Monodes

Monodes are a theoretical concept in algebraic topology, specifically in the field of homotopical algebra. The term, while not universally adopted or formally standardized, generally refers to a monoid object within a higher category, typically a bicategory or a more general (∞,1)-category.

In this context, a "monoid object" is an object equipped with a multiplication morphism and a unit morphism satisfying associativity and unitality axioms, but these axioms hold up to homotopy or higher-order equivalences, rather than strictly.

The precise definition of a monode depends on the chosen higher categorical framework. For example, in a bicategory, a monode might consist of an object X, a multiplication morphism m: X × X → X, and a unit morphism e: I → X (where I is the identity object), along with associativity and unitality 2-morphisms that are invertible. These 2-morphisms mediate the failure of the associativity and unitality axioms to hold strictly.

In the context of (∞,1)-categories, the definition involves similar concepts but expressed using the language of ∞-groupoids or simplicial sets. A monode would be an object X together with maps representing multiplication and a unit, satisfying homotopy coherent versions of the monoid laws.

The study of monodes is related to the broader study of higher algebraic structures and their applications in fields like homotopy theory, category theory, and mathematical physics. They provide a way to encode algebraic structure in settings where traditional, strict algebraic structures are not suitable.

The significance of monodes lies in their ability to capture more nuanced and flexible algebraic relationships than traditional monoids. They are used to model situations where composition laws are associative only up to homotopy or higher-order equivalences, which is common in many areas of mathematics.