Implication Details

Assumptions: exact filtered colimits

Conclusions: filtered-colimit-stable monomorphisms

Proof: This is because $f : X \longrightarrow Y$ is a monomorphism iff the diagram $\begin{CD} X @>{\id}>> X \\ @V{\id}VV @VV{f}V \\ X @>>{f}> Y \end{CD}$ is a pullback, and if a functor preserves finite limits, it preserves pullbacks in particular.

Show 19 categories using this implication

trivial category
category of Banach spaces with linear contractions
category of small categories
category of compact Hausdorff spaces
category of Hausdorff spaces
category of Jónsson-Tarski algebras
category of M-sets
category of metric spaces with non-expansive maps
category of metric spaces with ∞ allowed
category of pseudo-metric spaces with non-expansive maps
category of posets
category of prosets
category of sets
category of pairs of sets
category of sheaves
category of abelian sheaves
category of Z-functors
category of simplicial sets
walking isomorphism