Pullback dan Pushout di Kategori Modul Topologis

Abstract

A pullback of two morphisms with a common codomain f : A → C and g : B → C is the limit of a diagram consisting f and g . The dual notion of a pullback is called a pushout. A Pullback of morphisms f : A → C and g : B → C in R-MOD is a diagram that contains Ker ( p ∗ ) = { ( a, b ) ∈ A (cid:76) B | f ( a ) = g ( b ) } ⊂ A (cid:76) B where p ∗ = fπ 1 − gπ 2 : A (cid:76) B → C , and a pushout of morphisms k : A → B and l : A → C in R-MOD is a diagram that contains Coke ( q ∗ ) = B ⊕ C/Im ( q ∗ ) where q ∗ = ε 1 k + ε 2 l : A → B ⊕ C . In this paper, we inves-tigate existence of pullback and pushout in TopR-MOD as a subcategory of R-MOD by giving product topology τ prod on pullback and coproduct topology τ coprod on pushout. Furthermore, a pullback of two continuous homomorphisms f : A → C and g : B → C in TopR-MOD is a diagram that contains A × C B = { ( a, b ) ∈ A × B | f ( a ) = g ( b ) } ⊂ A × B with the subspace topology of τ prod on A × C B , and the pushout of two continuous homomorphisms k : A → B and l : A → C in TopR-MOD is a diagram that contains B (cid:76) A C = ( B (cid:76) C ) / ∼ where ∼ is the smallest equivalence relation containing the pairs ( k ( a ) , l ( a )) for all a ∈ A and topology on B (cid:76) C is coproduct topology τ coprod .