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Results tagged with limits-and-colimits
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user 75
For questions on limits and colimts in the sense of category theory, and related notions.
9
votes
Accepted
Maximum cardinality of a filtered limit of finite sets
You can get arbitrarily large cardinalities. For instance, let $X$ be any set and consider the poset $I$ of finite partitions of $X$, ordered by refinement. There is a "tautological" filtered system …
- 31.2k
21
votes
Accepted
Direct limit of compact topological spaces
A $T_1$ colimit $X$ of a sequence of compact spaces $X_n$ is compact iff there is some $n$ such that the map $X_n\to X$ is surjective. This condition is obviously sufficient; suppose that it fails. …
- 31.2k
13
votes
Is there a category in which finite limits and directed colimits *don't* commute
Consider the poset of closed subsets of $[0,1]$. Let $a=\{0,1\}$ and $b(r)=[0,r]$ for $r<1$. Then the (directed) colimit of the $b(r)$ is $b=[0,1]$, and the product (i.e., intersection) of $b$ and a …
- 31.2k