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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