Pullback & pushout

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1. Exemples et définition d'un pullback

16-pullback : In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X→ Z ← Y. The pullback is often written P = X ×Z Y. The categorical dual of a pullback is a called a pushout. Remarks opposite to the above apply: the pushout is a coproduct with additional structure.

17-universal-property-o-pullback : Explicitly, the pullback of the morphisms f and g consists of an object P and two morphisms p1 : P → X and p2 : P → Y for which the diagram Categorical pullback commutes. Moreover, the pullback (P, p1, p2) must be universal with respect to this diagram. That is, for any other such triple (Q, q1, q2) for which the following diagram commutes, there must exist a unique u : Q → P (called a mediating morphism) such that p_2 \circ u=q_2, \qquad p_1\circ u=q_1

18-exemple-de-pullback : In mathematics, a pullback bundle or induced bundle[1][2][3] is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′. The fiber of f*E over a point b′ in B′ is just the fiber of E over f(b′). Thus f*E is the disjoint union of all these fibers equipped with a suitable topology.

2. Pushout

20-definition-de-pushout : In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y. The pushout is the categorical dual of the pullback. 1. Suppose that X, Y, and Z as above are sets, and that f : Z → X and g : Z → Y are set functions. The pushout of f and g is the disjoint union of X and Y, where elements sharing a common preimage (in Z) are identified, together with the morphisms i1 , i2 from X and Y, i.e. P = X \coprod Y \Bigg/ \sim where ~ is the finest equivalence relation (cf. also this) such that i1 ∘f (z) ~ i2 ∘g(z). 2. The construction of adjunction spaces is an example of pushouts in the category of topological spaces. More precisely, if Z is a subspace of Y and g : Z → Y is the inclusion map we can "glue" Y to another space X along Z using an "attaching map" f : Z → X. The result is the adjunction space X \cup_{f} Y which is just the pushout of f and g. More generally, all identification spaces may be regarded as pushouts in this way.

21-propriété-universelle-de-pushout : Universal property Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i1 : X → P and i2 : Y → P such that the diagram Categorical pushout.svg commutes and such that (P, i1, i2) is universal with respect to this diagram. That is, for any other such set (Q, j1, j2) for which the following diagram commutes, there must exist a unique u : P → Q also making the diagram commute

3. Propriétés d'un pullback

19-propriétés d'un pullback : Properties Whenever X ×Z Y exists, then so does Y ×Z X and there is an isomorphism X ×Z Y ≅ Y ×Z X. Monomorphisms are stable under pullback: if the arrow f above is monic, then so is the arrow p2. For example, in the category of sets, if X is a subset of Z, then, for any g : Y → Z, the pullback X ×Z Y is the inverse image of X under g\circ p_2 . Isomorphisms are also stable, and hence, for example, X ×X Y ≅ Y for any map Y → X. Any category with pullbacks and products has equalizers.

 

Younes Derfoufi
CRMEF OUJDA

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