The categorical colimit that glues two objects along a common subobject: given maps A → B and A → C, the pushout B ⊔_A C is the universal object receiving maps from B and C that agree on A.

Principle

Characterized by the universal property of colimits: B ⊔_A C comes with canonical maps i_B: B → B ⊔_A C and i_C: C → B ⊔_A C such that i_B∘f = i_C∘g on A, and any other object receiving compatible maps from B and C factors uniquely through the pushout.

Demonstration

In the category of sets, the pushout of inclusions A ↪ B and A ↪ C is the disjoint union B ⊔ C modulo the identification of elements coming from A. In topology, pushouts glue spaces along a common subspace (e.g., attaching a cell by gluing its boundary to a space). In algebra, one often forms quotient constructions realizing amalgamated sums of rings or groups with extra relations.

Misapplication

Treating pushouts as naive set-theoretic unions in categories where additional structure matters (e.g., forgetting identifications, topological glueings, or algebraic relations). Another misuse is assuming pushouts preserve monomorphisms or embeddings automatically; they may introduce identifications that break injectivity.

Consequence

Pushouts implement gluings, amalgamations, and attaching constructions and are central to constructions of colimits, coequalizers, and diagrams. They allow one to combine objects while enforcing identifications dictated by a common subobject, producing universal amalgams used in algebra and topology.

Reversal

Dual to pullback: whereas the pushout is the universal cocone merging along a source, the pullback is the universal cone synchronizing along a target. Reversal swaps colimit behavior (amalgamation) for limit behavior (synchronization).

Boundary

Requires an ambient category with the relevant colimits; existence and form depend on the category (sets, topological spaces, modules, groups, rings behave differently). Not every pushout preserves properties like embeddings, finite presentation, or Hausdorff separation.

Semantic Tension

Confused with unions or naive concatenations because all ‘glue’ components, but pushout is defined by a universal property and can create new identifications or algebraic relations absent from plain unions. Tension also arises with fibered coproducts and coequalizers, which are closely related but differ in construction.

Synthesis

The pushout is the universal way to glue two objects along a shared subobject: it forms the cocone that identifies the images of the common part and mediates every compatible pair of maps into any other object.