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 $𝑋 𝑓 \times 𝑔 𝑌$ is the limit of the diagram on the left, whereas the fibre coproduct $𝑋 𝑓 ⨿ 𝑔 𝑌$ is the colimit of the diagram on the right: #m/def/cat

𝑋 𝑓 \to 𝑍 𝑔 \leftarrow 𝑌 𝑋 𝑓 \leftarrow 𝑍 𝑔 \to 𝑌

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 $𝑓 * 𝑔$ is the pullback of $𝑔$ along $𝑓$ , the latter we say $𝑓 * 𝑔$ is the pushout of $𝑔$ along $𝑓$ .

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

Fibre product and coproduct

Definition

Properties

Examples

See also