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Question 1.4 - Almost categories

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

  • 1.
    edited January 16

    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 \(| 1 - | 2 - 3 || = 0 \)

    Comment Source: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 \\)
  • 2.

    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?

    Comment Source: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?
  • 3.
    edited January 18

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

    Comment Source:I didn't know it's legitimate to define \(1-2\) as |-1| on the naturals.
  • 4.

    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!

    Comment Source: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!
  • 5.
    edited January 22

    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.

    Comment Source: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.
  • 6.
    edited January 23

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

    Comment Source:The intended operation can also be defined by f(x,y) = max(x,y) - min(x,y).
  • 7.
    edited January 23

    @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.

    Comment Source:@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.
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