H-object

In mathematics, specifically homotopical algebra, an H-object¹ is a categorical generalization of an H-space, which can be defined in any category ${\displaystyle {\mathcal {C}}}$ with a product ${\displaystyle \times }$ and an initial object ${\displaystyle *}$. 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 ${\displaystyle {\mathcal {C}}}$ with a product ${\displaystyle \times }$ and initial object ${\displaystyle *}$, an H-object is an object ${\displaystyle X\in {\text{Ob}}({\mathcal {C}})}$ together with an operation called multiplication together with a two sided identity. If we denote ${\displaystyle u_{X}:X\to *}$, the structure of an H-object implies there are maps

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

which have the commutation relations

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

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

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 Quillen¹ while constructing André–Quillen cohomology for commutative rings. For this section, let all algebras be commutative, associative, and unital. If we let ${\displaystyle A}$ be a commutative ring, and let ${\displaystyle A\backslash R}$ be the undercategory of such algebras over ${\displaystyle A}$ (meaning ${\displaystyle A}$-algebras), and set ${\displaystyle (A\backslash R)/B}$ be the associatived overcategory of objects in ${\displaystyle A\backslash R}$, then an H-object in this category ${\displaystyle (A\backslash R)/B}$ is an algebra of the form ${\displaystyle B\oplus M}$ where ${\displaystyle M}$ is a ${\displaystyle B}$-module. 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 ${\displaystyle \mu }$. Notice that in addition we have the other two structure maps given by

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

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

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

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