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consider a category N having only one object, let's call it *, and with one arrow! # # for every natural number. Moreover, suppose that the composition of two arrows m and n is given by addition, that is, mon corresponds to id: *→ * ? Check that the 3 axioms of composition are satisfied what is the natural number n that

consider a category N having only one object, let's call it *, and with one arrow! # # for every natural number. Moreover, suppose that the composition of two arrows m and n is given by addition, that is, mon corresponds to id: *→ * ? Check that the 3 axioms of composition are satisfied what is the natural number n that

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Understanding a Specific Category in Mathematics** **Question:** 3) Consider a category \( \mathbb{N} \) having only one object, let's call it \( * \), and with one arrow \( * \rightarrow * \) for every natural number. Moreover, suppose that the composition of two arrows \( m \) and \( n \) is given by addition, that is, \( m \circ n = m + n \). What is the natural number \( n \) that corresponds to \( \mathrm{id}_{*} : * \rightarrow * \)? Check that the 3 axioms of composition are satisfied. --- **Explanation:** In this exercise, we are asked to explore a mathematical category with specific properties. The category \( \mathbb{N} \) has a singular object and associates arrows with natural numbers. The composition operates through addition, reflecting a unique algebraic structure. Key points to consider are: 1. **Identity Element**: Determine the natural number that corresponds to the identity arrow. This is akin to finding which number, when added to another, leaves the other number unchanged. 2. **Composition Axioms**: Verify that the category satisfies: - **Associativity**: For any arrows \( m, n, \) and \( p \), \( (m \circ n) \circ p = m \circ (n \circ p) \). - **Identity Laws**: For any arrow \( m \), \( \mathrm{id}_{*} \circ m = m \) and \( m \circ \mathrm{id}_{*} = m \). This problem introduces fundamental principles of category theory using basic arithmetic operations within the familiar set of natural numbers.
Transcribed Image Text:**Title: Understanding a Specific Category in Mathematics** **Question:** 3) Consider a category \( \mathbb{N} \) having only one object, let's call it \( * \), and with one arrow \( * \rightarrow * \) for every natural number. Moreover, suppose that the composition of two arrows \( m \) and \( n \) is given by addition, that is, \( m \circ n = m + n \). What is the natural number \( n \) that corresponds to \( \mathrm{id}_{*} : * \rightarrow * \)? Check that the 3 axioms of composition are satisfied. --- **Explanation:** In this exercise, we are asked to explore a mathematical category with specific properties. The category \( \mathbb{N} \) has a singular object and associates arrows with natural numbers. The composition operates through addition, reflecting a unique algebraic structure. Key points to consider are: 1. **Identity Element**: Determine the natural number that corresponds to the identity arrow. This is akin to finding which number, when added to another, leaves the other number unchanged. 2. **Composition Axioms**: Verify that the category satisfies: - **Associativity**: For any arrows \( m, n, \) and \( p \), \( (m \circ n) \circ p = m \circ (n \circ p) \). - **Identity Laws**: For any arrow \( m \), \( \mathrm{id}_{*} \circ m = m \) and \( m \circ \mathrm{id}_{*} = m \). This problem introduces fundamental principles of category theory using basic arithmetic operations within the familiar set of natural numbers.
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