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Results tagged with homotopy-theory
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user 36625
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.
6
votes
Accepted
Is the discrete nerve of a small category a complete Segal space?
Your assertion $discnerve(\mathcal{C})=N\tilde{\mathcal{C}}$ is false.
Indeed, the category $id(\mathcal{C}^{[m]})$ (identities between chains of $m$ maps of $\mathcal{C}$), whose nerve is $discnerve( …
- 1,609
8
votes
1
answer
388
views
Unicity up to homotopy of simplicial enrichments
On the one hand, in their paper Simplicial structures on model categories and functors, Rezk, Schwede and Shipley proved that a simplicial model category structure on a given model category is unique …
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4
votes
1
answer
230
views
Factorization of morphisms in a diagram category
Let us suppose that $I$ is a small category and $\mathcal{E}$ a combinatorial model category. Then there exists two Quillen equivalent combinatorial model category structures on the diagram category $ …
- 1,609
8
votes
Definition of E-infinity operad
There exists several models of $E_{\infty}$-operads. One model which works in chain complexes over any ring is the Barratt-Eccles operad, defined by applying aritywise the normalized chain complex to …
- 1,609
8
votes
0
answers
240
views
Framed higher Hochschild cohomology
Given an $E_n$-algebra $A$, one can define its $E_n$-Hochschild complex $CH_{E_n}(A,A)$ by the formula $$Ch_{E_n}(A,A)=RHom_{Mod_A^{E_n}}(A,A)$$ where $Mod_A^{E_n}$ is the category of $A$-modules over …
- 1,609
23
votes
Why do people say DG-algebras behave badly in positive characteristic?
In characteristic zero, the model structure on commutative dg-algebras is obtained by transfer from the projective model structure on chain complexes, along the ajunction between the free algebra func …
- 1,609