The classification that decides whether a functor needs deriving
A functor carries objects to objects and morphisms to morphisms, preserving composition and identities. Between additive categories the useful ones are additive, preserving sums of morphisms. The critical classification is by exactness: exact functors need no repair, half-exact functors generate derived functors, and functors that are neither are outside the reach of the classical machinery.
Learning objectives
- Distinguish covariant and contravariant functors.
- Define additivity and explain why it is assumed.
- Place the standard functors in the exactness hierarchy.
- Explain faithfulness and fullness and what each buys.
Section 01Functors and variance
A covariant functor T assigns f: A → B to T(f): TA → TB; a contravariant functor reverses. Contravariance is not a separate theory: a contravariant functor on C is a covariant functor on Cop.
| Functor | Variance | Exactness |
|---|---|---|
| Hom(M, −) | Covariant | Left exact; exact iff M projective |
| Hom(−, N) | Contravariant | Left exact; exact iff N injective |
| M ⊗ − | Covariant | Right exact; exact iff M flat |
| Forgetful Mod → Ab | Covariant | Exact |
| Free: Set → Mod | Covariant | Not applicable — source is not abelian |
| Hn: Ch → Mod | Covariant | Neither; produces the long exact sequence instead |
Section 02Additivity
A functor between additive categories is additive when T(f + g) = T(f) + T(g) on each Hom group. Equivalently it preserves finite direct sums and the zero object.
Every functor that gets derived is assumed additive. Without it, a functor need not carry a split sequence to a split sequence, chain homotopies are not preserved, and the independence of derived functors from the chosen resolution breaks down. The assumption is so pervasive that it is often left unstated.
Section 03Faithful and full
Reflects equality of morphisms. The forgetful functor to sets is faithful, which is why algebraic categories behave like structured sets.
Every morphism between images comes from the source. A full and faithful functor is an equivalence onto its image.
A functor that is full, faithful and essentially surjective is an equivalence of categories — the correct notion of sameness, weaker than isomorphism of categories and far more useful. Morita equivalence of rings is exactly an equivalence of their module categories, and it shows that a ring is not determined by its modules.
An exact functor that is also faithful reflects exactness: if the image of a sequence is exact, so was the original. Hom(−, ℚ/ℤ) on abelian groups is the standard example, and it converts many statements into their duals.
ReferenceFrequently asked questions
Can a functor be neither left nor right exact?
Yes. Homology of a chain complex is such a functor, and so is the fixed-point functor on modules over a group in some formulations. When only half-exactness in a weaker sense holds, satellites rather than classical derived functors are the right tool.
Does a functor preserve isomorphisms?
Always — a functor preserves inverses because it preserves composition and identities. It need not reflect them, unless it is faithful and full, or conservative.
Why is variance such a common source of error?
Because Ext and Tor take two arguments with different variance, and long exact sequences change direction accordingly. The reliable habit is to write out which variable is fixed before writing any sequence.
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