View source for H-object

{{refimprove|date=February 2021}}

In [[mathematics]], specifically [[homotopical algebra]], an H-object<ref name=":0">{{Cite journal|last=Quillen|first=Dan|date=|title=On the (co-) homology of commutative rings|url=https://www.ams.org/books/pspum/017/|journal=Proceedings of Symposia in Pure Mathematics|volume=1970|pages=65–87|via=}}</ref> is a [[Categorification|categorical generalization]] of an [[H-space]], which can be defined in any [[Category theory|category]] <math>\mathcal{C}</math> with a [[Product (category theory)|product]] <math>\times</math> and an [[initial object]] <math>*</math>. 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 <math>\mathcal{C}</math> with a product <math>\times</math> and initial object <math>*</math>, an '''H-object''' is an object <math>X \in \text{Ob}(\mathcal{C})</math> together with an operation called multiplication together with a two sided identity. If we denote <math>u_X: X \to *</math>, the structure of an H-object implies there are maps<blockquote><math>\begin{align}
\varepsilon&: * \to X \\
\mu&: X\times X \to X
\end{align}</math></blockquote>which have the commutation relations<blockquote><math>\mu(\varepsilon\circ u_X, id_X) = \mu(id_X,\varepsilon\circ u_X) = id_X</math></blockquote>

== Examples ==

=== Magmas ===
All [[Magma (algebra)|magmas]] with [[Unity (mathematics)|units]] are H-objects in the category <math>\textbf{Set}</math>.

=== H-spaces ===
Another example of H-objects are H-spaces in the [[homotopy category]] of [[topological space]]s <math>\text{Ho}(\textbf{Top})</math>.

=== H-objects in homotopical algebra ===
In homotopical algebra, one class of H-objects considered were by Quillen<ref name=":0" /> while constructing [[André–Quillen cohomology]] for commutative rings. For this section, let all algebras be commutative, associative, and unital. If we let <math>A</math> be a commutative ring, and let <math>A\backslash R</math> be the [[Overcategory|undercategory]] of such algebras over <math>A</math> (meaning <math>A</math>-algebras), and set <math>(A\backslash R)/B</math> be the associatived overcategory of objects in <math>A\backslash R</math>, then an H-object in this category <math>(A\backslash R)/B</math> is an algebra of the form <math>B\oplus M</math> where <math>M</math> is a <math>B</math>-[[Module (mathematics)|module]]. These algebras have the addition and multiplication operations<blockquote><math>\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}</math></blockquote>Note that the multiplication map given above gives the H-object structure <math>\mu</math>. Notice that in addition we have the other two structure maps given by<blockquote><math>\begin{align}
u_{B\oplus M }(b\oplus m) &= b\\
\varepsilon (b) &= b\oplus 0
\end{align}</math></blockquote>giving the full H-object structure. Interestingly, these objects have the following property:<blockquote><math>\text{Hom}_{(A\backslash R)/B}(Y,B\oplus M) \cong \text{Der}_A(Y, M)</math></blockquote>giving an isomorphism between the <math>A</math>-derivations of <math>Y</math> to <math>M</math> and morphisms from <math>Y</math> to the H-object <math>B\oplus M</math>. In fact, this implies <math>B\oplus M</math> is an abelian group object in the category <math>(A\backslash R)/B</math> since it gives a contravariant functor with values in Abelian groups.

== See also ==

* [[André–Quillen cohomology]]
*[[Cotangent complex]] 
* [[H-space]]

== References ==
{{Reflist}}

[[Category:Category theory]]
[[Category:Homotopical algebra]]