39

While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still an especially important place in our heart. Especially when we say algebraic topology instead of homotopy theory. This says that while all of category theory and ...


33

Model categories capture the idea that in many cases you resolve an object by an equivalent object that is better behaved. The standard example is replacing a chain complex by a chain complex of projectives (or injectives), which is quasi-isomorphic. This is used both in building the derived categories and in deriving functors. Model categories are used ...


32

I'll work in the based category, and consider $S^1$ as $\{z\in\mathbb{C}:|z|=1\}$. Consider the maps $$\text{point}\xleftarrow{}S^1\xrightarrow{f}S^1, $$ where $f(z)=z^2$. Suppose that there is a pushout $P$. We would then have a natural isomorphism $[P,X]=\text{Hom}(\mathbb{Z}/2,\pi_1(X))$. On the other hand, the fibration $$ S^1 = B\mathbb{Z} \...


27

I like the following example because it is very close to the origins of homotopy theory (and also because I worked on it at the beginning of my career): proper homotopy theory. Objects are topological spaces, maps are proper maps, one can define proper homotopies via cylinders in the usual way, weak equivalences are proper homotopy equivalences, and ...


27

The letter may be found on Georges Maltsiniotis' webpage containing material related to Pursuing Stacks. (A direct link to the pdf.)


27

(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in). I'm going to try with a very naive answer, although I'm not sure I understand your question exactly. The (un)stable motivic ($\infty$-)category has a universal property. To be precise the following statements are true Theorem: Every ...


25

A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the structure of a model category. Here, we know that a counterexample must fail to be left proper, so start with a diagram$\require{AMScd}$ $$ \begin{CD} a @>\...


25

Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where $C$ is a small $\infty$-category, $R=\{f_i\colon X_i\to Y_i\}$ is a set of maps in $\mathrm{PSh}(C)=\mathrm{Fun}(C^\mathrm{op}, \mathrm{Gpd}_\infty)$, and $\...


22

Abstract homotopy theory allows one to use the tools of homotopy theory (e.g. inverting weak equivalences, computing homotopy colimits, doing Bousfield localization, taking fibrant and cofibrant replacements, etc) in many different settings. In some of these settings (e.g. homological algebra), it's not a huge surprise that you can do so. But settings ...


22

Here's a class of counterexamples for the pointed homotopy category of connected CW complexes (so even this restriction does not save you). Let $hCW_{\ast}$ denote this category, and let $\pi_{\bullet} : hCW_{\ast} \to \text{Set}_{\bullet}$ denote the functor taking a pointed CW complex to its homotopy groups $\pi_n$. By Whitehead's theorem, $\pi_{\bullet}$ ...


20

No. The projective model structure on chain complexes of modules over a ring is an abelian model category, and the homotopy category is the derived category, which is never abelian unless the ring is semisimple.


17

The following might help answer the last part of your post: In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy ...


17

This answer is an elaboration on Dylan's comments. 1) Let us define a homotopy theory to be a pair $(C, W)$, where $C$ is a category and $W$ is some class of morphisms called weak equivalences. (Let's say that $W$ satisfies some reasonable properties, e.g. it contains isomorphisms and is closed under composition.) Of course, the archetypal example is given ...


17

I just want to point out, with regards to your second question, that the fact that the $\mathbb P^1$ is not equivalent to a simplicial complex homotopic to the sphere built out of affine spaces (or whichever model for an un-Tate-twisted sphere) is exactly what you would expect from Grothendieck et. al.'s theory of motives. In fact having a homology or ...


16

Homotopy limits in any model category always coincide with limits in the associated $(\infty,1)$-category. To see this, you need to know the following (classical) facts: 1) given a cofibrant object $A$, the mapping space functor $Map(A,-)$ (constructed as in Hovey's book, say, using a Reedy cofibrant resolution of $A$ in the category of cosimplicial objects)...


16

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories with enough injective objects. Let me use the notation $D^+(\mathcal{A})$ and $D^+(\mathcal{B})$ to denote the stable $\infty$-categories whose homotopy categories are the (cohomologically bounded below) derived categories of $\mathcal{A}$ and $\mathcal{B}$, respectively (you can also consider ...


16

Today we understand that what we are really interested in when we talk about "homotopy theory" are in the end "$\infty$-categories". In fact I have even heard some peoples claim that maybe in the future model categories would not be needed anymore and we will only talk about $\infty$-categories (in this post $\infty$ means $(\infty,1)$) . Even if I do not ...


15

There is a whole slew of examples given by the $J$-semi model structures which arise in the study of operads and algebras over an operad. A $J$-semi model category satisfies most of the axioms of a model category, but two have to be tweaked. First, not all trivial cofibrations lift against fibrations; only those with cofibrant domain do. The lifting of ...


15

A very interesting example: consider semi-simplicial sets (alias $\Delta$-sets). These are simplicial sets without degeneracies, and there is an ``adjoin degeneracies'' functor from semi-simplicial sets to simplicial sets that is left adjoint to the evident forget degeneracies functor. One can compose this with geometric realization or one can define ...


15

One specific statement that people are likely referring to when they say things about fibrations and cofibrations being "the same" in spectra is that a homotopy pushout square of spectra is also a homotopy pullback square (considering squares with one corner trivial gives homotopy fibration and cofibration sequences). A brief explanation of this is given by ...


15

I'd like to expand on Matthias's answer a little bit. There is some unfortunate terminology going around. Being a cofibration resp. fibration is a particular property a map can have; for spaces Hurewicz cofibrations and Serre cofibrations (relative CW-complexes) are ones that get most use, and they have associated notions of fibrations; the concept of a ...


14

One key point is that if you let $S/2$ denote the cofibre of twice the identity on the sphere spectrum (also known as the mod $2$ Moore spectrum, or $\Sigma^{-1}\mathbb{R}P^2$), then twice the identity map on $S/2$ is nonzero. This cannot happen in the homotopy category of simplicial abelian groups, or (roughly speaking) any other triangulated category ...


14

The answer is yes: see the paper of Chu-Haugseng-Heuts, "Two models for the homotopy theory of ∞-operads", arXiv:1606.03826. In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial operads", arXiv:1109.1004) proved a Quillen equivalence between simplicial operads and dendroidal sets. In the paper of Cisinski and Moerdijk that you link to, ...


14

In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and Ulmer's?) theorem that it is impossible for a category and its opposite both to be locally presentable, unless they are both posets. Indeed, if $A$ is a set ...


13

One fundamental difference concerns the behavior of Postnikov towers, or the relationship between the spectrum/simplicial abelian group and its homotopy groups. In simplicial abelian groups all Postnikov towers are splittable, since there are no higher Ext's between abelian groups; thus every simplicial abelian group is equivalent to a product of K(A,n)'s. ...


13

It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a recent result of Lennart Meier, a certain "double cosubdivision" of the Rezk classification diagram of a model category is a complete Segal space, so (by a result of Joyal and Tierney) we can take degreewise 0-...


13

In response to Ryan Budney's comment, let me try to say something about topological data analysis, and other recent applications of algebraic topology outside of traditional mathematics. Applied Algebraic Topology has been around in various forms for many years. I first learned about it in my training in computer science from Rob Ghrist's work. In fact, I ...


13

Model categories provide a powerful framework for commuting (homotopy) limits and colimits, and, more generally, for commuting left adjoint functors and (homotopy) limits, as well as right adjoint functors and (homotopy) colimits. (In what follows, I omit the adjective “homotopy” before limits and colimits.) In finitely presentable ∞-categories filtered ...


12

An even more compelling reason is that stable natural transformations between generalized cohomology theories on spaces form the wrong category, i.e. are not the same as the morphisms of the stable category. Let $E,F$ be generalized cohomology theories. Let $Z_n$ be the terms of the $\Omega$-spectrum of a CW model of $E$. Then, more or less by definition, ...


12

For an introductory textbook I will recommend Homotopy theories and model categories by Dwyer and Spalinski. This 56-page paper is one chapter of the book "Handbook of algebraic topology" and gives a reader-friendly and comprehensive introduction to model category. I think the book "Model Category" by Hovey, as tetrapharmakon and David has already ...


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