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.

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. \end{align} which have the commutation relations \mu(\varepsilon\circ u_X, id_X) = \mu(id_X,\varepsilon\circ u_X) = id_X (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.

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