Search Results

The following theorem is proved in EGA IV 2.4.6: Every morphism of schemes, which is flat and locally of finite presentation, is open. I've already seen some applications of this theorem, so I want …

Where can I find a reference for the following fact: If $C$ is a small category with finite colimits, then $\text{Ind}(C)$ is a category with all colimits, and it is the universal one in the followin …

A category with just one object is a monoid. A category with two objects (which are distinguished) can be described by the following data (imagine the picture $\stackrel{M}{\curvearrowright} \bullet …

Let $X$ be a scheme. Does the forgetful functor $$\mathrm{EtSch}/X \to \mathrm{Sch}/X$$ have a right adjoint $Z \mapsto \tilde{Z}$? One might call $\tilde{Z}$ the étalification of $Z$. So this is an …

The notion of a "compact category" was introduced by Isbell$\color{red}{^{1,2}}$. A locally small category $\mathcal{C}$ is called compact when every functor $\mathcal{C} \to \mathcal{D}$ into any cat …

If $\mathscr{X}$ is a stack over some base ring $k$ (if you are not familiar with stacks, read "schemes" here), we may consider it as a pseudofunctor $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{Gpd}$ …

Here is a cute observation: Let $F,G : \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functors between symmetric monoidal categories, and let $\eta : F \to G$ be a monoidal transformation. Then …

This is probably well-known: Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed? - Is every profinite group isomorphi …

Let $k$ be a ring (perhaps a field). Let $M$ be the "set" of isomorphism classes of $k$-algebras and regard it as a commutative monoid with multiplication $\otimes_k $ and unit element $k$. There is a …

Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an …

The following statement is true* and not hard to prove. Let $X$ be a quasi-compact and separated scheme. Then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a free $\mathcal{O}_X …

Every monoidal category $(\mathcal{C},\otimes)$ can be seen as a one-object bicategory: the morphisms are the objects of $\mathcal{C}$, and the $2$-morphisms are the morphisms of $\mathcal{C}$. In eve …

Let $\{X_1 \rightrightarrows X_0\}$ be a smooth groupoid object in the category of affine schemes ($X_0 \to X_1$, $X_1 \to X_1$ and $X_1 \times_{X_0} X_1 \to X_1$ also belong to the datum). Equivalent …

Let $X$ be a scheme and $I \subseteq \mathcal{O}_X$ be a quasi-coherent ideal of finite type. The blowing up $\mathrm{Bl}_I(X)$ has the following universal property: It comes with a morphism $p : \mat …

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is cocomple …