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Results tagged with homotopy-theory
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Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
2
votes
Accepted
Fibrations in a model structure for homotopy $n$-types of simplicial sets
No, this is not true. A counterexample is provided in Hirschhorn's book, Example 2.1.6 on page 36. See also the text on page 71: "Unfortunately, Example 2.1.6 shows that not all S-local trivial cofibr …
- 30.3k
9
votes
Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
In the ten years since this question was asked, there has been a lot of progress in algebraic $K$-theory. For example, Achim Krause, Ben Antieau, and Thomas Nikolaus came up with an algorithm to compu …
- 30.3k
2
votes
Accepted
Simplicial enrichment on unbounded algebras over an operad
There is no obstruction. If $M$ is a simplicial monoidal model category, and $O$ is an operad in $M$, then the category of $O$-algebras is simplicially enriched, tensored, and cotensored. If it's a mo …
- 30.3k
2
votes
Accepted
“Geometric” vs Homotopical completion
Yes, there's a way to relate the two. First, it's helpful to think of both in terms of universal properties. Since geometric completion is a fiber product, it's a pullback. Meanwhile, the homotopical …
- 30.3k
3
votes
Accepted
Homotopical Combinatorics
Back when this question was asked, several people suggested they were not sure what was meant by "homotopical combinatorics." Well, now there's a subfield known as homotopical combinatorics. It concer …
- 30.3k
9
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Homotopy theory and algebraic topology last 10 years. Is it a dying field?
No, it's not dying at all. If anything, now is the best time to do homotopy theory. Thanks to the recent work of Lurie and others, homotopy theory is easier than ever to get into (advances have lowere …
6
votes
Bar construction in commutative algebras is calculated by pushout
Welcome to MathOverflow! First, let me point out that what you're asking is already true at the 1-categorical level. The pushout in the category of commutative rings is computed by the tensor product. …
- 30.3k
2
votes
Reference request for equivalences between different models of lax limits
This is a great question. Let me start with limits and discuss lax limits later. Given a $D$-shaped diagram $X$ of model categories (where $D$ is a small category), one can ask whether the two ways (B …
- 30.3k
5
votes
Accepted
Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection
This sounds like a very exciting phase of your studies! As you point out, there's plenty of intersections between algebraic geometry and homotopy theory. One intersection is motivic homotopy theory, a …
- 30.3k
13
votes
Accepted
Plus construction on Simplicial Sets?
The answer is yes. This is spelled out in the book The local structure of algebraic K-theory by Bjørn Ian Dundas, Thomas G. Goodwillie and Randy McCarthy. Check out Section 1.6.1 on page 26, where the …
- 30.3k
12
votes
Accepted
Model categories as a tool to resolve size issues for localizing categories
I guess I'm the canonical person to answer this question. I wrote those notes as a PhD student, a long time ago, to go along with a talk I was giving at a grad student conference. They were basically …
- 30.3k
7
votes
Accepted
Is the mapping cylinder a replacement for morphism by cofibration in model categories?
The short answer is "yes," it is true that the induced map $X\to M_f$ is a cofibration. I refer you to Section IX of Williamson's thesis Cylindrical model structures, page 114 of the pdf. He says that …
- 30.3k
1
vote
Accepted
Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
This question was already answered in the comments, but I don't want it to linger forever on the unanswered queue, so I'm making a CW answer summarizing the comments and adding my own example.
Tyrone …
5
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Non-examples of model structures, that fail for subtle/surprising reasons?
This week, we learned that another example is the Strøm (aka Hurewicz) model structure on the category of simplicial sets. Specifically, there is no model structure on $sSet$ whose class of weak equiv …
- 30.3k
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A category with weak equivalences that is not a model category
This week, we learned that another example is the category of simplicial sets, and the class of weak equivalences the simplicial homotopy equivalences. All credit to Tom Goodwillie, Tim Campion, and T …