# Documentation

Mathlib.CategoryTheory.Limits.FinallySmall

# Finally small categories #

A category given by (J : Type u) [Category.{v} J] is w-finally small if there exists a FinalModel J : Type w equipped with [SmallCategory (FinalModel J)] and a final functor FinalModel J ⥤ J.

This means that if a category C has colimits of size w and J is w-finally small, then C has colimits of shape J. In this way, the notion of "finally small" can be seen of a generalization of the notion of "essentially small" for indexing categories of colimits.

Dually, we have a notion of initially small category.

• final_smallCategory : S x F,

There is a final functor from a small category.

A category is FinallySmall.{w} if there is a final functor from a w-small category.

Instances
theorem CategoryTheory.FinallySmall.mk' {J : Type u} {S : Type w} (F : ) :

Constructor for FinallySmall C from an explicit small category witness.

An arbitrarily chosen small model for a finally small category.

Instances For
noncomputable instance CategoryTheory.smallCategoryFinalModel (J : Type u) :
noncomputable def CategoryTheory.fromFinalModel (J : Type u) :

An arbitrarily chosen final functor FinalModel J ⥤ J.

Instances For
• initial_smallCategory :

There is an initial functor from a small category.

A category is InitiallySmall.{w} if there is an initial functor from a w-small category.

Instances
theorem CategoryTheory.InitiallySmall.mk' {J : Type u} {S : Type w} (F : ) :

Constructor for InitialSmall C from an explicit small category witness.

An arbitrarily chosen small model for an initially small category.

Instances For
noncomputable instance CategoryTheory.smallCategoryInitialModel (J : Type u) :
noncomputable def CategoryTheory.fromInitialModel (J : Type u) :

An arbitrarily chosen initial functor InitialModel J ⥤ J.

Instances For