Understanding Direct Limits: $\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})$

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In summary, a direct limit is a mathematical concept that describes the union of a sequence of objects and their mappings while preserving their structure and properties. It differs from a union by also considering the relationships between the objects being united. An example of a direct limit is $\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})$, which combines the integers and odd integers while preserving their properties. Understanding direct limits is important because they are fundamental in mathematics and have many applications in fields such as algebraic topology. In research, direct limits can be applied in areas such as algebraic geometry, algebraic topology, and category theory, and can potentially lead to new discoveries and solutions in various fields.
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I'm trying to understand direct limits so consider the direct limit

[itex]\lim_\rightarrow (\mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \cdots)[/itex]
where each map is multiplication by 2.
I concluded that the solution is [itex]\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})[/itex]. Is this correct?
 
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OK, note first that the direct limit of groups should again be a group.
In this particular example it is isomorphic to [tex]\mathbb{Z}[1/2][/tex], i.e. the multiplicative group of rational numbers of the form [tex]a/2^b[/tex], where a,b are integers, [tex]b\geq0[/tex].
 

1. What is a direct limit?

A direct limit is a mathematical concept that describes the union of a sequence of objects and their mappings in a way that preserves the structure and properties of those objects.

2. How is a direct limit different from a union?

A direct limit is different from a union in that it also takes into account the mappings between the objects being united, whereas a union simply combines the objects without considering their relationships.

3. What is $\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})$?

$\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})$ is a specific example of a direct limit, where $\mathbb{Z}$ represents the integers and $\mathbb{Z}-2\mathbb{Z}$ represents the odd integers. This direct limit combines these two sets and preserves their properties, resulting in a new set that contains both the integers and the odd integers.

4. Why is it important to understand direct limits?

Understanding direct limits is important because they are a fundamental concept in mathematics, particularly in the field of algebraic topology. They allow us to study the properties of objects that are constructed from smaller, simpler objects, and have many applications in various areas of mathematics and science.

5. How can I apply my understanding of direct limits in my research?

Direct limits have many applications in research, particularly in areas such as algebraic geometry, algebraic topology, and category theory. They can be used to study the properties of various mathematical structures and can also be applied in physics, computer science, and other fields. By understanding direct limits, you can potentially use them to solve problems or make new discoveries in your research.

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