In category theory, the concept of [[pullback]] is a way to describe and analyze relationships between objects and morphisms in a category. It provides a universal construction that allows us to compare and relate different parts of a category. Given two objects, A and B, in a category, the pullback of A and B (also known as a fiber product) is an object P along with two morphisms: one from P to A, and another from P to B. This construction guarantees that certain commutative diagrams involving these objects can be made to commute. ## Pushout is called coequalizer The pullback has a complement called the [[pushout]] (or [[coequalizer]]). While the pullback relates objects going into A and B, the pushout relates objects coming out of A and B. It provides a dual construction that allows us to compare the outputs or results of different processes or operations. ## But pullback is NOT called equalizer No, a pullback is not called an equalizer in category theory. The terms "pullback" and "equalizer" refer to different concepts in category theory. An equalizer is a limit of a pair of parallel morphisms, which means it is a universal object that represents the most general way to "equalize" those morphisms. It consists of an object and a morphism such that any other object with a morphism to the endpoints of the original pair can be uniquely factored through the equalizer. On the other hand, a pullback is a construction that allows one to "pull back" objects and morphisms along another morphism. Given two morphisms with a common codomain, the pullback consists of an object and two morphisms such that they form a commutative square with the original pair of morphisms. Furthermore, any other object with two morphisms forming such a square can be uniquely factorized through the pullback. While there may be some similarities between these concepts in specific cases, they are distinct concepts in category theory with different properties and purposes. # Usefulness in combinatorics Now, how is this useful in combinatorics? Combinatorics deals with [[counting]], arranging, and structuring discrete objects or configurations. Category theory provides a framework to abstractly study various mathematical structures, including combinatorial structures. By using pullbacks (and pushouts), we can identify common patterns or relationships between different combinatorial structures. For example, let's consider counting problems. We can represent counting problems as objects in a category where morphisms represent transformations or operations on these objects. The pullback allows us to compare different counting problems by considering how they relate in terms of inputs or initial configurations. Similarly, we can use pushouts to analyze combinatorial structures based on their outputs or final configurations. By studying these universal constructions in category theory, we gain insights into combinatorics by understanding how various structures are connected or related based on certain transformations or operations. Overall, the concepts of pullbacks (and pushouts) provide powerful tools for studying relationships between objects in category theory. In combinatorics, they allow us to analyze and compare different combinatorial structures based on their inputs and outputs, providing a deeper understanding of the underlying combinatorial phenomena. # References ```dataview Table title as Title, authors as Authors where contains(subject, "pullback") or contains(subject, "Pullback") ```