Mathematics.

foundations and logic

Topos Theory

Category Theory120 minDifficulty10 out of 10

You should know: limits and colimits, adjoint functors, natural transformations, propositional logic

Overview

A topos is a category that behaves like a universe of sets — but generalised far beyond the category Set. Every topos has finite limits, exponential objects (internal function spaces), and crucially a subobject classifier Ω that plays the role of the two-element set {true, false}. Toposes arise as sheaf categories on a site (Grothendieck toposes, central to algebraic geometry) and as categories satisfying pure categorical axioms (elementary toposes, supporting an internal intuitionistic logic).

Intuition

Think of a topos as a 'generalised universe of sets'. In Set, a subset S ⊆ A is classified by its characteristic function χ_S: A → {true, false}. In a topos, there is an object Ω (the subobject classifier) playing the role of {true, false}: any subobject m: S → A corresponds bijectively to a morphism χ_m: A → Ω. This single axiom (plus finite limits and exponentials) is enough to recover most of set-theoretic mathematics internally.

Formal Definition

Definition

An elementary topos is a category E satisfying three axioms:

E has all finite limits\text{E has all finite limits}
finite-limits
E is Cartesian closed: Hom(A×B,C)≅Hom(A,CB) naturally\text{E is Cartesian closed: } \text{Hom}(A \times B, C) \cong \text{Hom}(A, C^B) \text{ naturally}
cartesian-closed
∃ Ω∈E,  true:1→Ω such that Sub(A)≅Hom(A,Ω) naturally in A\exists\, \Omega \in E,\; \text{true}: 1 \to \Omega \text{ such that } \text{Sub}(A) \cong \text{Hom}(A, \Omega) \text{ naturally in } A

Subobject classifier: every subobject m: S↪A has a unique characteristic morphism χ_m: A→Ω pulling true back to m

subobject-classifier
P(A):=ΩA(power object = internal powerset)\mathcal{P}(A) := \Omega^A \quad (\text{power object = internal powerset})
power-object

Properties

Every topos has a subobject classifier

Sub(A)≅Hom(A,Ω) naturally in A.\text{Sub}(A) \cong \text{Hom}(A, \Omega) \text{ naturally in } A.

Set is a topos

In Set, Ω={false,true} and χS(x)=true iff x∈S.\text{In Set, } \Omega = \{\text{false}, \text{true}\} \text{ and } \chi_S(x) = \text{true iff } x \in S.

Presheaf toposes

For any small category C, the presheaf category [Cop,Set] is a Grothendieck topos.\text{For any small category } \mathcal{C}\text{, the presheaf category } [\mathcal{C}^{\text{op}}, \text{Set}] \text{ is a Grothendieck topos.}

Law of excluded middle fails internally

In Sh(X) for a non-discrete X, the proposition Ω need not satisfy p∨¬p=true — the internal logic is intuitionistic.\text{In } \text{Sh}(X) \text{ for a non-discrete } X\text{, the proposition } \Omega \text{ need not satisfy } p \lor \neg p = \text{true} \text{ — the internal logic is intuitionistic.}

Theorems

Theorem 1: Giraud's Theorem
A category E is a Grothendieck topos iff it has a small set of generators, all small colimits, universal colimits, disjoint coproducts, and effective equivalence relations.\text{A category } E \text{ is a Grothendieck topos iff it has a small set of generators, all small colimits, universal colimits, disjoint coproducts, and effective equivalence relations.}
Theorem 2: Diaconescu's Theorem
Geometric morphisms Set[Cop]→E from the presheaf topos to E correspond naturally to flat functors C→E.\text{Geometric morphisms } \text{Set}[\mathcal{C}^{\text{op}}] \to E \text{ from the presheaf topos to } E \text{ correspond naturally to flat functors } \mathcal{C} \to E.
Theorem 3: Internal Logic
Every topos E has an internal higher-order intuitionistic logic: propositions are subobjects of 1, connectives are operations on Ω, and quantifiers arise from adjoint pairs involving P.\text{Every topos } E \text{ has an internal higher-order intuitionistic logic: propositions are subobjects of 1, connectives are operations on } \Omega \text{, and quantifiers arise from adjoint pairs involving } \mathcal{P}.

Worked Examples

  1. A presheaf F: C^{op} → Set is a functor. Subobjects of F are natural transformations with componentwise injections.

  2. The subobject classifier Ω assigns to each object c ∈ C the set of sieves on c (downward-closed sets of morphisms into c).

    Ω(c)={sieves on c}={S⊆Hom(−,c)∣S downward-closed}\Omega(c) = \{\text{sieves on } c\} = \{S \subseteq \text{Hom}(-,c) \mid S \text{ downward-closed}\}

  3. The morphism true: 1 → Ω picks the maximal sieve at each c: true_c = Hom(−,c).

  4. For a natural transformation m: G ↪ F, its classifying map χ_m(c)(x) = {f: d→c | F(f)(x) ∈ G(c)}.

Answer: In [C^{op}, Set], the subobject classifier is the presheaf of sieves: Ω(c) = {all sieves on c}. This generalises {false, true} = {∅, max sieve} from the case C = 1.

Practice Problems

Difficulty 8/10

Identify the subobject classifier in Set and verify it classifies subsets correctly.

Difficulty 9/10

What does it mean for a topos to have a 'geometric morphism' f: E → F? Give an example.

Difficulty 10/10

Explain how the power object P(A) = Ω^A in a topos plays the role of the powerset.

Common Mistakes

Common Mistake

Thinking every topos has classical logic internally.

Only Boolean toposes (where Ω is a Boolean algebra, e.g., Sh(X) for a discrete X or Set itself) have classical internal logic. Sheaves on a non-discrete space have an intuitionistic logic where LEM fails.

Common Mistake

Confusing Grothendieck toposes with elementary toposes.

Every Grothendieck topos is an elementary topos, but not conversely. Grothendieck toposes have a site description and satisfy Giraud's axioms; elementary toposes only satisfy the three categorical axioms (finite limits, Cartesian closure, subobject classifier).

Quiz

The subobject classifier Ω in an elementary topos satisfies:
The internal logic of a topos is:
Which of the following is a Grothendieck topos?

Historical Background

Grothendieck introduced toposes in the 1960s as sheaf categories to enable étale cohomology for algebraic varieties. Lawvere and Tierney extracted the purely categorical axioms in 1969–70, revealing that toposes support an internal higher-order intuitionistic logic — unifying algebraic geometry, categorical logic, and foundations of mathematics.

  1. 1963

    Grothendieck introduces toposes as sheaf categories on a site

    Alexander Grothendieck

  2. 1966

    Giraud's theorem characterises Grothendieck toposes

    Jean Giraud

  3. 1969

    Lawvere–Tierney formulate elementary topos axioms

    William Lawvere, Myles Tierney

  4. 1977

    Johnstone's 'Topos Theory' consolidates the field

    Peter Johnstone

  5. 2002

    Johnstone's 'Sketches of an Elephant' gives the comprehensive reference

    Peter Johnstone

Summary

  • An elementary topos is a Cartesian closed category with finite limits and a subobject classifier Ω.
  • The subobject classifier Ω generalises {true, false}: subobjects of A biject with morphisms A → Ω.
  • Grothendieck toposes are sheaf categories Sh(C, J) on a site (C, J); they satisfy Giraud's axioms.
  • Every topos supports an internal higher-order intuitionistic logic; classical logic holds iff the topos is Boolean.
  • Geometric morphisms E → F are adjoint pairs f* ⊣ f_* with f* left exact — the categorical notion of a 'continuous map' between toposes.

References

  1. WebsiteWikipedia — Topos