nLab locally closed subtopos

Idea

A locally closed subtopos generalizes the concept of a locally closed subspace from topology to toposes.

Definition

A subtopos $\mathcal{E}_j\hookrightarrow\mathcal{E}$ is called locally closed if it is the meet $\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c$ of an open subtopos $\mathcal{E}_o$ and a closed subtopos $\mathcal{E}_c$ in the lattice of subtoposes of $\mathcal{E}$.

Properties

• Since $\mathcal{E}$ is closed as well as open in itself, its closed and open subtoposes are trivially locally closed.

• Since open and closed subtoposes are complemented in the lattice of subtoposes, the join of the complements $\bar\mathcal{E}_{o}\cup\bar\mathcal{E}_{c}$ provides a complement for a locally closed subtopos $\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c$.

• $\mathcal{E}_j$ is locally closed iff $\mathcal{E}_j\hookrightarrow\mathcal{E}$ can be factored into an open followed by a closed inclusion.

• A locally closed subtopos of an exponentiable topos is itself exponentiable (cf. Johnstone (2002, p.749)).

Examples

• A way to realize a topos $\mathcal{E}_j$ as a locally closed subtoposes is by repeated Artin gluing to appropriate left exact functors $f_1:\mathcal{E}_j\to \mathcal {F}_1$ and $f_2:\mathcal{F}_2\to Gl(f_1)$. Then $\mathcal{E}_j$ is open in $Gl(f_1)$ which itself is closed in $Gl(f_2)$ whence by the above remark on factorisations $\mathcal{E}_j\hookrightarrow Gl(f_2)$ is locally closed.

• Locally closed subspaces $Y$ of topological spaces $X$ yield locally closed subtoposes of the corresponding sheaf topos $Sh(Y)\hookrightarrow Sh(X)$ e.g. let $X$ be the space on $\{a,b,c\}$ with non-trivial open subsets $\{a\}$, $\{a,b\}$. Then $\{b\}$ as the intersection of $\{a,b\}$ with the closed subset $\{b,c\}$ is a locally closed subset to which a neither closed nor open but locally closed copy of $Set$ corresponds in the lattice of subtoposes of $Sh(X)$.

Remark

Bunge and Niefield (2000) consider general notions of locally closed subcategory and locally closed geometric morphism.

Related entries

• open subtopos
• closed subtopos
• dense subtopos
• locally closed set
• co-Heyting boundary

References

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (Exposé IV, ex.9.4.9., pp.462-463)

• M. Bunge, S. Niefield, Exponentiability and single universes , JPAA 148 (2000) pp.217-250.

• Peter Johnstone, Conditions Related to de Morgan’s Law , pp.479-491 in LNM 753 Springer Heidelberg 1979.

• Peter Johnstone, Sketches of an Elephant vol. 2 , Cambridge UP 2002.

• A. Kock, T. Plewe, Glueing analysis for complemented subtoposes , TAC 2 (1996) pp.100-112. (pdf)