Questions tagged [homotopy-theory]
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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LS category of the quotient of a manifold by its involution
$\DeclareMathOperator\cat{cat}$Let $\cat(X)$ denote the Lusternik–Schnirelmann (LS) category of $X$. This homotopy invariant has been well-studied for CW complexes: it is $0$ if and only if $X$ is ...
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Two definitions of a monad on an ∞-category
In the literature on $\infty$-categories (quasi-categories) I found two different definitions of a monad on an $\infty$-category, and I don't understand the relation between them.
The first ...
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Is there a theory of fundamental groups for $C^*$-algebras instead of topological spaces?
Is it possible to construct a theory of fundamental groups $\pi_1 (A,a_0)$ for pointed $C^*$-algebras $(A,a_0)$ instead of pointed topological spaces $(X,x_0)$ : $\pi_0 (X,x_0)$ ?
If the answer is yes,...
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Spaces in the spectrum THH(R)
Let $R$ be a ring spectrum. Then we can form the topological Hochschild Homology of $R$ as the spectrum
$$THH(R) = R \otimes S^1 \simeq R \wedge _{R \wedge R^{op}} R.$$
What is known about the spaces ...
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How to determine the LS category of branched covers?
Define the (normalized) Lusternik-Schnirelmann (LS) category of a space $X$, denoted $\mathsf{cat}(X)$ to be the least integer $n$ such that $X$ can be covered by $n+1$ number of open sets $U_i$ each ...
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Pushouts vs contractible colimits
Suppose that $C$ has all weakly contractible colimits, i.e. colimits of functors $F: I \rightarrow C$ where the geometric realization $|I|$ is weakly contractible. Then $C$ has pushouts and filtered ...
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Proof of the equivalence of spectra $(\mathbb{S}^{-1} \otimes \mathbb{S}^{-1})_{h \Sigma_2} \cong \Sigma^{-1} \mathbb{RP}_{-1}^{\infty}$
$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\Th}{Th}$I am trying to give a hands-on proof of the equivalence of spectra in the title. I am using the definitions $\mathbb{RP}^{\infty}_{-...
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Simplicial right Kan extensions and Cartesian transformations
I will write the concrete question first, in case the answer is clear independently of the context:
Question: Given an $\infty$-topos $\mathfrak{X}$ and a diagram $F\colon\Delta^1\times\Delta_+^{op}\...
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How to get an $E_\infty$-ring from a commutative differential graded ring?
I want to figure out the following question: How to get an $E_\infty$-ring from a commutative differential graded ring?
More precisely, let $\operatorname{cdga}$ be the ($1$-)category of cdgas, let $...
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Is the standard model structure on reduced simplicial sets cofibrantly generated?
Let $\mathrm{sSet}_0$ be the category of simplicial sets with a single zero cell, also known as reduced simplicial sets. It is a well known fact (due to Quillen) that $\mathrm{sSet}_0$ supports a ...
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A few questions about Priddy’s construction of $BP$
In A Cellular Construction of BP and Other Irreducible Spectra, Priddy gives an interesting approach to constructing the Brown-Peterson spectrum $BP$. His result is often summarized as
If you start ...
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Reference for Reedy weak factorization systems, not Reedy model structure?
What paper/book is appropriate as the standard reference for Reedy weak factorization systems, not Reedy model structure? Specifically, I would like a reference for the following Propositions 1 and 2:
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Are isomorphisms of fundamental quandles canonical?
Given an (oriented) knot projection, we can create a quandle by assigning each curve segment a generator, and a relation $a \triangleright b = c$ for each intersection. If two projections are related ...
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Examples of cyclic A-infinity algebra
I am wondering about (references to) examples of cyclic A-infinity algebras- especially including explicit descriptions of the structure maps and pairing.
Thanks a lot!
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Identifying a map in a fiber sequence
Let $Q = \Omega^{\infty} \Sigma^{\infty}$ be the stabilization functor. Suppose we have a sequence of maps $Q \mathbb{RP}^{n-1} \to Q \mathbb{RP}^{n} \to QS^n$
and suppose we know that it is a fiber ...
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