Functors and Their Exactness Properties

Executive summary

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 C^op.

Functors that appear constantly
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
H_n: 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.

Additivity is a standing hypothesis

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

FaithfulInjective on each Hom set

Reflects equality of morphisms. The forgetful functor to sets is faithful, which is why algebraic categories behave like structured sets.

FullSurjective on each Hom set

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.

Faithfully exact functors reflect exactness

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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Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Functors and Their Exactness Properties. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Functors and Their Exactness Properties as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—functors, functor, section, exactness, covariant—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Functors and Their Exactness Properties?