Limits and colimits

Builds on

Required prerequisites, all the way down: 20

You will learn to

• Compute equalisers, pullbacks and pushouts in Set, Grp and Top.
• Prove that a category with all small products and equalisers has all small limits.
• Show that hom functors Hom(A, −) preserve limits, and describe the p-adic integers as an inverse limit.

Definition

• A diagram of shape J in C is a functor D : J → C.
• A cone over D with apex X is a natural transformation from the constant functor at X to D; a limit of D is a terminal cone.
• A colimit is a limit in the opposite category, that is, an initial cocone.

Theorem

• A category with all small products and equalisers has all small limits.
• Set, Grp, Ab, Top and Vect_k have all small limits and colimits.
• Hom(A, −) sends limits in C to limits in Set.
• The p-adic integers are the inverse limit of the rings ℤ/pⁿℤ.

Common misconceptions

• Identifying categorical limits with limits of sequences. They are related (the supremum of an increasing sequence of reals is a colimit in the poset (ℝ, ≤)) but they are different notions.
• Expecting forgetful functors to preserve colimits as they preserve limits; Grp → Set preserves products but not coproducts.

Leads to

Optional discoveries

• Adjunctions Limits are right adjoints: taking limits of J-shaped diagrams is right adjoint to the diagonal functor C → [J, C], and right adjoints preserve all limits.Wonder Trail →
• ← Universal properties
• ← Quotient groups and quotient rings

Further reading

• Emily Riehl, Category Theory in Context (2016)
• Tom Leinster, Basic Category Theory (2014)