algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on operads model structure on algebras over an operad homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Introduction to Basic Homotopy Theory Introduction to Abstract Homotopy Theory geometry of physics – homotopy types Definitions Paths and cylinders Homotopy groups Basic facts Theorems An -space (“H” for Hopf, as in Hopf construction) is a magma internal to the classical homotopy category of topological spaces Ho(Top), or in the homotopy category of pointed topological spaces, which has a unit up to homotopy. Similarly: An -monoid is a monoid object in Ho(Top), hence an -space is an -monoid if the product of the magma is associative up to homotopy. An -group (or grouplike space) is a group object in Ho(Top), so an -monoid is an -group if it also has inverses up to homotopy (cf. Arkowitz 2011 Def. 2.2.1). An -ring is a ring object in (pointed) Ho(Top). To continue in this pattern, one could say that an H-category is a category internal to Ho(Top). Notice that here the homotopies for units, associativity etc. are only required to exist for an H-space, not required to be equipped with higher coherent homotopies. An -monoid equipped with such higher and coherent homotopies is instead called a strongly homotopy associative space or -space for short. If it has only higher homotopies up to level , it is called an -space. A better name for an -space might be -unitoid, but it is rarely used. The stands for Heinz Hopf, and reflects the sad fact that the natural name ‘homotopy group’ was...