H-object

In mathematics, specifically homotopical algebra, an H-object is a categorical generalization of an H-space, which can be defined in any category formula_1 with a product formula_2 and an initial object formula_3. 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.
Definition.
In a category formula_1 with a product formula_2 and initial object formula_3, an H-object is an object formula_7 together with an operation called multiplication together with a two sided identity. If we denote formula_8, the structure of an H-object implies there are mapsformula_9which have the commutation relationsformula_10
Examples.
Magmas.
All magmas with units are secretly H-objects in the category formula_11.
H-spaces.
Another example of H-objects are H-spaces in the homotopy category of topological spaces formula_12.
H-objects in homotopical algebra.
In homotopical algebra, one class of H-objects considered were by Quillen while constructing André–Quillen cohomology for commutative rings. For this section, let all algebras be commutative, associative, and unital. If we let formula_13 be a commutative ring, and let formula_14 be the undercategory of such algebras over formula_13 (meaning formula_13-algebras), and set formula_17 be the associatived overcategory of objects in formula_14, then an H-object in this category formula_17 is an algebra of the form formula_20 where formula_21 is a formula_22-module. These algebras have the addition an multiplication operationsformula_23Note that the multiplication map given above gives the H-object structure formula_24. Notice that in addition we have the other two structure maps given byformula_25giving the full H-object structure. Interestingly, these objects have the following property:formula_26giving an isomorphism between the formula_13-derivations of formula_28 to formula_21 and morphisms from formula_28 to the H-object formula_20. In fact, this implies formula_20 is an abelian group object in the category formula_17 since it gives a contravariant functor with values in Abelian groups.