# Questions tagged [homotopy-theory]

Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called a homotopy group can be obtained from the equivalence classes. The simplest homotopy group is the fundamental group. Homotopy groups are important invariants in algebraic topology.

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### Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
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### Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
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### What is the essential difference between a sheaf and a fibration with lifting property?

Are there cases where the two notions coincide? By sheaf I mean a pre-sheaf ($\mathcal{C}^{op} \rightarrow \mathcal{Set}$) satisfying the sheaf condition. The sheaf condition says that, for every ...
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### Many definitions of Hochschild homology and cyclic homology

It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some ...
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### Difference between $[pt/G]$ and $BG$

Let $G$ be a finite group. In topological category, we have the quotient stack of a point by $G$, denoted by $[pt/G]$. We also have the classifying space $BG$, which is a topological space. I am a ...
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### Map $\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y))$ bijection?

I have two questions. How do I see that the map$$\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y)), \quad [f] \mapsto f_*,$$is a bijection if $X$ is an $(n - 1)$-connected CW complex and $Y$ is ...
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