Bar homotopy

Bar homotopy is a concept from algebraic topology that studies how two shapes or paths can be continuously deformed into each other within a space. Imagine stretching or bending a flexible object, like a rubber band, so that one shape becomes another without tearing or cutting. Bar homotopy specifically involves algebraic structures called "bar complexes," which encode information about the space's paths and loops. If two such algebraic structures can be connected through a series of transformations maintaining certain properties, they are said to be bar homotopic, indicating they are equivalent in a topological or algebraic sense.