Given any abelian category, Mathlib supplies some ring and a way to represent the category inside that ring's module category. Distinct arrows remain distinct, every arrow between represented objects comes from the original category, and finite limit and finite colimit constructions are preserved.
abelian category
module category
full functor
faithful functor
finite limits
finite colimits
exact embedding
Exact theorem
Exact Mathlib statement
theorem CategoryTheory.Abelian.freyd_mitchell (C : Type u) [Category.{v} C] [Abelian C] :
∃ (R : Type (max u v)) (_ : Ring R) (F : C ⥤ ModuleCat.{max u v} R),
F.Full ∧ F.Faithful ∧ PreservesFiniteLimits F ∧ PreservesFiniteColimits F
The theorem at a glance
Freyd–Mitchell at a glance
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
Freyd–Mitchell at a glance
Loading full-resolution image…
The endpoint is an exact embedding into modules over some ring, not an equivalence with every module.Open original image in a new tab
The endpoint is an exact embedding into modules over some ring, not an equivalence with every module. Explanatory diagram.Detailed visual description
The poster pairs a one-way categorical embedding diagram with short boundary-safe copy. The source category reappears as a highlighted proper subdiagram inside the larger module field; the exact declared properties are listed below, followed by an explicit non-equivalence boundary.
A full, faithful functor embeds the abelian category into modules over some ring and preserves finite limits and finite colimits.Open original image in a new tab
A full, faithful functor embeds the abelian category into modules over some ring and preserves finite limits and finite colimits. Explanatory scientific diagram.Detailed visual description
This target indexes Mathlib's existential Freyd–Mitchell endpoint for a category C equipped with an Abelian instance: there exist a ring R and a functor F : C ⥤ ModuleCat R that is full, faithful, preserves finite limits, and preserves finite colimits. The universe of R and the module category is max u v. The statement is an embedding into a module category, not an equivalence onto that whole category.
Why it matters
A mathematical landmark
The Freyd–Mitchell theorem connects abstract abelian categories with concrete module categories. It explains why many arguments about kernels, cokernels, exact sequences, and diagram chasing can be understood through module-like models without identifying the entire abstract category with every module over a ring.
Accepted Atlas resultNot recorded for the preferred artifact
Mathlib is the source of the theorem; the local Lean replay and page review are separate.
Claim boundary
No new theorem is claimed
This target indexes Mathlib's existential Freyd–Mitchell endpoint for a category C equipped with an Abelian instance: there exist a ring R and a functor F : C ⥤ ModuleCat R that is full, faithful, preserves finite limits, and preserves finite colimits. The universe of R and the module category is max u v. The statement is an embedding into a module category, not an equivalence onto that whole category.
Proof Atlas did not originate the Freyd–Mitchell theorem or Mathlib's declaration.
The selected theorem does not assert that C is equivalent to the entire category of modules over R.
The existentially supplied ring is not claimed to be a fixed familiar ring or canonical for every purpose.
The endpoint states preservation of finite limits and finite colimits, not arbitrary limits and colimits.
The generated explanation and visuals are not proof evidence.
