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For H\mathbf{H} a topos, then an essential subtopos H lH\mathbf{H}_l \hookrightarrow \mathbf{H} is called a level of H\mathbf{H}. This is equivalently the inclusion of the right adjoint of an adjoint cylinder/adjoint modality.

The terminology is due to the fact (Kelly-Lawvere 89) that the essential subtoposes of a topos, or more generally the essential localizations of a suitably complete category, form a complete lattice.

If for two levels H 1H 2\mathbf{H}_{1} \hookrightarrow \mathbf{H}_2 the second one includes the modal types of the idempotent comonad of the first one, and if it is minimal with this property, then Lawvere speaks of “Aufhebung” (see there for details) of the unity of opposites exhibited by the first one.



The lattice of essential localizations of a category is contained in the lattice of all localizations. While the suprema of localizations coincide in both lattices, the infinimum in the lattice of all localizations of even a pair of essential localizations need not be an essential localization (Kelly-Lawvere 89).

Recall (e.g. from Borceux 1994, p.106) that adjunctions compose: if GF:𝒜G\dashv F:\mathcal{A}\to\mathcal{B} and KH:𝒞K\dashv H:\mathcal{B}\to\mathcal{C} then GKHF:𝒜𝒞G\circ K \dashv H\circ F:\mathcal{A}\to\mathcal{C}. Applying this twice to a pair of essential localizations yields:


Given two essential localizations i !i *i *:𝒜i_!\dashv i^\ast \dashv i_\ast :\mathcal{A}\to\mathcal{B} and q !q *q *:𝒞q_!\dashv q^\ast \dashv q_\ast :\mathcal{B}\to\mathcal{C} , hence in the situation:

𝒜𝒞 \mathcal{A}\overset{\hookrightarrow}{\underset{\hookrightarrow}{\to}} \mathcal{B}\overset{\hookrightarrow}{\underset{\hookrightarrow}{\to}}\mathcal{C}

there results a composed essential localization q !i !i *q *q *i *:𝒜𝒞q_!\cdot i_!\dashv i^\ast \cdot q^\ast \dashv q_\ast\cdot i_\ast :\mathcal{A}\to\mathcal{C}. \qed

Here , of course, i *q *q *i *i^\ast \cdot q^\ast \dashv q_\ast\cdot i_\ast is just the adjunction involved in the composition qiq\cdot i of the two geometric morphisms qq and ii (provided the categories involved are toposes), the important thing is that the essentialities q !q_! and i !i_! compose as well.

Since this just states that inclusions as well as essential geometric morphism are closed under composition and therefore so are essential inclusions, our primary interest here is to extract an explicit formula for the modalities corresponding to the composition:

The corresponding adjoint modalities are (q !i !)(i *q *)(q *i *)(i *q *):𝒞𝒞(q_!\cdot i_!)\cdot (i^\ast \cdot q^\ast) \dashv (q_\ast\cdot i_\ast)\cdot (i^\ast \cdot q^\ast) :\mathcal{C}\to\mathcal{C}.


  • M.Artin, A.Grothendieck, J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4 , Springer LNM 269 Heidelberg 1972. (sec. IV 7.6., pp.414-416)

  • F. Borceux, Handbook of Categorical Algebra vol. 1 , Cambridge UP 1994.

  • P. Johnstone, Remarks on Quintessential and Persistent Localizations , TAC 2 no.8 (1996) pp.90-99. (pdf)

  • G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations , Bull.Soc.Math. de Belgique XLI (1989) 289-319 [pdf]

  • C. Kennett, E. Riehl, M. Roy, M. Zaks, Levels in the toposes of simplicial sets and cubical sets , JPAA 215 no.5 (2011) pp.949-961. (arXiv:1003.5944)

  • R. Lucyshyn-Wright, Totally Distributive Toposes , arXiv.1108.4032 (2011). (pdf)

  • F. Marmolejo, M. Menni, Level ϵ\epsilon , arXiv:1909.12757 (2019). (abstract)

  • E. M. Vitale, Essential Localizations and Infinitary Exact Completions , TAC 8 no.17 (2001) pp.465-480. (pdf)

Last revised on March 27, 2023 at 07:19:14. See the history of this page for a list of all contributions to it.