Higher algebra

Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts




Co-H-spaces are the Eckmann-Hilton duals of H-spaces. They are co-H-objects in the category of pointed topological spaces. Thus a co-H-space (X,ϕ)(X, \phi) is a pointed space, XX, together with a map ϕ:XXX\phi: X \to X \vee X (the wedge sum), such that p iϕp_i \circ \phi is homotopic to 1 X1_X, where p i,i=1,2p_i, i = 1, 2, are the projections XXXX \vee X \to X. Alternatively, (X,ϕ)(X, \phi) is a co-H-space if and only if jϕj \circ \phi is homotopic to Δ\Delta, where j:XXX×Xj: X \vee X \to X \times X is the inclusion and Δ:XX×X\Delta: X \to X \times X is the diagonal map.

The importance of the notion is that XX is a co-H-space if and only if for every space YY, [X,Y][X, Y] has a binary operation with unit. Further properties of ϕ\phi are of interest, in particular being (co)associative and having right and left (co)inverses. In this case XX is a cogroup. The suspension of a topological space is a cogroup.

Every co-H-space is path-connected, and its fundamental group is free.


  1. Martin Arkowitz, Co-H-spaces, chapter 23 of Handbook of Algebraic Topology, Ioan James (ed.).

Revised on July 8, 2013 16:22:05 by Urs Schreiber (