Implication Details

Claim: If a category has finite powers and has sequential limits, then it has countable powers.

Proof: We can write $X^{\IN}$ as the limit of the sequence $\cdots \to X^3 \to X^2 \to X \to 1$ with transition morphisms $f_n : X^{n+1} \to X^n$ , $(x_1,\dotsc,x_{n+1}) \mapsto (x_1,\dotsc,x_n)$ , i.e., $p_i f_n = p_i$ for $1 \leq i \leq n$ .

Show 19 categories using this implication

category of coproducts of Euclidean spaces
category of finite abelian groups
category of finite sets
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 finitely generated free abelian groups
category of countable groups
category of smooth manifolds
category of metric spaces with non-expansive maps
category of pseudo-metric spaces with non-expansive maps
category of finitely generated projective modules over the ring of dual numbers
category of schemes
category of countable sets
category of combinatorial species
category of countable-dimensional vector spaces
category of large vector spaces over a large field with a small basis