What Model Categories Model

In any normal curriculum on homotopy theory, you learn about model categories first and -categories much later (if ever). There are good reasons for why this order makes sense, but for a while you will be left wondering: What is it that model categories model?

In this post I will discuss some intuition about homotopy theory and model categories that I found helpful. It is a slightly odd discussion, since it goes over basics while already assuming solid understanding of the theory of model categories and -categories. It often makes sense in mathematics to go back over earlier material (earlier both in a didactic and in a historical sense) and see how it can be expressed differently with the tools that you have learned since. This gives you the chance to see your complicated and heavyweight machinery in action on examples that you already understand. But it also gives you an opportunity to compress and remember a lot of knowledge that previously had to be presented in an awkward way.

Localisation

In classical homotopy theory we say that a map between topological spaces is a weak homotopy equivalence if it induces isomorphisms for all . This notion of equivalence is much broader than that of categorical isomorphism or homotopy equivalence. In general, a weak homotopy equivalence does not even admit any form of generalised inverse. The topological spaces merely serve as a presentation of some underlying concept, a homotopy type. A weak homotopy equivalence then is a map of presentations which becomes an equivalence after passing to homotopy types.

This is the setting of abstract of homotopy theory. We want to study concepts that are difficult to mathematically capture directly by finding a presentation and noting which transformations of these presentations do not change the underlying concept. As a framework, abstract homotopy theory applies to much more than topological spaces, homotopy types, cups and donuts.

We start out with a category of presentations and single out a collection of maps in which we call the weak equivalences. We say that the pair is a relative category if contains at least the identity maps. Topological spaces with weak homotopy equivalences are one example of a relative category. So are chain complexes and quasi-isomorphisms, simplicial sets and Kan equivalences, and many more. We say that a functor between two relative categories is homotopical when it sends weak equivalences in to weak equivalences in . The relative categories and homotopical functors then form a category .

For every category there is a relative category in which the weak equivalences are exactly the isomorphisms. The musical symbol is a notational pun, indicating that this constitutes the natural choice of weak equivalences for the category . Since every functor preserves isomorphisms, we see that defines a functor by which is a subcategory of .

The weak equivalences in a relative category are not invertible; but what if we made them? Universal constructions in category theory allow us to attack such questions, and this is quite a common practice across mathematics. In commutative algebra we can take a subset of a commutative ring and ask for the ring obtained from by forcing the elements of to be invertible. The elements of are formal fractions and follow the same rules for fractions you have learned in school. In fact, we can obtain the rational numbers through this process by starting with the integers and asking for every non-zero integer to have a multiplicative inverse. Due to the role of this process in algebraic geometry, it has come to be called localisation.

In analogy to the localisation of rings, we can define a localisation of categories which freely makes a collection of morphisms invertible. This localisation can be compactly characterised by observing that the functor has a left adjoint:

The functor sends a relative category to its localisation (also called the homotopy category). The localisation satsifies the universal property that every functor that sends weak equivalences in to isomorphisms in induces a unique functor :

In other words, there is a homotopical functor (obtained as the unit of the adjunction) such that any other homotopical functor factors uniquely through . You can in principle create a concrete model for the localisation by equivalence classes of zigzags of morphisms, but in practice some powerful tools are necessary to practically work with the localisation.

-Localisation

The homotopy category is introduced early in every treatment of model categories, but it has a fundamental problem: it loses way to much information. There exist examples of model categories that are not Quillen equivalent but have equivalent homotopy categories. So while it is useful, it can not serve as the “basis-independent” description of a homotopy theory.

The problem arises since in the homotopy category all maps that are homotopic (in the appropriate sense) become equal, throwing away the higher homotopical structure. In some cases, the homotopy category can be augmented with extra structure to encode that higher structure in the objects (e.g. in the form of generalised mapping cylinders). To obtain a more intrinsic encoding, we can take the localisation in -categories instead.

The homotopy category and the -localisation have a very similar definition; however in the latter we allow the result to be an -category. The mapping spaces of the -localisation remember much more information than the homsets of the homotopy category. In particular, they do not just conflate two homotopic maps but keep track of the particular homotopies between them, and the homotopies between homotopies, etc.

When given a model category we now get an -category together with the localising functor . We have gained something very valuable here. Both the model category and its localisation are -categories and the localisation functor encodes their relationship. Notably everything is now codified as a mathematical object. This is very different to the situation I was presented with when first learning about model categories: the “homotopy theory” that a model category was supposed to represent was never made explicit and neither was the exact nature of that relationship. Thereby many claims about the adequacy of a construction in capturing the nature of the model category’s homotopy theory were meta-mathematical.

Diagrams and Rectification

The homotopy category and -localisation only need a notion of weak equivalences, but are in general very difficult to work with. Model categories come with much more structure and this structure can be used to make the localisation controllable.

When is a relative category and is a diagram, we automatically obtain a diagram in the localisation by postcomposition with the localisation map . By letting the diagram vary, this process becomes a functor

We can make into a relative category itself, whose weak equivalences are the natural transformations whose components are all weak equivalences in . Because sends weak equivalences to equivalences, the postcomposition functor induces a functor out of the localisation of the functor category at the natural weak equivalences:

When is a model category, this functor becomes an equivalence:

Any equivalence of -categories is in particular essentially surjective. It therefore follows that, up to equivalence, any diagram in the localisation can be rectified to be a diagram in :

This result is quite remarkable. For a model category , a diagram in only commutes up to a coherent system of homotopies. Yet it can always be rearranged in such a way that it is induced by a diagram in that commutes strictly on-the-nose.

The rectification result requires only half of the structure of a model category: It already works for sufficiently nice cofibration categories; and thereby by duality also for sufficiently nice fibration categories. The result works by using this structure to push homotopical information into the objects, generalising constructions like mapping cylinders that are used manually in classical homotopy theory.

Derived functors

A functor between relative categories only induces a functor between localisations if it is homotopical. However many interesting functors in homotopy theory aren’t homotopical, but admit homotopical “approximations” from one side or the other.

A functor is the left derived functor of if it is the right Kan extension of along the localisation map , i.e. if there is a natural equivalence

Dually, a functor is the right derived functor of if it is the left Kan extension of along the localisation map , inducing a natural equivalence between the functor -categories

Left or right derived functors do not always exist in general, and neither do they in general agree with each other. As a special case, when is already homotopical, the left and right derived functors exist and agree. Whether for some non-homotopical functor we chose to pick the left or right derived functor depends on what we intend the functor on localisations to do; because can detect aspects of the presentation that go away after localisation, there is no canonical choice.

These definitions, while general, are quite abstract and not very convenient to work with in practice. Most examples of derived functors arise when fails to be homotopical on all objects of but preserves weak equivalences between particularly “good” objects so that every object is weakly equivalent (from the left or from the right) to a “good” object in a coherent way. For example, when is a commutative ring and an -module, the tensor product functor does not preserve quasi-isomorphisms in general but becomes homotopical when restricted to chain complexes of projective modules.

When is a full subcategory we say that a functor is a left deformation if there is a natural weak equivalence . A homotopical left deformation induces an equivalence . If restricts to a homotopical functor , we can compute its left derived functor as the composite

To verify that this is in fact a model of the left derived functor, we can follow the sequence of natural equivalences

By defining the left derived functor abstractly via Kan extensions instead of directly going the route of left deformations we now see without additional effort that the left derived functor does not depend on the choice of left deformation. We are therefore free to pick any left deformation that is convenient for us.

For right derived functors, there is a dual notion of a right deformation with a natural weak equivalence . A similar verification as above shows that we can obtain right derived functors via right deformations.

When working with a model category , a cofibrant replacement functor provides a homotopical left deformation. Dually a fibrant replacement functor provides a homotopical right deformation. We can always produce such cofibrant and fibrant replacement functors as long as the model structure’s factorisation systems are functorial. When they aren’t, we can still use cofibrant or fibrant replacements to compute derived functors but the process is a bit more manual.

Quillen Adjunctions and Equivalences

Adjunctions are the bread and butter of category theory, and this is no different in the theory of -categories. It is therefore worthwhile to study when an adjunction of relative categories induces an adjunction of the localised -categories. This is the case for Quillen adjunctions between model categories.

When is a Quillen adjunction, then is not homotopical in general. However preserves weak equivalences between cofibrant objects, and so its left derived functor exists and can be computed via cofibrant replacement. Dually, preserves weak equivalences between fibrant objects, and so its right derived functor exists and is computed via fibrant replacement. These left and right derived functors are then again adjoints:

Quillen adjunctions therefore give us a tool to create adjunctions between -categories purely from -categorical data. For example, the identity functor of the category of simplicial sets forms a Quillen adjunction between the Joyal and Kan model structures

that induces a derived adjunction between the -categories of -categories and spaces:

The right adjoint is the inclusion, while the left adjoint sends an -category to its classifying space obtained by freely inverting all maps.

I quite like this example since it produces a non-trivial construction from trivial inputs. It is also part of a more general pattern: Whenever a model category is a left Bousfield localisation of , the identity functor on forms a Quillen adjunction

so that the derived adjunction is a reflective localisation (i.e. the right adjoint is fully faithful). It is this fact that motivates the name of the left Bousfield localisation.

The adjunction can be produced by the same mechanism as well. We can equip the category of marked simplicial sets with two different model structures: in the relative model structure a marked simplicial set is fibrant if is a quasicategory, contains the equivalences of , and is closed under composition and homotopy. In the cartesian model structure we require that is a quasicategory and that contains exactly the equivalences of . Then the identity adjunction on is a Quillen adjunction

and the derived adjunction is the localisation adjunction from above:

This is overall a bit circular since we are using the -localisation functor to define the derived adjunction in the first place. So this is a place where you would have to be quite careful that everything is bootstrapped properly and that you take proper care of size issues.

Another famous example is the nerve-realisation adjunction between simplicial sets and topological spaces, which forms a Quillen adjunction

This Quillen adjunction is special in that its unit and counit are both natural weak equivalences, i.e. it is a Quillen equivalence. Quillen equivalences are aptly named, for their derived adjunctions induce an adjoint equivalence between the -categories that they represent:

This gives a mathematical meaning to the slogan that two model categories represent the same homotopy theory when they are connected by Quillen equivalences.

Homotopy Limits

The localisation functor neither preserves nor reflects limits in general. However, the model categorical machinery gives us some tools to identify sufficient conditions for specific limit diagrams in to become limit diagrams in after localisation. We call such diagrams homotopy limit diagrams.

For example, the pullback of a map along a map in is a homotopy pullback when and are fibrant and is a fibration. This is a very useful criterion and allows us to calculate concrete models of pullbacks in the -category . Some introductory texts use this criterion to define homotopy pullbacks. That is a pragmatic choice at first but is conceptually unsatisfying.

Let us explore how one could arrive at the criteria to check if a limit diagram is in fact a homotopy limit diagram. To begin, we characterise how we can detect limits in an -category . Suppose that is a diagram and is an object of . For to be the limit of , we would need a limit cone consisting of a coherent system maps from to the diagram ; this can be packed into a natural transformation from the constant diagram to . Moreover we would need that any other cone factors through the limit cone up to a contractible space of factorisations. Taken all together, this data is equivalent to a natural equivalence

Now suppose that has all -shaped limits so that we can associate a limit to every diagram . We would then have an equivalence

that is both natural in and . In other words:

Earlier we have discussed how we can obtain adjunctions of -categories from Quillen adjunctions. So suppose that is a model category and is a model structure on whose weak equivalences are pointwise and so that the adjunction of -categories

is a Quillen adjunction. We then obtain a derived adjunction

Via rectification, we have an equivalence

The constant diagram functor is already homotopical, and so the left-derived functor is simply the constant diagram functor

Our derived adjunction therefore is equivalently of the form

By uniqueness of right adjoints it then follows that the right derived functor computes limits in the -category . In particular, if a diagram is already fibrant in the diagram model structure , taking the -categorical limit computes a homotopy limit.

It then remains to find a diagram model structure so that the adjunction becomes Quillen as we assumed above. If and are sufficiently nice, we can use the injective or Reedy model structures for this purpose. By the above argument, we need then just to unpack what it means for a diagram to be fibrant in order to get a sufficient criterion for when limits in compute limits in .

There is a dual story for colimits, with the colimit functor being the left adjoint to the constant diagram functor. The projective and Reedy model structures then provide criteria to detect homotopy colimits.

Shut up and calculate

In principle, you will be able to unpack many arguments and definitions that involve -categories to be purely on the level of the model categories. In fact, for many concrete calculations, you will find it to be incredibly helpful to use the machinery of model categories. So what do we gain by this -categorical perspective?

I knew that Quillen equivalent model categories represented the same homotopy theory. Yet somehow it still tripped me up. When I first learned of Segal spaces, I needlessly worried about why one source required them to be Reedy fibrant while the other only asked for them to be projectively fibrant. Now I know that a Segal space is a particular kind of functor into the -category of spaces, and you can choose whatever model of that which is most convenient for your purposes. During a course on stable homotopy theory, I got confused about the zoo of model categories of spectra; sequential spectra, symmetric spectra, orthogonal spectra, S-modules, etc. At least for me, they now make a lot more sense when I can see them as presentations with different tradeoffs of the same -category of spectra.

For me, the -categories makes the formalism a closer match with the abstract ideas that it seeks to represent. The computational affordances and explicit resolutions provided by a model category are additional structure that is not an essential part of the homotopy theory. I want to have this additional structure around just-in-time when I need it to compute something, but I want to get it out of the way otherwise. That way I find it easier to keep track of what is actually a property of the thing I ultimately care about and what is just an artifact of the specific representation. I also find that by dipping in and out of model categories only when needed allows to pick the most convenient representation for any specific argument instead of trying to make a global choice.