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Results tagged with homotopy-theory
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user 8818
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.
13
votes
2
answers
1k
views
Correspondence between operads and $\infty$-operads with one object
Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose hom-spac …
- 1,738
10
votes
1
answer
556
views
Loops and suspensions of higher categories
Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ Du …
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8
votes
$RO(G)$-graded homotopy groups vs. Mackey functors
I can answer your first question in some special cases.
Let $p$ be a prime and $G=C_p$ the cyclic group of order $p$. If $p=2$, the answer to your question is yes and if $p$ is odd, then it is no.
…
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5
votes
Accepted
The homotopy of universal Thom spectrum
Assume that $R$ is a connective $E_\infty$ ring spectrum. Typically $GL_1(R)$ denotes the set of components in $\Omega^\infty R$ which span $GL_1(\pi_0 R)=\pi_0 R^\times$. I would call the unit compon …
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4
votes
Group actions in a homotopy category
The spectral sequence I constructed with Niles Johnson was precisely designed to handle questions of this sort (here is a version that is closer to the publication version: T-algebra SS). A special ca …
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4
votes
Accepted
Loops and suspensions of higher categories
First let me thank Urs, Karol, and Rune Haugseng for helpful comments.
Now note that the inclusion, $i$, of $\infty$-groupoids into $(\infty,n)$-categories has an $\infty$-categorical left adjoint, $ …
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4
votes
Weak operad and deloopings
Some clarifications:
1) You need that $X$ is grouplike (so the induced multiplication makes $\pi_0 X$ a group). This condition is always satisfied for a loop space, but not satisfied by the discrete …
- 1,738