
Since a CW-complex is a Hausdorff space, by prop. and prop. its -closed subsets are precisely those whose intersection with every compact subspace is closed.
Since a CW-complex is a colimit in Top over attachments of standard n-disks (its cells), by the characterization of colimits in (prop.) a subset of is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the -disks are compact, this implies one direction: if a subset of intersected with all compact subsets is closed, then is closed.
For the converse direction, since a CW-complex is a Hausdorff space and since compact subspaces of Hausdorff spaces are closed, the intersection of a closed subset with a compact subset is closed.
For completeness we record further classes of examples:
The category of compactly generated topological spaces includes
all first-countable topological spaces,
hence in particular
Recall that by corollary , all colimits of compactly generated spaces are again compactly generated.
The product topological space of a CW-complex with a compact CW-complex, and more generally with a locally compact CW-complex, is compactly generated.
(Hatcher “Topology of cell complexes”, theorem A.6)
More generally:
For a compactly generated space and a locally compact Hausdorff space, then the product topological space is compactly generated.
e.g. (Strickland 09, prop. 26)
Finally we check that the concept of homotopy and homotopy groups does not change under passing to compactly generated spaces:
For every topological space , the canonical function (the adjunction counit) is a weak homotopy equivalence.
By example , example and lemma , continuous functions and their left homotopies are in bijection with functions and their homotopies .
(model structure on compactly generated topological spaces)
The restriction of the model category structure on from theorem along the inclusion of def. is still a model category structure, which is cofibrantly generated by the same sets (def. ) and (def. ). The k-ification coreflection of cor. is a Quillen equivalence (def. )
By example , the sets and are indeed in . By example all arguments above about left homotopies between maps out of these basic cells go through verbatim in . Hence the three technical lemmas above depending on actual point-set topology, topology, lemma , lemma and lemma , go through verbatim as before. Accordingly, since the remainder of the proof of theorem of follows by general abstract arguments from these, it also still goes through verbatim for (repeatedly use the small object argument and the retract argument to establish the two weak factorization systems).
Hence the (acyclic) cofibrations in are identified with those in , and so the inclusion is a part of a Quillen adjunction (def. ). To see that this is a Quillen equivalence (def. ), it is sufficient to check that for a compactly generated space then a continuous function is a weak homotopy equivalence (def. ) precisely if the adjunct is a weak homotopy equivalence. But, by lemma , is the same function as , just considered with different codomain. Hence the result follows with prop. .
Compactly generated weakly Hausdorff topological spaces
While the inclusion of def. does satisfy the requirement that it gives a cartesian closed category with all limits and colimits and containing all CW-complexes, one may ask for yet smaller subcategories that still share all these properties but potentially exhibit further convenient properties still.
A popular choice introduced in (McCord 69) is to add the further restriction to topological spaces which are not only compactly generated but also weakly Hausdorff. This was motivated from (Steenrod 67) where compactly generated Hausdorff spaces were used by the observation ((McCord 69, section 2)) that Hausdorffness is not preserved my many colimit operations, notably not by forming quotient spaces.
On the other hand, in above we wouldn’t have imposed Hausdorffness in the first place. More intrinsic advantages of over are the following:
every pushout of a morphism in along a closed subspace inclusion in is again in
in quotient spaces are not only preserved by cartesian products (as is the case for all compactly generated spaces due to being a left adjoint, according to cor. ) but by all pullbacks
in the regular monomorphisms are the closed subspace inclusions
We will not need this here or in the following sections, but we briefly mention it for completenes:
A topological space is called weakly Hausdorff if for every continuous function
out of a compact Hausdorff space , its image is a closed subset of .
Every Hausdorff space is a weakly Hausdorff space, def. .
For a weakly Hausdorff topological space, def. , then a subset is -closed, def. , precisely if for every subset that is compact Hausdorff with respect to the subspace topology, then the intersection is a closed subset of .
e.g. (Strickland 09, lemma 1.4 (c))
So far the classical model structure on topological spaces which we established in theorem , as well as the projective model structures on topologically enriched functors induced from it in theorem , concern the hom-sets, but not the hom-spaces (def. ), i.e. the model structure so far has not been related to the topology on hom-spaces. The following statements say that in fact the model structure and the enrichment by topology on the hom-spaces are compatible in a suitable sense: we have an “enriched model category”. This implies in particular that the product/hom-adjunctions are Quillen adjunctions, which is crucial for a decent discusson of the derived functors of the suspension/looping adjunction below.
Let and be morphisms in , def. . Their pushout product
is the universal morphism in the following diagram
If and are inclusions, then their pushout product from def. is the inclusion
For instance
is the inclusion of two adjacent edges of a square into the square.
The pushout product with an initial morphism is just the ordinary Cartesian product functor
i.e.
The product topological space with the empty space is the empty space, hence the map is an isomorphism, and so the pushout in the pushout product is . From this one reads off the universal map in question to be :
With
the generating cofibrations (def. ) and generating acyclic cofibrations (def. ) of (theorem ), then their pushout-products (def. ) are