“Equifier” Explained

“Equifier” Explained

What’s in a name?

According to the nLab, an equifier is a generalization of the concept of an equalizer, which is a limit of a diagram in a category that captures the notion of two morphisms having the same image or “equalizing” some other morphisms in the diagram.

More specifically, given a diagram D in a category C and a class of morphisms S in C, an equifier of D with respect to S is an object E along with a morphism e: E -> D such that for any morphism f: X -> D in S, the pullback of f along e exists and is an isomorphism. That is, for any f: X -> D in S, there exists a unique morphism u: X -> E such that f = e ∘ u, and the pullback of f along e, which is a morphism from the pullback of f to E, is an isomorphism.

Intuitively, an equifier is a limit of a diagram in which a given class of morphisms are made equal, rather than just a pair of morphisms as in the case of an equalizer. Equifiers have applications in algebraic geometry, algebraic topology, and higher category theory.

p.s. don’t ask me more.. I am still working on the basics of category theory…