Reflective Subcategory: Cat

Cat is a reflective subcategory of SSet, by which we mean that Cat is a full subcategory whose inclusion functor has a left adjoint. The inclusion functor i is called the reflection and the left adjoint is called the reflector.

Reflectors tend to be meaningful, and in this case, the functor which reflects SSet to Cat is called the homotopy category functor of SSet.

Similarly, reflections tend to be meaningful too, and in this case, the functor which includes Cat into SSet is called the nerve functor of Cat.

In terms of stuff, structure, and properties, objects of SSet are like objects of Cat but with extra stuff; this is the reflection of the fact that, because Cat is a full subcategory, objects of Cat are like objects of SSet but with extra properties.

reflective_subcategories: Cat, SSet

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