# Tag Info

Accepted

### Examples of a categories without products

For fields the most basic problem is fields of different characteristic. If $K$ and $L$ are fields, then the product $K\times L$ (if it existed) would have to be a field that can map into both $K$ ...
Accepted

### Compact subset in colimit of spaces

As you suggest, choose a sequence of points $x_n\in K\cap X_n\setminus X_{n-1}$ (possibly replacing $(X_n)$ with a subsequence). Let $A=\{x_n\}$. Then if $B\subseteq A$, then $B\cap X_n$ is finite ...
Accepted

### Calculating (co)limits of ringed spaces in $\mathbf{Top}$

The forgetful functor $\mathsf{LRS} \to \mathsf{RS}$ has a right adjoint. The right adjoint "$\mathrm{Spec}$" is a rather direct generalization of the spectrum of a commutative ring. You can find the ...

Accepted

### Example of complete category with no initial object

None of the examples of categories with no initial object you describe are complete. It is actually tricky to write down an example here. The difficulty is that by the adjoint functor theorem, a ...
Accepted

### Any criteria for a category to have all connected limits?

A category has all connected limits iff it has all pullbacks, equalizers, and filtered limits. First any limit can be built as a filtered limit of finite limits, and if the original limit was ...

### Is a direct limit of Noetherian rings necessarily Noetherian?

In fact, every ring is a direct limit of Noetherian rings (here by "ring" I mean "commutative ring"). For any ring $R$, the set of finitely generated subrings of $R$ is a directed set under inclusion,...
Accepted

### Why is $\mathbb{Z}[1/p]$ the direct limit of $\mathbb{Z}\xrightarrow{p}\mathbb{Z}\xrightarrow{p}\mathbb{Z}\to...$?

There's a better way to see why $\Bbb Z[1/p]$ is the direct limit (in my opinion): set up an isomorphism from the original direct limit to another one that's easier to understand: \require{AMScd} \...
Accepted

Accepted

### The direct limit of roots of unity

Yes, send $a/b \in \mathbb{Q}/\mathbb{Z}$ to $\zeta_{b}^{a}$, where $\zeta_{b}$ is a primitive $b$-th root of unity.

### What are the end and coend of Hom in Set?

The most important example of an end is $\int_{c : \mathcal{C}} \mathcal{D} (F c, G c)$, where $F$ and $G$ are functors $\mathcal{C} \to \mathcal{D}$. If you unfold the definition you will find that ...
There is a rather silk proof which requires some observations: Let $F$ be a diagram in $\mathcal{C}$ indexed by $\mathscr{J}$. Then, a limiting cone $(\lim F, \mu\colon \Delta \lim F \to F)$ ...