locally closed subtopos


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


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


  • 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 ¯ o¯ c\bar\mathcal{E}_{o}\cup\bar\mathcal{E}_{c} provides a complement for a locally closed subtopos j= o c\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c.

  • j\mathcal{E}_j is locally closed iff j\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)).


  • A way to realize a topos j\mathcal{E}_j as a locally closed subtoposes is by repeated Artin gluing to appropriate left exact functors f 1: j 1f_1:\mathcal{E}_j\to \mathcal {F}_1 and f 2: 2Gl(f 1)f_2:\mathcal{F}_2\to Gl(f_1). Then j\mathcal{E}_j is open in Gl(f 1)Gl(f_1) which itself is closed in Gl(f 2)Gl(f_2) whence by the above remark on factorisations jGl(f 2)\mathcal{E}_j\hookrightarrow Gl(f_2) is locally closed.

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


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


  • 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)

Last revised on March 9, 2018 at 08:00:27. See the history of this page for a list of all contributions to it.