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Options

*Almost categories.*

(a) Give an example of some data – objects, morphisms, composition, and identities– that satisfies the associative laws but not the unit law.

(b) Give an example of some data – objects, morphisms, composition, and identities– that satisfies the unit laws but not the associative law.

## Comments

I'll kick this off with my solution because this is a fun one to see a bunch of different answers

a) A single object category whose morphisms are the Natural numbers, composition given by \(x ; y = x * y \), and the identity morphism on the sole object is \(0 \).

This obeys the associative law because \((x * y) * z = x * (y * z) \)

This fails to obey the unit law as you can see by the example \(0 * 1 \neq 1 \)

b) A single object category whose morphisms are the natural numbers, composition given by \(x ; y = |x - y| \), and the identity morphism on the sole object is \(0 \).

This obeys the unit law because \(|0 - f| = f \) and \(|f - 0| = f \) when \(f \geq 0 \), which is true because \(f \in \mathbb{N} \)

This fails to obey the associative law as you can see by example \(||1 - 2| - 3| = 2 \) and \(| 1 - | 2 - 3 || = 0 \)

`I'll kick this off with my solution because this is a fun one to see a bunch of different answers a) A single object category whose morphisms are the Natural numbers, composition given by \\(x ; y = x * y \\), and the identity morphism on the sole object is \\(0 \\). This obeys the associative law because \\((x * y) * z = x * (y * z) \\) This fails to obey the unit law as you can see by the example \\(0 * 1 \neq 1 \\) b) A single object category whose morphisms are the natural numbers, composition given by \\(x ; y = |x - y| \\), and the identity morphism on the sole object is \\\(0 \\). This obeys the unit law because \\(|0 - f| = f \\) and \\(|f - 0| = f \\) when \\(f \geq 0 \\), which is true because \\(f \in \mathbb{N} \\) This fails to obey the associative law as you can see by example \\(||1 - 2| - 3| = 2 \\) and \\(| 1 - | 2 - 3 || = 0 \\)`

Can anyone think of an "almost-category" that satisfies everything but the right identity law? (for any \(f : c \to d \), the equation \(id_c ; f = f \) holds but \(f ; id_d = f \) doesn't)

How about an "almost-category" satisfying everything but the left identity law?

`Can anyone think of an "almost-category" that satisfies everything but the right identity law? (for any \\(f : c \to d \\), the equation \\(id_c ; f = f \\) holds but \\(f ; id_d = f \\) doesn't) How about an "almost-category" satisfying everything but the left identity law?`

I didn't know it's legitimate to define (1-2) as |-1| on the naturals.

`I didn't know it's legitimate to define \(1-2\) as |-1| on the naturals.`

Hmm, I suppose you could look at it like I'm only defining one operation which is "absolute value of x - y" so all we have is |1 - 2| = 1, not the concept of -1. But if there is something wrong with this logic, I would like to know!

`Hmm, I suppose you could look at it like I'm only defining one operation which is "absolute value of x - y" so all we have is |1 - 2| = 1, not the concept of -1. But if there is something wrong with this logic, I would like to know!`

It was regarding your formulation as "only defining one operation" which I found problematic. Coding it would take 2 primitive operations, one of which cannot be coded on the naturals in this example.

`It was regarding your formulation as "only defining one operation" which I found problematic. Coding it would take 2 primitive operations, one of which cannot be coded on the naturals in this example.`

The intended operation can also be defined by f(x,y) = max(x,y) - min(x,y).

`The intended operation can also be defined by f(x,y) = max(x,y) - min(x,y).`

@JakeGillberg wrote:

Here is one based ideas from the web. Let the morphisms of our one-object category be the integers \(\mathbb{Z}\).

Define the binary operation by a;b = |a|*b.

It's easy to show that this is associative. (So, a semigroup.)

1 is a left identity. But there's no right identity.

`@JakeGillberg wrote: > How about an "almost-category" satisfying everything but the left identity law? Here is one based ideas from the web. Let the morphisms of our one-object category be the integers \\(\mathbb{Z}\\). Define the binary operation by a;b = |a|*b. It's easy to show that this is associative. (So, a semigroup.) 1 is a left identity. But there's no right identity.`