Categories

Which of these are categories (with the obvious composition)? Choose all that apply.

Regard the set $\{1, 2, 3, 6\}$, ordered by divisibility, as a category. How many morphisms does it have in total, including the identities?

Let $M = (\mathbb{Z}/12, \times, 1)$ be the monoid of residues modulo $12$ under multiplication, and $\mathbf{B}M$ its one-object category. How many of the $12$ morphisms of $\mathbf{B}M$ are isomorphisms?

In the proof that a left inverse $g$ and a right inverse $h$ of $f : A \to B$ are equal, $g = g \circ \mathrm{id}_B = g \circ (f \circ h) = (g \circ f) \circ h = \mathrm{id}_A \circ h = h,$ which axiom justifies the step $g \circ (f \circ h) = (g \circ f) \circ h$?

Which of these are epimorphisms that are not surjective? Choose all that apply.

Theorem: in every category, if $g \circ f$ is a monomorphism then $f$ is a monomorphism. What does the duality principle give us for free?

Mathlib writes composition in diagrammatic order: a ≫ f means "first a, then f", i.e. $f \circ a$. Which statement says that f : X ⟶ Y is a monomorphism?

A function between finite sets $\{0, \dots, m-1\} \to \{0, \dots, n-1\}$ is stored as a tuple f of length $m$ with values in range(n). Write is_epi(f, n) that decides whether f is an epimorphism directly from the definition, by testing cancellation against all pairs of maps $k, l : \{0, \dots, n-1\} \to \{0, 1\}$ (that is, k ∘ f == l ∘ f must force k == l). Do not test surjectivity directly!