Pullback dan Pushout di Kategori Modul Topologis

Abstract

A pullback of two morphisms with a common codomain $f\colon A\to C$ and $g\colon B\to C$ is the limit of a diagram consisting $f$ and $g$. The dual notion of a pullback is called a pushout. A pushout of two morphisms with a common domain $k\colon A\to B$ and $l\colon A\to C$ is the colimit of a diagram consisting $k$ and $l$. The pullback and the pushout of two morphisms need not exists. In this paper, we constructed a pullback and a pushout of two morphism in category of topological modules. A pullback of two continuous homomorphisms $f\colon A\to C$ and $g\colon B\to C$ in category of topological modules is a diagram that contains $A\times _{C} B=\{(a,b)\in A\times B \mid f(a)=g(b)\}\subset A\times B$ with the subspace topology on $A\times _{C} B$. Furthermore, the pushout of two continuous homomorphisms $k\colon A\to B$ and $l\colon A\to C$ in category of topological modules is a diagram that contains $B\bigoplus_{A} C=(B\bigoplus C)/\sim$ where $\sim$ is the smallest equivalence relation containing the pairs $(k(a),l(a))$ for all $a\in A$ and topology on $B\bigoplus C$ is coproduct topology $\tau_{coprod}$