Implication Details

Claim: If a category has a parametrized natural numbers object and is pointed, then it is trivial.

Proof: Let $(N,z,s)$ be a parametrized natural numbers object in a category with a zero object, denoted $0$ . The morphism $z : 0 \to N$ must be zero. Applying the universal property with $A=1$ shows that $s : N \to N$ is an initial object in the category of endomorphisms. This initial object exists and is given by the identity $0 \to 0$ . Therefore, $N = 0$ . Now, Lemma 3 here implies that $\id_X = 0$ for every object $X$ , so that $X = 0$ .

Show 57 categories using this implication

category of abelian groups
category of finitely generated abelian groups
category of Banach spaces with linear contractions
category of commutative monoids
category of small categories
category of cochain complexes of abelian groups
category of filtered vector spaces
category of finite abelian groups
category of finite groups
category of finite-dimensional vector spaces
category of finite-dimensional vector spaces [countable field]
category of finite-dimensional vector spaces [finite field]
category of finite-dimensional vector spaces [uncountable field]
category of free abelian groups
category of groups
category of countable groups
category of Hausdorff spaces
category of locally ringed spaces
category of M-sets
category of smooth manifolds
category of measurable spaces
category of metric spaces with continuous maps
category of metric spaces with ∞ allowed
category of monoids
category of sets with a distinguished subset
partially ordered set of extended natural numbers
category of partially ordered sets
category of preordered sets
category of left modules over a ring
category of left modules over a division ring
category of left modules over a non-semisimple ring
category of sets and relations
category of rngs
category of schemes
category of sequences of abelian groups
category of sets
category of pointed sets
category of set functions and commutative squares
category of countable sets
category of non-empty sets
category of pairs of sets
category of sheaves
category of abelian sheaves
category of topological spaces
category of pointed topological spaces
category of torsion abelian groups
category of torsion-free abelian groups
category of vector spaces
category of Z-functors
preordered set of integers w.r.t. divisibility
category of graded abelian groups
category of graded modules over a graded ring
real interval [0,1]
category of simplicial sets
walking commutative square
walking composable pair
walking morphism