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

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.

and initial object

, an H-object is an object

together with an operation called multiplication together with a two sided identity.

, the structure of an H-object implies there are maps

which have the commutation relations

μ ( ε ∘

All magmas with units are H-objects in the category

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

{\displaystyle {\text{Ho}}({\textbf {Top}})}

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.

be a commutative ring, and let

be the undercategory of such algebras over

be the associatived overcategory of objects in

, then an H-object in this category

is an algebra of the form

These algebras have the addition and multiplication operations

{\displaystyle {\begin{aligned}(b\oplus m)+(b'\oplus m')&=(b+b')\oplus (m+m')\\(b\oplus m)\cdot (b'\oplus m')&=(bb')\oplus (bm'+b'm)\end{aligned}}}

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:

{\displaystyle {\text{Hom}}_{(A\backslash R)/B}(Y,B\oplus M)\cong {\text{Der}}_{A}(Y,M)}

giving an isomorphism between the

In fact, this implies

is an abelian group object in the category

since it gives a contravariant functor with values in Abelian groups.