This book presents a study of homotopy invariant algebraic structures on topological spaces, focusing on the theory of PROPs and their applications to infinite loop spaces. The authors develop a framework for understanding how algebraic structures can be defined on topological spaces and how these structures can be preserved under homotopy equivalences. The theory is based on the concept of "homotopy invariant structures," which are algebraic structures that remain unchanged under homotopy equivalences. The authors introduce the notion of PROPs, which are categories of operators that allow for the definition of algebraic structures on topological spaces. They also define the bar construction for theories, which is a key tool in the study of homotopy invariant structures. The book discusses various aspects of homotopy invariant structures, including the concept of homotopy homomorphisms, which are maps between spaces that preserve algebraic structures up to homotopy. The authors also explore the relationship between homotopy invariant structures and infinite loop spaces, which are spaces that can be iteratively looped. They show that certain algebraic structures, called E-structures, characterize infinite loop spaces. The authors also compare their approach with other methods used to study infinite loop spaces, such as those developed by Segal, Beck, and May. They highlight the advantages and disadvantages of different approaches and emphasize the importance of homotopy invariant structures in understanding the algebraic topology of topological spaces. The book concludes with a discussion of homotopy colimits and their applications in homotopy theory.This book presents a study of homotopy invariant algebraic structures on topological spaces, focusing on the theory of PROPs and their applications to infinite loop spaces. The authors develop a framework for understanding how algebraic structures can be defined on topological spaces and how these structures can be preserved under homotopy equivalences. The theory is based on the concept of "homotopy invariant structures," which are algebraic structures that remain unchanged under homotopy equivalences. The authors introduce the notion of PROPs, which are categories of operators that allow for the definition of algebraic structures on topological spaces. They also define the bar construction for theories, which is a key tool in the study of homotopy invariant structures. The book discusses various aspects of homotopy invariant structures, including the concept of homotopy homomorphisms, which are maps between spaces that preserve algebraic structures up to homotopy. The authors also explore the relationship between homotopy invariant structures and infinite loop spaces, which are spaces that can be iteratively looped. They show that certain algebraic structures, called E-structures, characterize infinite loop spaces. The authors also compare their approach with other methods used to study infinite loop spaces, such as those developed by Segal, Beck, and May. They highlight the advantages and disadvantages of different approaches and emphasize the importance of homotopy invariant structures in understanding the algebraic topology of topological spaces. The book concludes with a discussion of homotopy colimits and their applications in homotopy theory.