Here is an example of such pair $T$ and $R$. $k$ is a finite field. $G$ is the Lie group $SO(2)$ and $G^{\delta}$ the same group but with dicrete topology. $R$ is the group algebra $k[G^{\delta}]$. $...

Adeel answer is perfect, I will be more basic. There is many notions of "2-category" structure here around. 1) the category of small dg-categories $\mathbf{dgCat}$ is symmetric monoidal closed ...

Sketch of proof. I will use the following ingredients 0) the category of spaces will be the category of simplicial sets. All the computation are in the derived sense. 1) use the Quillen adjunction $...

Let start with notations: $Y$ is a spectra, $H\mathbb{Z}$ is the Eilenberg-Mac Lane spectra associated to $\mathbb{Z}$ The singular homology of $Y$ is given by $H_{i}(Y)=\pi_{i}^{stable}(Y\wedge H\...

May be I going to say something obvious, but I hope it will give some clarification since I feel (Maybe I'm wrong) that there is some confusion about where you are taking the homotopy cofiber ...

May be I misunderstood your question. $C^{\ast}(X)$ is an $E_{\infty}$-algebra. For pointed simply connected space $X$ there is an equivalent of $E_{\infty}$-algebra between $BC^{\ast}(X)$ and $C^{\...

I will try to give a example in topology and then in dg-world. Suppose that $G$ and $H$ are topological groups. There is a fiber sequence $$Map_{\ast}(BG,BH)\rightarrow Map(BG,BH)\rightarrow BH$$ ...

First of all Muro's answer is perfect. This a long comment which I hope it will be helpful, it is more about intuition. I think Rezk's answer Group completion theorem is closely related to your ...

I did not understand exactly your question. What is important here, is the ring $k$. If $k=\mathbf{Z}$ then the problem you are asking for is hard, if $k$ is any commutative ring then the problem is ...

The spiral of prime numbers (white dots) the "pattern" is amazing, for an explanation of the picture you can take a look to this short youtube video.

I think the right way to call this functors is "quasi-representable" functors (in the language of Toen) and "potentially distinguished" functors in the language of Dwyer-Hess. At least when $\mathbf{...

I think you will have different answers to your questions. As you noticed there is more about Hochschild Homology in n-lab page and you also pointed out the interpretation of (positive) low dimension ...

$\textbf{This is a point of view which means that it is only one side of the story}$. Let me start with Gel'fand theorem. The (opposite) category of compact Hausdorff spaces is equivalent to the ...

in the rational case, the forgetful functor U: dgCAlg-------> dgAlg induces a faithful functor in homotopy category under the following conditions. R is connected (connective) differential commutative ...

I will try to answer the question. As I said in a comment, the Thomason model structure on $Cat$ is not simplicial model structure. Let $C$ be a small category, we will view it as a topological ...

I'm just reformulating your question in simplicial case. If you consider the category of simplicial sets $\mathbf{sSet}$ you can formulate your question as follows: Is the diagonal functor $diag: [\...