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In mathematics, specifically homotopical algebra, an H-object [1] is a categorical generalization of an H-space, which can be defined in any category with a product and an initial object . These are useful constructions because they help export some of the ideas from algebraic topology and homotopy theory into other domains, such as in commutative algebra and algebraic geometry.
In a category with a product and initial object , an H-object is an object together with an operation called multiplication together with a two sided identity. If we denote , the structure of an H-object implies there are maps
which have the commutation relations
All magmas with units are secretly H-objects in the category .
Another example of H-objects are H-spaces in the homotopy category of topological spaces .
In homotopical algebra, one class of H-objects considered were by Quillen [1] while constructing André–Quillen cohomology for commutative rings. For this section, let all algebras be commutative, associative, and unital. If we let be a commutative ring, and let be the undercategory of such algebras over (meaning -algebras), and set be the associatived overcategory of objects in , then an H-object in this category is an algebra of the form where is a - module. These algebras have the addition and multiplication operations
Note that the multiplication map given above gives the H-object structure . Notice that in addition we have the other two structure maps given by
giving the full H-object structure. Interestingly, these objects have the following property:
giving an isomorphism between the -derivations of to and morphisms from to the H-object . In fact, this implies is an abelian group object in the category since it gives a contravariant functor with values in Abelian groups.