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The Lumenary
Advanced undergraduateCategory TheoryPlanned

Limits and colimits

Diagrams, cones and limits: equalisers, pullbacks and inverse limits and their duals, completeness of Set, Grp and Top, the construction of all limits from products and equalisers, and preservation of limits by functors.

math.cat.limits-colimits

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The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.

Builds on

Products, coproducts, initial and terminal objectsNatural transformations

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

SheavesAbelian categoriesElementary topos theoryKan extensions

Optional discoveries

Further reading

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