(Span). The two maps with the same domain are called span:

(Homotopy Pushout). The homotopy pushout or homotopy colimit (denoted as ) of the span diagram:

where is an equivalence relation and for . If is a based space with basepoint , we add the relation for all .

data pushout (A B C: U) (f: C -> A) (g: C -> B) = po1 (_: A) | po2 (_: B) | po3 (c: C) <i> [ (i = 0) -> po1 (f c) , (i = 1) -> po2 (g c) ]


(Homotopy Fibers).

hofiber (A B: U) (f: A -> B) (y: B): U = pullback A unit B f (\(x: unit) -> y)

(Fiber Pullback).

fiberPullback (A B: U) (f: A -> B) (y: B) : pullbackSq (hofiber A B f y)

(Homotopy Cofiber).

cofiber (A B: U) (f: B -> A): U = pushout A unit B f (\(x: B) -> tt)


kernel (A B: U) (f: A -> B): U = pullback A A B f f


cokernel (A B: U) (f: B -> A): U = pushout A A B f f


[1]. Brian Munson and Ismar Volić. Cubical Homotopy Theory