H-object

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.

Definition

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

Examples

Magmas

All magmas with units are secretly H-objects in the category .

H-spaces

Another example of H-objects are H-spaces in the homotopy category of topological spaces .

H-objects in homotopical algebra

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 an 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.

See also

References

  1. Quillen, Dan. "On the (co-) homology of commutative rings". Proceedings of Symposia in Pure Mathematics. 1970: 65–87.
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