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 $$\begin{align} \varepsilon&: * \to X \\ \mu&: X\times X \to X \end{align}$$ which have the commutation relations"$\mu(\varepsilon\circ u_X, id_X) = \mu(id_X,\varepsilon\circ u_X) = id_X$"

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 $$\begin{align} (b\oplus m)+(b'\oplus m') &= (b + b')\oplus (m+m') \\ (b\oplus m)\cdot(b'\oplus m') &= (bb')\oplus(bm' + b'm) \end{align}$$ 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 $$\begin{align} u_{B\oplus M }(b\oplus m) &= b\\ \varepsilon (b) &= b\oplus 0 \end{align}$$ 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.