Best demonstrated by query which returns no results when run on e.g. AskUbuntu: select * from posts where posttypeid = 1 and tags like '%<.%' This should return, for example, the 1,080 questions tagged .desktop, but none of them show up. These tags do appear in the Tags and PostTags tables, so i...

Suppose we have a small category $\mathcal{I}$, a diagram $D : \mathcal{I} \to \mathcal{C}$, and a functor $F : \mathcal{C} \to \mathcal{D}$. De know that the functor $\hat{F} : \text{Cone}_{\mathcal{C}}(D) \to \text{Cone}_\mathcal{D}(F \circ D)$ preserves the terminal object(s). In categories wi...

This is something that is glossed over far too many times, from what I can tell. I'd like to dig into the weeds a little bit, and my approach is a bit from the logician's side (meaning will use little natural language). I will address only one direction, since the other follows immediately. We wr...

Given some index category $\mathcal{I}$ and a complete category $\mathcal{C}$, we consider the diagonal functor $\Delta : \mathcal{C} \to [\mathcal{I}, \mathcal{C}]$ to be the functor that sends all objects $C$ of $\mathcal{C}$ to the constant diagram $\Delta(C)$ that maps each $I$ to the fixed o...

I’ve already known some examples that a finitely complete category can have not any infinite product. But I can’t find an example that a finitely complete category may not have infinite multiple equalizers, i.e. the limit of an infinite family of parallel arrows. But how could it be? That assumpt...