##  [Pushout](/pushout-0) 

 Definition

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

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

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

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

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

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

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

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

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.