I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263. Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-schemes, then $\operatorname{Lif}(X,\tilde{S})$ is the gerbe of liftings to $\tilde{S}=S(\mathbb{Z...

I've just stumbled on something that seems either too good to be true, or else too good for me not to have heard of it before. It has to do with the basepoint forgetting map $$ u: [A, M] \to \langle A, M \rangle, $$ where $A$ is a pointed space, $M$ is a topological monoid, and $\langle A, M\ran...

Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\sim$$ where $(a,b)\sim (a',b')$ if and only if $a+b'=a'+b$. It has been quite surprizing find ou...

The question is from the definition of to topological group. I can find an example such that the inversion map is continuous but the group operation is not continuous, but I cannot find an example such that the group operation is continuous but the inversion map is not continuous. I guess that su...