#Contents# * table of contents {:toc} ## Idea ## H-spaces are simply types equipped with the structure of a magma (from classical Algebra). They are useful classically in constructing fibrations. ## Definition ## A H-Space consists of * A type $A$, * A basepoint $e:A$ * A binary operation $\mu : A \to A \to A$ * for every $a:A$, equalities $\mu(e,a)=a$ and $\mu(a,e)=a$ ## Properties ## Let $A$ be a connected H-space. Then for every $a:A$, the maps $\mu(a,-),\mu(-,a):A \to A$ are equivalences. ## See also ## [[synthetic homotopy theory]] [[hopf fibration]] ### On the nlab ### Classically, an [[nLab:H-space]] is a [[nLab:homotopy type]] equipped with the structure of a [[nLab:unitality|unital]] [[nLab:magma]] in the [[nLab:homotopy category]] (only). ## References ## [[HoTT book]] category: homotopy theory