H-object
# 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 ${\mathcal {C}}$ with a product $\times$ 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 ${\mathcal {C}}$ with a product $\times$ and initial object $*$, an H-object is an object $X\in {\text{Ob}}({\mathcal {C}})$ together with an operation called multiplication together with a two sided identity. If we denote $u_{X}:X\to *$, the structure of an H-object implies there are maps

   ${\begin{aligned}\varepsilon &:*\to X\\\mu &:X\times X\to X\end{aligned}}$

which have the commutation relations

   $\mu (\varepsilon \circ u_{X},id_{X})=\mu (id_{X},\varepsilon \circ u_{X})=id_{X}$

## Examples

### Magmas

All magmas with units are secretly H-objects in the category ${\textbf {Set}}$.

### H-spaces

Another example of H-objects are H-spaces in the homotopy category of topological spaces ${\text{Ho}}({\textbf {Top}})$.

### 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 $A$ be a commutative ring, and let $A\backslash R$ be the undercategory of such algebras over $A$ (meaning $A$-algebras), and set $(A\backslash R)/B$ be the associatived overcategory of objects in $A\backslash R$, then an H-object in this category $(A\backslash R)/B$ is an algebra of the form $B\oplus M$ where $M$ is a $B$-module. These algebras have the addition and multiplication operations

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

Note that the multiplication map given above gives the H-object structure $\mu$. Notice that in addition we have the other two structure maps given by

   ${\begin{aligned}u_{B\oplus M}(b\oplus m)&=b\\\varepsilon (b)&=b\oplus 0\end{aligned}}$

giving the full H-object structure. Interestingly, these objects have the following property:

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

giving an isomorphism between the $A$-derivations of $Y$ to $M$ and morphisms from $Y$ to the H-object $B\oplus M$. In fact, this implies $B\oplus M$ is an abelian group object in the category $(A\backslash R)/B$ since it gives a contravariant functor with values in Abelian groups.

