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It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it. I'm sure this can be done not only for $\text{Se …

Let $\mathcal{C}$ be a small category; let $\mathcal{S}$ be any family of maps in $\text{Psh}\left(\mathcal{C}\right)$. Call $X\in \text{Psh}\left(\mathcal{C}\right) $ an $\mathcal{S}$-sheaf when $\ …

Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$ Then, being able to find a suitable $h$ i …

Let $\mathcal{M}$ be a locally finitely presentable model category, cofibrantly generated by two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial cofibrations with presentable domain …

I am reading about the Reedy model structure from Hovey's book and I was wondering if the Reedy model structure on $\mathcal{M}^{\Delta}$ is cofibrantly generated by a small set of arrows and permits …

The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of in …