nLab level of a topos

Idea

For $\mathbf{H}$ a topos, then an essential subtopos $\mathbf{H}_l \hookrightarrow \mathbf{H}$ is called a level of $\mathbf{H}$. 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 $\mathbf{H}_{1} \hookrightarrow \mathbf{H}_2$ 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.

Examples

• Every open subtopos is a level.

• For $\mathbf{H} =$ sSet the topos of simplicial sets, then the tower of n-coskeleton-inclusions for all $n$ 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 $n$th order infinitesimal objects, for each $n$.

• 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 $i_!\dashv i^\ast \dashv i_\ast :\mathcal{A}\to\mathcal{B}$ where $i_!\simeq i_\ast$ is called a quintessential localization (cf. Johnstone 1996 and the discussion at quality type).

Properties

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 $G\dashv F:\mathcal{A}\to\mathcal{B}$ and $K\dashv H:\mathcal{B}\to\mathcal{C}$ then $G\circ K \dashv H\circ F:\mathcal{A}\to\mathcal{C}$. Applying this twice to a pair of essential localizations yields:

Proposition

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

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

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

Here , of course, $i^\ast \cdot q^\ast \dashv q_\ast\cdot i_\ast$ is just the adjunction involved in the composition $q\cdot i$ of the two geometric morphisms $q$ and $i$ (provided the categories involved are toposes), the important thing is that the essentialities $q_!$ and $i_!$ 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 $(q_!\cdot i_!)\cdot (i^\ast \cdot q^\ast) \dashv (q_\ast\cdot i_\ast)\cdot (i^\ast \cdot q^\ast) :\mathcal{C}\to\mathcal{C}$.

Related concepts

• rigid topology

• quality type

• subtopos

• open subtopos

• essential sublocale

• essential subtopos

• essential geometric morphism

• Aufhebung

• Hegelian taco

References

• 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 $\epsilon$ , arXiv:1909.12757 (2019). (abstract)

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