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 𝑋𝑓×𝑔𝑌 is the limit of the diagram on the left, whereas the fibre coproduct 𝑋𝑓⨿𝑔𝑌 is the colimit of the diagram on the right: cat

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

For the first diagram we say that 𝑝𝑋 is the pullback of 𝑔 along 𝑓, the latter we say 𝑖𝑋 is the pushforward of 𝑔 along 𝑓.

Examples

• The union of non-disjoint sets (or topological spaces)
• Amalgamated free product

Properties

• Fibre product is the equalizer of a product

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