Universal construction

Fibre product and coproduct

The fibre product is a categorical generalization of the intersection of two sets, whereas the fibre coproduct is the generalization of the union of two non-disjoint sets.

Definition

The fibre product is the limit of the diagram on the left, whereas the fibre coproduct is the colimit of the diagram on the right: #m/def/cat

Notationally the and are usually omitted, and diagramatically the fibre product or coproduct is denoted with a right angle symbol.

For the first diagram we say that is the pullback of along , the latter we say is the pushforward of along .

Properties

• To explicitly construct a fibre (co)product, we can appeal to Fibre product is the equalizer of a product and its dual result.

Examples

• Amalgamated free product

See also

• Subobject classifier

#state/tidy | #lang/en | #SemBr