Pullbacks, Pushouts and General Limits

Executive summary

Two constructions that build every other limit

A pullback is the universal object mapping compatibly to two objects over a common target; a pushout is its dual. In module categories both are explicit — a submodule of a direct sum, and a quotient of one. They matter here because the pushout is exactly how the Baer sum of extensions is built, and because products together with equalisers generate all limits, so checking two cases establishes completeness.

Learning objectives

Construct pullbacks and pushouts in a module category.
State the universal property of a general limit.
Explain why products and equalisers suffice.
State the exactness of filtered colimits and why it matters.

Section 01Pullbacks and pushouts

Given f: A → C and g: B → C, the pullback is

A ×_C B = { (a, b) : f(a) = g(b) } ⊆ A ⊕ B

Dually, for f: C → A and g: C → B, the pushout is the quotient of A ⊕ B by the submodule of elements (f(c), −g(c)).

Where the Baer sum comes from

Given two extensions of C by A, pull back along the diagonal of C and push out along the addition map of A. The result is a new extension, and this operation is the group law on Ext¹. Pullback and pushout are therefore not incidental — they are the arithmetic of extensions.

Properties worth knowing
Property	Statement
Pullback preserves monos	If g is mono, so is its pullback along f
Pushout preserves epis	If g is epi, so is its pushout
Pullback of a split epi	Is again split
In an abelian category	A square is both a pullback and a pushout exactly when a certain short sequence is exact

Section 02Limits and colimits

A limit of a diagram is a universal cone over it; a colimit is a universal cocone. Products, equalisers, pullbacks, kernels and inverse limits are all limits; coproducts, coequalisers, pushouts, cokernels and directed colimits are all colimits.

Limits and colimits by shape of diagram
Diagram shape	Limit	Colimit
Discrete	Product	Coproduct
Parallel pair	Equaliser	Coequaliser
Cospan / span	Pullback	Pushout
Single arrow to 0	Kernel	Cokernel
Directed poset	Inverse limit	Direct limit
Empty	Terminal object	Initial object

Two constructions suffice

A category with all products and all equalisers has all limits, since any limit can be built as an equaliser of two maps between products. Verifying completeness therefore reduces to two checks.

Section 03Exactness of colimits

Filtered colimits of modules are exact; inverse limits are not. This asymmetry is one of the most consequential facts in the subject.

Exact

Filtered colimits

Direct limits over a directed system are exact in module categories. Hence homology commutes with directed colimits, and finitely generated cases often suffice.

Not exact

Inverse limits

Only left exact. The failure is measured by lim¹, which appears in the Milnor sequence and in the convergence of spectral sequences over a filtration.

Consequence

Completion arguments

Whenever a filtration is completed, a lim¹ term must be shown to vanish before the answer can be read off.

lim<sup>1</sup> is easy to forget

A spectral sequence that converges at each stage may still fail to compute the intended object if the inverse system of filtration quotients has non-vanishing lim¹. Convergence statements always carry a hypothesis controlling this, and omitting it is a genuine error rather than a technicality.

ReferenceFrequently asked questions

Is the pullback the same as the fibre product?

Yes — the terms are interchangeable. Fibre product is preferred in geometry, where the fibres of the two maps over each point are being multiplied; pullback is preferred in categorical and homological contexts.

Do limits exist in every abelian category?

Finite ones do, by the axioms. Arbitrary products and coproducts require additional axioms — the Grothendieck AB axioms — and categories satisfying them, together with a generator, are the ones where injective resolutions can be constructed in general.

Why are filtered colimits exact but arbitrary ones not?

Because in a filtered system any finite collection of elements can be brought into a single stage, so an exactness check reduces to one stage where it holds by hypothesis. Non-filtered colimits offer no such reduction and can destroy exactness.

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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 Pullbacks, Pushouts and General Limits. 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 Pullbacks, Pushouts and General Limits as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—colimits, limits, pullbacks, pushouts, section—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 Pullbacks, Pushouts and General Limits?