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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.
88
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Is there an accepted definition of $(\infty,\infty)$ category?
For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with s …
- 30.3k
12
votes
6
answers
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What is a good basic reference on model categories?
I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for somet …
- 5,146
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Does abstract nonsense of model categories determine the "nonlinear" morphisms of $L_\infty$...
Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a …
- 118k
11
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2
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508
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How to interpret topologically that the equalizer in Groupoids of ${\rm id}, {\rm id}: BG \r...
Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object. Consider the diagram in (the 2-category) Groupoids with one vertex, labeled $1/G$, the one arrow from that vertex …
- 54.6k
6
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1
answer
362
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Does "simplicial" commute with "Bousfield localization"?
Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the left Bousfield localization $L_SM$ of $M$ with respect …
- 52.7k
10
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What suffices to check completeness in an n-fold Segal space?
Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the
Segal condition: For each $j$, the map
$$ X_j \to \underbrace{X …
- 54.6k
12
votes
1
answer
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I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-categ...
I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need …
- 27.5k
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1
answer
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Does there exist a model of chains on oriented manifolds with both a strict intersection pai...
Let $M$ be a smooth oriented $n$-dimensional manifold. My favorite model of $\operatorname{Chains}_\bullet(M) \otimes \mathbb R$ is the space of smooth compactly-supported de Rham forms on $M$, shift …
- 54.6k
17
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How are these algebraic and geometric notions of homotopy of maps between manifolds related?
Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and …
- 1,696
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Is there a "derived" Free $P$-algebra functor for an operad $P$?
Recall that an operad (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S_n$, with maps $P …
- 24.1k
7
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1
answer
452
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How equivalent are the theories of reduced and groupal $\infty$-groupoids?
I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this …
- 31.2k
27
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2
answers
4k
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What's the current state of the classification of not-fully-extended TQFTs?
Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some …
- 17.8k