What does coequalizer mean in category theory? In category theory, a coequalizer is a concept that generalizes the notion of equivalence or identification between objects. Given two morphisms (arrows) f and g from an object A to an object B in a category, a coequalizer is an object C along with a morphism q: B → C, such that the compositions q ∘ f and q ∘ g are equal. Intuitively, this means that the morphism q "collapses" or "identifies" the images of f and g in object B into a single object C, making them indistinguishable from each other. The coequalizer C is considered to be the "coarsest" or "most general" object where f and g become equal. The coequalizer also satisfies a universal property: for any other object D and morphism h: B → D such that h ∘ f = h ∘ g, there exists a unique morphism u: C → D such that u ∘ q = h. In summary, a coequalizer in category theory provides a way to identify or collapse objects based on the equality of their images under certain morphisms. It plays an important role in various areas of mathematics and computer science, providing tools for studying quotient structures, equivalence relations, and many other concepts. # Coequalizer and pushout In category theory, a [[coequalizer]] can be considered as a type of [[pushout]]. In general, a [[pushout]] is a universal construction that combines two morphisms in a category into a single morphism. It is often visualized as a diagram where two arrows share the same starting point and converge at a common endpoint. A [[coequalizer]] specifically refers to the pushout of two morphisms that have the same target object. This means that the starting point of both morphisms is different, but they both end at the same object. The coequalizer then creates a new object that represents the "merge" of these two starting points, along with an arrow that connects this new object to the common endpoint. # Coequalizer and equalizer Please see [[equalizer]]. # Conclusion In summary, while not all pushouts are coequalizers, every coequalizer can be thought of as a special case of a pushout where the target objects of the morphisms being combined are equal. # References ```dataview Table title as Title, authors as Authors where contains(subject, "coequalizer") ```