circle

Circle C consists of the following: (1)

• Subject kind ob(C)
• The analogue of arrow hom(C) between the subject
  • Each projection f ∈ hom(C) is accompanied by an object a ∈ ob(C), called the starting region, and an object b ∈ ob(C), called the ending region, and we say that “f is the projection from a to b” and write f: a → b.
• The class of the projection from a to b (hom-class; hom class) hom(a, b) is the class formed by the whole projection from a to b.

Then, for any three objects a, b, c ∈ ob(C), there exists a binary operation hom(a, b) × hom(b, c) → hom(a, c); (f, g) ↦ g ∘ f called the composition of arrays, satisfying the following axioms: .

• If the coupling laws: f: a → b, g: b → c, h: c → d then h ∘ (g ∘ f) = (h ∘ g) ∘ f holds.
• Unit rule: for each object x ∈ ob(C)
  • There exists a self-projection $i{d}_x=1_x:x\to x$ called the identity projection of x,
  • For any arrays f: a → x and g: x → b, $1_x\circ f=f$ and $g\circ 1_x=g$.

Sphere (mathematics) - Wikipedia

Comparison with monoid | | Sphere monoid. | | — | — | — | | | Target class ob(C) Set S | | | Composition of projectiles Multiplication | | * | hom(a, b) × hom(b, c) → hom(a, c) | S × S → S | | | associative law | | | Existence of identity projection Existence of unitary source |

---

This page is auto-translated from /nishio/圏 using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I’m very happy to spread my thought to non-Japanese readers.