For a topos, then an essential subtopos is called a level of . 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 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.
Every open subtopos is a level.
For sSet the topos of simplicial sets, then the tower of n-coskeleton-inclusions for all is a sequence of levels (the extra left adjoint exhibiting the essentialness being the simplicial skeleton). The Aufhebung in this example is discussed in detail in (KRRZ 11).
The lowest level is the inclusion of the trivial subtopos as the terminal object. See also at unity of opposites the section Werden.
Often the level above this will be cohesion and the level above that, if it exists, differential cohesion.
Here differential cohesion itself typically comes in a countable tower of levels, given by the th order infinitesimal objects, for each .
The Hegelian taco abstracts a certain stacking of three levels viewed as a diagram in a 2-category into a graphic monoid.
An essential localization where is called a quintessential localization (cf. Johnstone 1996 and the discussion at quality type).
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 and then . Applying this twice to a pair of essential localizations yields:
Given two essential localizations and , hence in the situation:
there results a composed essential localization .
Here , of course, is just the adjunction involved in the composition of the two geometric morphisms and (provided the categories involved are toposes), the important thing is that the essentialities and 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 .
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 , 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 November 1, 2024 at 12:21:03. See the history of this page for a list of all contributions to it.