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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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