Homotopy Type Theory H-spaceoid > history (Rev #3, changes)

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The oidification of an H-space


An H-spaceoid AA consists of the following.

  • A type A 0A_0, whose elements are called objects. Typically AA is coerced to A 0A_0 in order to write x:Ax:A for x:A 0x:A_0.

  • For each a,b:Aa,b:A, a type hom A(a,b)hom_A(a,b), whose elements are called arrows or morphisms.

  • For each a:Aa:A, a morphism 1 a:hom A(a,a)1_a:hom_A(a,a), called the identity morphism.

  • For each a,b,c:Aa,b,c:A, a function

    hom A(b,c)hom A(a,b)hom A(a,c)hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)

    called composition, and denoted infix by gfgfg \mapsto f \mapsto g \circ f, or sometimes gfgf.

  • For each a,b:Aa,b:A and f:hom A(a,b)f:hom_A(a,b), we have f=1 bff=1_b \circ f and f=f1 af=f\circ 1_a.

See also

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