Modeling Homotopy Theories Julia E. Bergner Throughout mathematics, each field has fundamental ob- which induce isomorphisms on all homology groups. In jects of study, whether it be real-valued functions or vector these examples, there is no reason to assume that such a spaces or Lie algebras or smooth manifolds. Furthermore, function has any kind of meaningful function going in the we are not only interested in what these objects are but reverse direction at all. These two situations provide clas- also in appropriate kinds of functions between them, such sical examples of “homotopy theories.” as linear transformations of vector spaces or smooth func- Our aim in this article is to give an overview of the devel- tions between manifolds. Of particular interest are those opment of the notion of homotopy theory, starting with functions that tell us that two specific objects are the same these classical examples and leading up to more modern as each other in some critical way. For example, we typ- perspectives. As we discuss briefly at the end, we have cho- ically do not distinguish between two different sets that sen one of several possible entry points into this subject each have three elements, since they are in bijection with and in particular do not say much about the higher categor- one another. Similarly, we ignore the difference between ical aspects of the theory. We simply remark that the “ho- groups that are isomorphic or topological spaces that are motopy theories” that we discuss here coincide with the homeomorphic. In each of these examples, the criterion “(∞, 1)-categories” that are being used widely in a num- for “sameness” is quite rigid: there is a function of the ap- ber of related fields. propriate flavor (function of sets, group homomorphism, or continuous map) that admits an inverse. Topological Spaces and Chain Complexes In other situations, we have notions of “sameness” that We begin our investigations with the classical homotopy are not quite so strict. For instance, since it is generally theory of topological spaces. Let us first recall the precise extremely difficult to determine whether there is a homeo- definition of homotopy equivalence. morphism between two topological spaces, one might ask 푓∶ 푋 → 푌 instead whether there is a homotopy equivalence between Definition 1. A continuous map of topological them, namely, a map that may not have an inverse on the spaces is a homotopy equivalence if there exists a continuous 푔∶ 푌 → 푋 푔∘푓 nose but only up to homotopy. map such that is homotopic to the identity 푋 푓 ∘ 푔 Many further examples can be described via invariants. map of and is homotopic to the identity map of 푌 One might consider weak homotopy equivalences of topo- . logical spaces, which induce isomorphisms on all homo- To define the related notion of weak homotopy equiva- topy groups, or quasi-isomorphisms of chain complexes, lence, we need to consider the homotopy groups of a topo- logical space. The most familiar of these groups is the fol- Julia Bergner is a professor of mathematics at the University of Virginia. Her lowing. email address is [email protected]. For permission to reprint this article, please contact: Definition 2. Let 푋 be a topological space with a chosen [email protected]. basepoint 푥0. Then its fundamental group is 휋1(푋, 푥0) = 1 DOI: https://doi.org/10.1090/noti1957 [푆 , 푋]∗, the set of homotopy classes of basepoint- OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1423 that the composite 푊 → ∗ → 푊 is homotopic to the identity on 푊. However, we had to use a rather exotic example of a topological space to see the difference between our two notions of equivalence, so we might ask if they agree for maps between spaces that are sufficiently nice. A good an- swer comes from the following two theorems. Whitehead’s Theorem. Any weak homotopy equivalence be- tween CW complexes is a homotopy equivalence. A CW complex is a topological space that, roughly speaking, is built by gluing together cells of different Figure 1. A quasi-circle. dimensions. Such spaces are considered well-behaved be- cause they can be described so explicitly, but the following 푆1 → 푋 preserving maps which has a natural group op- theorem also tells us that they are sufficiently general to eration. capture much of the behavior of topological spaces. While not as frequently presented in an introductory CW Approximation Theorem. Given any topological space course, higher homotopy groups can be defined analogous- 푋, there exists a weak homotopy equivalence 푌 → 푋 with 푌 a ly. CW complex. 푋 Definition 3. Let be a topological space with basepoint Taking these two theorems together, we can transition 푥 푛 ≥ 1 푛 푋 0. Then for any , the th homotopy group of is from the usual category of topological spaces and contin- 휋 (푋, 푥 ) = [푆푛, 푋] 푛 0 ∗. uous maps to a new category in which weak homotopy This definition also works when 푛 = 0, but in that case equivalences become isomorphisms. This category still 휋0(푋) is simply the set of path components of 푋 and does has all topological spaces as objects, but given two spaces not have a natural group structure. 푋 and 푌, we define the set of maps 푋 → 푌 to consist of When 푛 ≥ 1, the group 휋푛(푋, 푥0) does not depend homotopy classes of maps between CW replacements of on the specific basepoint 푥0 but only on the path compo- 푋 and 푌. Since homotopy equivalences have inverses up nent from which it is chosen. Hence, in what follows we to homotopy, their corresponding homotopy classes have implicitly assume that appropriate statements hold for all inverses in this new category, which we call the homotopy path components and simply write the homotopy groups category of spaces. as 휋푛(푋). Now let us look at an algebraic example for which the An important feature of homotopy groups is that they same kind of phenomenon occurs. Let 푅 be a ring, and are functorial: given a continuous map of topological consider nonnegatively graded chain complexes of spaces 푓∶ 푋 → 푌, there are induced group homomor- 푅-modules phisms 푓∗ ∶ 휋푛(푋) → 휋푛(푌). 휕1 휕2 퐶• = (퐶0 ← 퐶1 ← 퐶2 ← ⋯). Definition 4. A continuous map of topological spaces Each 퐶푖 is an 푅-module, and each composite 푋 → 푌 is a weak homotopy equivalence if, for every 푛 ≥ 0, 휕푖 휕푖+1 the map 퐶푖−1 ← 퐶푖 ← 퐶푖+1 휋푛(푋) → 휋푛(푌) is the zero map. Given another such chain complex 퐷•, a is an isomorphism. chain map 푓∶ 퐶• → 퐷• consists of 푅-module homomor- 푓 ∶ 퐶 → 퐷 푖 ≥ 0 One can check, using functoriality of homotopy groups, phisms 푖 푖 푖 for each so that the diagram o o o that any homotopy equivalence is a weak homotopy equiv- 퐶0 퐶1 퐶2 ⋯ alence. However, the converse does not hold, as the fol- 푓0 푓1 푓2 lowing example illustrates. o o o Example 5. Let 푊 be the quasi-circle in which an arc of 퐷0 퐷1 퐷2 ⋯ 1 1 푆 is replaced by a sin( 푥 ) curve and its limit interval, as commutes. depicted in Figure 1. Then the unique map 푊 → ∗ is a Just as for the example of topological spaces, there are weak homotopy equivalence, since the homotopy groups two natural ways to think of equivalences between chain of both spaces are all trivial. However, this map is not a complexes. First, there is the notion of chain homotopy homotopy equivalence, since there is no map ∗ → 푊 such equivalence, which means having an inverse up to chain 1424 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 homotopy. There is also the weaker notion of quasi- following the primary example of topological spaces, are isomorphism or map that induces isomorphisms on all ho- called fibrations and cofibrations, which are subject to several mology groups 퐻푖(퐶•) = ker(휕푖)/ im(휕푖+1). So, again, axioms. The following definition first appeared in [15], we see two flavors of equivalence: one that produces iso- although we follow more closely the one given in [6]. morphisms after taking appropriate homotopy classes of ℳ maps, and one defined by an algebraic invariant. Definition 8. A model category is a category together with a choice of three distinguished classes of morphisms: So what is the comparable kind of object to CW com- ∼ plexes here? The following theorem provides an answer. weak equivalences (→), cofibrations (↪), and fibrations (↠), each closed under composition and containing all isomor- Theorem 6. Any quasi-isomorphism between nonnegatively phisms and such that the following five axioms hold. graded chain complexes made up of projective modules is a chain (MC1) The category ℳ has all small limits and colimits. homotopy equivalence. (MC2) All three classes of morphisms are closed under But, as before, chain complexes of projective modules retracts. are also quite general. (MC3) The class of weak equivalences satisfies the two-out- of-three property: if 푓∶ 푋 → 푌 and 푔∶ 푌 → 푍 are Theorem 7. Any nonnegatively graded chain complex has a morphisms in ℳ such that two of 푓, 푔, and 푔푓 projective resolution, namely, a quasi-isomorphic chain complex are weak equivalences, then so is the third. made up entirely of projective modules. (MC4) Consider any commutative square Once again, we can define a category whose objects are / 퐴 _ > 푋 푅 } nonnegatively graded chain complexes of -modules and } whose maps are chain homotopy classes of maps between 푖 } 푝 } projective resolutions. The resulting category is often / 퐵 푌. called the derived category in homological algebra.