Implication Details

Claim: If a category has countable products and has equalizers, then it has sequential limits.

Proof: See Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case. Namely, the limit of $\cdots \to X_2 \to X_1 \to X_0$ is the equalizer of two suitable endomorphisms of $\prod_{n \geq 0} X_n$ .

Show 32 categories using this implication

category of small categories
category of cochain complexes of abelian groups
category of filtered vector spaces
category of finite ordered sets
category of Hausdorff spaces
category of Jónsson-Tarski algebras
category of locally ringed spaces
category of M-sets
category of measurable spaces
category of metric spaces with continuous maps
category of metric spaces with ∞ allowed
category of sets with a distinguished subset
category of partially ordered sets
category of preordered sets
category of sequences of abelian groups
category of sets
category of set functions and commutative squares
category of sets with finite-to-one maps
category of non-empty sets
category of pairs of sets
category of sheaves
category of abelian sheaves
category of topological spaces
category of torsion abelian groups
category of torsion-free abelian groups
category of uniform 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