Topos Theory — Foundations¶
Job: State elementary topos axioms; develop the subobject classifier and internal logic; identify sheaf toposes; record geometric morphisms in one clean package.
Core¶
Elementary topos axioms¶
An elementary topos \(\mathcal{E}\) is a category such that:
- Finite limits exist (equivalently: terminal object + pullbacks).
- Exponentials: for all \(A,B\), an object \(B^A\) with \(\mathcal{E}(C\times A,B)\cong\mathcal{E}(C,B^A)\) natural in \(C\) (cartesian closed + finite limits \(\Rightarrow\) finite products).
- Subobject classifier: object \(\Omega\) and mono \(\mathsf{true}:1\rightarrowtail\Omega\) such that every mono \(m:A\rightarrowtail B\) is the pullback of \(\mathsf{true}\) along a unique characteristic morphism \(\chi_m:B\to\Omega\): $$ \begin{CD} A @>>> 1\ @VmVV @VV\mathsf{true}V\ B @>>\chi_m> \Omega \end{CD} $$
Power objects: \(PA:=\Omega^A\); mono \(\in_A\rightarrowtail A\times PA\) classifies relations. Elementary topos \(\Leftrightarrow\) finite limits + power objects (alternative packaging).
| Consequence | Note |
|---|---|
| Finite colimits | exist in any elementary topos |
| Image factorizations | epi-mono factorization |
| Internal Heyting algebra | \(\Omega\) carries \(\land,\lor,\Rightarrow,\bot,\top\) |
| Natural numbers object | not automatic; often assumed (NNO) |
Subobject classifier in examples¶
| Topos | \(\Omega\) |
|---|---|
| \(\mathbf{Set}\) | \(\{0,1\}\); \(\mathsf{true}\mapsto 1\) |
| \(\mathbf{Set}^{C^{\mathrm{op}}}\) | sieves: \(\Omega(c)=\{\)sieves on \(c\}\) |
| \(\mathbf{Sh}(X)\) | sheaf of open sets: \(\Omega(U)=\{V\text{ open}:V\subseteq U\}\) |
| \(G\text{-}\mathbf{Set}\) | \(\{0,1\}\) with trivial action (if \(G\) group) — Boolean |
Mono \(\leftrightarrow\) subobject: subobjects of \(B\) are equivalence classes of monos into \(B\); \(\mathrm{Sub}(B)\cong\mathcal{E}(B,\Omega)\).
Sheaf toposes¶
Site \((C,J)\): small category \(C\) + Grothendieck topology \(J\) (covering sieves stable under pullback, etc.).
Presheaf \(P:C^{\mathrm{op}}\to\mathbf{Set}\). Sheaf: for every covering sieve \(S\) on \(c\), $$ P(c)\xrightarrow{\sim}\mathrm{Cone}(S,P) $$ (matching families glue uniquely). Sheafification \(a:\mathbf{PSh}(C)\to\mathbf{Sh}(C,J)\) left adjoint to inclusion — left exact localization.
Theorem (Giraud). A category is a Grothendieck topos (equivalent to some \(\mathbf{Sh}(C,J)\)) iff it satisfies Giraud’s axioms: cocomplete, exactness of colimits/pullbacks, small generating set, disjoint universal coproducts, effective equivalence relations.
Geometric examples: \(\mathbf{Sh}(X)\) for topological \(X\); étale topos of a scheme; classifying toposes of geometric theories.
Internal logic¶
Mitchell–Bénabou language: types = objects; terms = morphisms; formulas interpreted as subobjects (via \(\{x\mid \varphi(x)\}\rightarrowtail A\)).
Kripke–Joyal semantics (forcing-style clauses for sheaves):
- \(U\Vdash \varphi\land\psi\) iff both
- \(U\Vdash \varphi\Rightarrow\psi\) iff for all \(V\to U\), \(V\Vdash\varphi\Rightarrow V\Vdash\psi\)
- \(U\Vdash \forall x:A.\,\varphi\) iff for all \(V\to U\) and \(a:V\to A\), \(V\Vdash\varphi(a)\)
- \(U\Vdash \exists x:A.\,\varphi\) iff there is a cover \((U_i\to U)\) and \(a_i:U_i\to A\) with \(U_i\Vdash\varphi(a_i)\)
Logic is intuitionistic in general (Heyting, not necessarily Boolean). Boolean topos: \(\Omega\) is internal Boolean algebra (e.g. \(\mathbf{Set}\), atomic Boolean cases). Two-valued: global elements of \(\Omega\) only \(\bot,\top\).
Internal reasoning: prove statements in intuitionistic higher-order logic; they hold in every topos. Enables “synthetic” arguments for sheaves without unwrapping.
Geometric morphisms (one package)¶
A geometric morphism \(f:\mathcal{F}\to\mathcal{E}\) is an adjunction $$ f^\dashv f_:\mathcal{E}\to\mathcal{F} $$ with \(f^*\) left exact (preserves finite limits). \(f^*\): inverse image; \(f_*\): direct image.
| Example | \(f^*\) / \(f_*\) |
|---|---|
| Continuous \(\varphi:X\to Y\) | \(\varphi^*\), \(\varphi_*\) on sheaves |
| Point \(x:1\to\mathcal{E}\) | stalk functors / global sections adjunction variants |
| Sheafification | \(a\dashv i:\mathbf{Sh}\hookrightarrow\mathbf{PSh}\) is geometric |
Essential geometric morphism: \(f^*\) has a left adjoint \(f_!\). Bounded / étale morphisms encode relative smallness and local homeomorphism analogues.
Logical functors preserve topos structure more strictly (finite limits + \(\Omega\)); geometric morphisms preserve geometric logic (colimits + finite limits in the inverse image).
Lawvere–Tierney topology¶
Internal locality: mono \(j:\Omega\to\Omega\) with \(j\circ\mathsf{true}=\mathsf{true}\), \(j\circ j=j\), \(j\circ\land=\land\circ(j\times j)\). Corresponds to localizations / sheaf subtoposes of an elementary topos — elementary stand-in for Grothendieck topologies.
See it¶
Forcing as sheaves. Cohen forcing poset \(\mathbb{P}\) with double-negation topology: \(\mathbf{Sh}(\mathbb{P},\neg\neg)\) is a Boolean topos whose internal \(\mathbf{Set}\) behavior yields independence of CH — forcing extensions are geometric presentations of Boolean-valued models / sheaf toposes.
Étale space vs sheaf. For topological \(X\), sheaves \(\leftrightarrow\) local homeomorphisms \(E\to X\) (étalé spaces). Stalks \(F_x\) are fibers; geometric morphism \(\mathbf{Set}\to\mathbf{Sh}(X)\) from a point \(x\in X\) has \(f^*=\) stalk at \(x\).
Check¶
Prove that in a topos, every mono is a kernel pair equalizer of its cokernel pair? (idea)
Toposes are regular/exact in the appropriate sense: monos are equalizers (of \(\chi_m\) and \(\mathsf{true}\circ !\)), and equivalence relations are effective — part of the elementary package / Giraud for Grothendieck case. Concretely: \(m\) is the pullback of \(\mathsf{true}\) along \(\chi_m\), hence an equalizer of \(\chi_m\) and \(\top\circ !\).
Why does \(f^*\) in a geometric morphism preserve finite limits but not necessarily \(\Omega\)?
Geometric morphisms correspond to “continuous maps” / inverse image of sheaves: open sets pull back, finite limits of sheaves preserved. Power objects / \(\Omega\) encode logic of the topos and need not map to \(\Omega\); logical functors preserve \(\Omega\). Example: non-open continuous maps still induce geometric morphisms of sheaf toposes without preserving the full higher-order logic.
Pitfalls¶
- Presheaf topos vs sheaf topos: all presheaf categories are toposes; imposing \(J\) changes \(\Omega\) and colimits (sheafification identifies locally equal sections).
- \(\exists\) is local: existence statements glue only along covers — classic sheaf-theoretic subtlety.
- Generators: without a small generating family, elementary toposes need not be Grothendieck (no sheaf representation on a small site).
- Internal \(\mathbf{N}\): without NNO, arithmetic is unavailable; analysis inside toposes needs more structure (e.g. natural number object + Cauchy reals caveats).
Refs¶
Mac Lane–Moerdijk; Johnstone, Elephant; Johnstone, Topos Theory (1977); Borceux III; Caramello, Theories, Sites, Toposes (classifying toposes).
Next¶
Index · Category theory · Logic · Algebraic geometry · Homotopy type theory · Condensed mathematics — condensed sets are sheaves on profinite sets, a Grothendieck topos · Classifying & Geometric · Boolean & Forcing · Cohesive & Higher · Synthetic & Infinitesimal