Lawvere–Tierney topology

If E is a topos, then a topology on E is a morphism j from the subobject classifier Ω to Ω such that j preserves truth (${\displaystyle j\circ {\mbox{true}}={\mbox{true}}}$), preserves intersections (${\displaystyle j\circ \wedge =\wedge \circ (j\times j)}$), and is idempotent (${\displaystyle j\circ j=j}$).

Commutative diagrams showing how j-closure operates. Ω and t are the subobject classifier. χ_s is the characteristic morphism of s as a subobject of A and ${\displaystyle j\circ \chi _{s}}$ is the characteristic morphism of ${\displaystyle {\bar {s}}}$ which is the j-closure of s. The bottom two squares are pullback squares and they are contained in the top diagram as well: the first one as a trapezoid and the second one as a two-square rectangle.

Given a subobject ${\displaystyle s:S\rightarrowtail A}$ of an object A with classifier ${\displaystyle \chi _{s}:A\rightarrow \Omega }$, then the composition ${\displaystyle j\circ \chi _{s}}$ defines another subobject ${\displaystyle {\bar {s}}:{\bar {S}}\rightarrowtail A}$ of A such that s is a subobject of ${\displaystyle {\bar {s}}}$, and ${\displaystyle {\bar {s}}}$ is said to be the j-closure of s.

Some theorems related to j-closure are (for some subobjects s and w of A):

• inflationary property: ${\displaystyle s\subseteq {\bar {s}}}$
• idempotence: ${\displaystyle {\bar {s}}\equiv {\bar {\bar {s}}}}$
• preservation of intersections: ${\displaystyle {\overline {s\cap w}}\equiv {\bar {s}}\cap {\bar {w}}}$
• preservation of order: ${\displaystyle s\subseteq w\Longrightarrow {\bar {s}}\subseteq {\bar {w}}}$
• stability under pullback: ${\displaystyle {\overline {f^{-1}(s)}}\equiv f^{-1}({\bar {s}})}$.

Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets over C.

• Lawvere, F. W. (1971), "Quantifiers and sheaves" (PDF), Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 1, Paris: Gauthier-Villars, pp. 329–334, MR 0430021, S2CID 2337874, archived from the original (PDF) on 2018-03-17
• Mac Lane, Saunders; Moerdijk, Ieke (2012) [1994], Sheaves in geometry and logic. A first introduction to topos theory, Universitext, Springer, ISBN 978-1-4612-0927-0
• McLarty, Colin (1995) [1992], Elementary Categories, Elementary Toposes, Oxford Logic Guides, vol. 21, Oxford University Press, p. 196, ISBN 978-0-19-158949-2