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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.
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Products of representables are regular on a regular skeletal Reedy category?
The definition of a (pre)regular skeletal Reedy category in the sense of Cisinski generalizes the intuition behind the Eilenberg-Zilber factorization for simplicial sets, specifically the property tha …
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Cartesian-closed category of spaces with the Whitehead property?
I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes). …
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Abstract classes of anodyne maps (relative to an interval) in a presheaf category are stable...
Let $A$ be a small category, and let $X:=Psh(A)$ denote the category of presheaves on $A$. It is a theorem that for any such category $X$, there exists a small set $M$ of monomorphisms admitting the …
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A question about fibrations of simplicial sets and their fibers
I couldn't think of a title for this, but here we go:
Fix $p:S\rightarrow T$, a left fibration of simplicial sets, and an edge $f:\Delta^1 \rightarrow T$. Let $t$ be the first vertex of $f$, and $t' …
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In what generality does the following statement hold: A fibration is acyclic if and only if ...
This may not be precise enough for MO, but I'll give it a go.
Let $M$ be a symmetric closed monoidal model category with unit $u\in Ob(M)$. We define the vertices of an object $A$ to be points $x\in …
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Does derived algebraic geometry allow us to take quotients with reckless abandon?
So one of the major problems with the categories of schemes and algebraic spaces is that the "correct quotients" are oftentimes not schemes or algebraic spaces. The way I've seen this sort of thing r …
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When are "diagrams of cofibrations" projectively cofibrant?
Let $P$ be a small category, and let $F:P\to M$ be a diagram in a left-proper combinatorial model category $M$. We say that $F$ is a diagram of cofibrations if for every object $p\in P$, $F(p)$ is co …
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A Grothendieck fibration over a weakly contractible category with weakly contractible fibers...
Let $B$ be a weakly contractible category (that is, it has a weakly contractible nerve), and let $F:E\to B$ be a Grothendieck fibration. Suppose further that the ordinary fibers of the Grothendieck f …
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Pointwise evaluation of the Beck-Chevalley map in $\infty$-categories
In Higher Algebra Lemma 6.1.6.3, most of the proof is pretty straightforward, but after thinking I understood it all correctly, I realized I had a gap in my understanding.
Suppose we have a homotopy …
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Derivators (in English)
Grothendieck, before he disappeared, was working on a manuscript called "Les Derivateurs", which detailed the theory of derivators. Prof. Cisinski has done work with them as he mentioned in this post …
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The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres
Since Ronnie Brown and his collaborators have come up with a general proof of the higher Van Kampen theorems, what impediments are there to using these to compute the unstable homotopy groups of spher …
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Excellent monoidal model categories admit enriched fibrant replacement functors?
Let $\mathbf{S}$ be an excellent model category in which all objects are cofibrant, viewed as an $\mathbf{S}$-enriched category by its canonical self-enrichment. Then we know that there is an obvious …
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Computing homotopies
Oftentimes, in the standard algebraic topology books (May, Switzer, Whithead, for instance), there are tricky little proofs that depend on proving that two maps are homotopic. This is comparable to t …
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What is the "universal problem" that motivates the definition of homotopy limits/colimits (a...
The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan extens …
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Product-preserving fibrant replacement functor for the Joyal model structure
There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\cd …
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