It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.2K
- Applied Category Theory Course 355
- Applied Category Theory Seminar 4
- Exercises 149
- Discussion Groups 49
- How to Use MathJax 15
- Chat 480
- Azimuth Code Project 108
- News and Information 145
- Azimuth Blog 149
- Azimuth Forum 29
- Azimuth Project 189
- - Strategy 108
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 711
- - Latest Changes 701
- - - Action 14
- - - Biodiversity 8
- - - Books 2
- - - Carbon 9
- - - Computational methods 38
- - - Climate 53
- - - Earth science 23
- - - Ecology 43
- - - Energy 29
- - - Experiments 30
- - - Geoengineering 0
- - - Mathematical methods 69
- - - Meta 9
- - - Methodology 16
- - - Natural resources 7
- - - Oceans 4
- - - Organizations 34
- - - People 6
- - - Publishing 4
- - - Reports 3
- - - Software 21
- - - Statistical methods 2
- - - Sustainability 4
- - - Things to do 2
- - - Visualisation 1
- General 39

Options

**Cost**-categories)? How do we interpret this in terms of distances?

## Comments

In a Lawvere metric space, \(I=0\). So, an identity element is a map \(\mathrm{id}_x:0\to\mathcal{X}(x,x)\). In terms of distances, this means that \(0\geq d(x,x)\) (since \(0\) is the smallest element of \([0,\infty]\), this means \(d(x,x)=0\) (this was already observed in the text between the definition of a Lawvere metric space as a

Cost-category and the example of \(\mathbb{R}\) as a Lawvere metric space, in chapter 2), which is the statement of definiteness for a Lawvere metric).`In a Lawvere metric space, \\(I=0\\). So, an identity element is a map \\(\mathrm{id}_x:0\to\mathcal{X}(x,x)\\). In terms of distances, this means that \\(0\geq d(x,x)\\) (since \\(0\\) is the smallest element of \\([0,\infty]\\), this means \\(d(x,x)=0\\) (this was already observed in the text between the definition of a Lawvere metric space as a **Cost**-category and the example of \\(\mathbb{R}\\) as a Lawvere metric space, in chapter 2), which is the statement of definiteness for a Lawvere metric).`