Category Theory/Subcategories

subcategory Let be a category. Then a subcategory of is a category such that and . full A subcategory of a category is called full iff for all , we have . limits are preserved when restricting to a full subcategory Let be a category, let be a diagram in , and let be a full subcategory of . Suppose that is a limit over in such that and all targets of the are in . Then is a limit over in . Certainly, the underlying cone of is contained within , because the subcategory is full. Now let another cone in over the diagram (which, analogously, is a diagram in ) be given. By the universal property of in , there exists a unique morphism which satisfies for all . Since is full, is in . Analogously, we have: colimits are preserved when restricting to a full subcategory Let be a category, let be a diagram in , and let be a full subcategory of . Suppose that is a colimit over in such that and all domains of the are in . Then is a colimit over in . This follows from its "dual" proposition, reversing all arrows in its statement and proof except the direction of the diagram functor.