# Questions tagged [limits-colimits]

For questions about categorical limits and colimits, including both questions about limits and colimits of general diagrams and questions about specific special kinds of (co)limits such as (co)products or (co)equalizers.

365 questions
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### How to Find limits and co-limits of diagrams over Vector?

I am having trouble understanding how to find limits and colimits of specific diagrams over the category of finite dimension vector spaces. I understand the definitions of cones, terminal objects, ...
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### I am stuck with an example of direct limits and need help to prove $\mathbb{Q/Z} = \varinjlim \mathbb{Z}/(i)$.

Let $N$ denote the set of positive integers that are ordered as $m \le n \iff m | n$. Let $X_m = \mathbb{Z}/(m)$ denote the set of integers $x \mod m$. I want to show that the direct limit of $X_i$, ...
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### How does a map between inverse systems induce the inverse limit of its components?

I encountered the following definition in the book Profinite Groups by Ribes and Zallesski: Let $\{X_i, \varphi_{ij} \}$ and $\{ X_i', \varphi_{ij}' \}$ be inverse systems of topological spaces over ...
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### Inverse limit of compact Hausdorff spaces is nonempty and compact

Suppose that $\left\{ X_n \right\}_{n=1}^{\infty}$ is a sequence of compact Hausdorff spaces and for each $n, f_n : X_{n+1} \rightarrow X_n$ is a continuous function (not necessarily onto). Show ...
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### Question about $Q = \varinjlim \frac{1}{n} \mathbb{Z}$

I am a little confused by one of Vakil's exercises, which reads Interpret the statement $Q = \varinjlim \frac{1}{n} \mathbb{Z}$ Does it mean $Q = \bigcup \frac{1}{n} \mathbb{Z}$? But what is the ...
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