# Colimits

### Some articles on colimits:

Limit (category Theory) - Functors and Limits - Examples
... Hom functors need not preserve colimits ... functor C → Set preserves limits (but not necessarily colimits) ... The forgetful functor U Grp → Set creates (and preserves) all small limits and filtered colimits however, U does not preserve coproducts ...
Topos - Grothendieck Topoi (topoi in Geometry) - Equivalent Definitions - Giraud's Axioms
... a small set of generators, and admits all small colimits ... Furthermore, colimits commute with fiber products ... Since C has colimits we may form the coequalizer of the two maps R→X call this X/R ...
Category Of Rings - Properties - Limits and Colimits
... Ring is both complete and cocomplete, meaning that all small limits and colimits exist in Ring ... forgetful functor U Ring → Set creates (and preserves) limits and filtered colimits, but does not preserve either coproducts or coequalizers ... Examples of limits and colimits in Ring include The ring of integers Z forms an initial object in Ring ...
Limit (category Theory) - Examples - Colimits
... Examples of colimits are given by the dual versions of the examples above Initial objects are colimits of empty diagrams ... Coproducts are colimits of diagrams indexed by discrete categories ... Copowers are colimits of constant diagrams from discrete categories ...
Category Of Topological Spaces - Limits and Colimits
... The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top ... → Set uniquely lifts both limits and colimits and preserves them as well ... Dually, colimits in Top are obtained by placing the final topology on the corresponding colimits in Set ...