# Are there any theorems about functors that reflect exactness?

Suppose $F:\mathbf{A}\to \mathbf{B}$ is an additive functor between two abelian categories, we say $F$ is exact iff it preserves short exact sequences.

Is there a name for a functor $F$ that reflects short exact sequences? For example the forgetful functor $\mathbf{Mod_R}\to\mathbf{Ab}$ both preserves and reflects short exact sequences.

Also, we know for example right adjoints are left exact; are there any similar theorems giving us functors that reflect left exactness and so on?

• A functor that preserves and reflects exact sequences is said to be faithfully exact. – Zhen Lin Jan 26 '15 at 19:06
• The dual notion to an exact functor is an exact functor (in the same way that the dual notion to a right exact functor is a left exact functor). – Qiaochu Yuan Jan 27 '15 at 1:35

## 1 Answer

Every functor $F : \mathcal A → \mathcal B$ between Abelian categories which reflects left (or dually: right) exactness must be faithful (if $f, g : A → B$ are mapped by $F$ to the same morphism, then $\mathrm{id} : FA → FA$ is the kernel of $Ff-Fg = 0$, but then $\mathrm{id} : A → A$ must be the kernel of $f-g$, which implies $f = g$).

Now if $F$ is faithful it is also conservative (because $\mathcal A$ is balanced), and conservative functors reflect all (co)limits they preserve. So faithful left exact functors reflect left exact sequences, and this is in particular true for forgetful functors, as you mentioned.

I don't know if there are other or weaker sufficient conditions to reflect exactness.