Modalities, Closure and Reflection
The concept of adjunction as such expresses a duality. The stronger concept of an adjoint cylinder or adjoint modality is specifically an adjunction between idempotent (co-)monads and is meant to express specifically a duality between opposites.
In terms of the corresponding adjoint triple of (co-)reflections and localizations the concept was suggested in (Lawvere 91, p. 7, Lawvere 94, p. 11) to capture the phenomena of “Unity and Identity of Opposites” as they appear informally in Georg Hegel’s Science of Logic. (One might therefore say the notion is meant to capture the idea of “dialectic”, though there is some debate as to whether Hegel’s somewhat mythical “creation out of paradox” should really go by this term, see this Wikipeda entry ).
In terms of adjoint pairs of modal operators in the context of modal logic/modal type theory and thought of as Galois connections the concept appears in (Reyes-Zolfaghari 91). Further developments along these lines include (DJK 14).
In (Lawvere 94) an adjoint cylinder is defined to be an adjoint triple such that the outer two adjoints are full and faithful functors. This means equivalently that the induced adjoint pair on the codomain of these inclusions consists of an idempotent monad and comonad (adjoint monads). One may also consider the situation where the middle functor of the adjoint triple is fully faithful, hence one has adjoint modal operators either of the form
or of the form
A category equipped with an adjoint modality of the second form is called a category of being in (Lawvere 91). If the category is a topos then this is also called a level of a topos.
Given any such, we may say that the “unity” expressed by the two opposites is exhibited by the canonical natural transformation
which is the composite of the counit of the comodality and the unit of the modality.
If for two levels the next one contains the modal types of the idempotent comonad of the former, then Lawvere speaks of “Aufhebung” (see there for more).
One can consider longer sequences of such adjoints of co/modalities, but the longer they get, the less likely they are to be non-trivial. The longest that still has good nontrivial models seems to be adjoint triples of modalities. Of these there is then similarly either the form
(the “Yin triple”) as for instance in the definition of cohesion and
(the “Yang triple”) as for instance in the definition of differential cohesion.
Since adjoint triples are equivalently adjunctions of adjunctions (Licata-Shulman, section 5), it is suggestive to denote these as
Simple illustrative examples
The following simple illustrative example of an adjunction of the form has been suggested in (Lawvere 00).
Consider the two inclusions of the even and the odd integers, i.e. the maps and , respectively.
Both are adjoint to the operation of forming the of the result of dividing by two, this is right adjoint to the inclusion of even numbers, and left adjoint to the inclusion of odd numbers.
The adjoint modalities, , are then the composites and
In the same vein there is an example for an adjunction of the form :
Consider the inclusion of the integers into the real numbers, both regarded linear orders. This inclusion has a left adjoint given by and a right adjoint civen by . The composite is an idempotent monad and the composite is an idempotent comonad on . Both express a moment of integrality in an real number, but in opposite ways, each real number sits in between its floor and celling
Indeed the moments form an adjunction
Werden : Sein Nichts
For a topos/(∞,1)-topos consider the “initial topos”, the terminal category (category of sheaves on the empty site).
There is then an adjoint triple
given by including the initial object and the terminal object into .
In the type theory of this corresponds to the adjoint pair of modalities
which are constant on the initial object/terminal object, respectively.
The induced unity transformation is
hence the unique factorization of the unique function through any other type.
Looking through (Hegel 1812, vol 1, book 1, section 1, chapter 1) one might call “nothing”, call “being” and then call this unity of opposites “becoming”. In particular in §174 of Science of Logic it says
there is nothing which is not an intermediate state between being and nothing
which seems to be well-captured by the above unity transformation.
Quantity : discreteness continuity
The adjoint modality in a local topos is that given by flat modality sharp modality
Capturing discrete objects/codiscrete objects.
The corresponding unity transformation is
According to (Lawvere 94, p. 6) this unity captures the duality that in a set all elements are distinct and yet indistinguishable, an apparent paradox that may be traced back to Georg Cantor.
Looking through Hegel’s Science of Logic at On discreteness and repulsion one can see that matches with what Hegel calls
(par 398) Quantity is the unity of these moments of continuity and discreteness
Continuuum : repulsion cohesion
For a cohesive topos/cohesive (∞,1)-topos the shape modality flat modality constitute an adjoint cylinder
The corresponding unity-transformation is the points-to-pieces transform
Looking through (Hegel 1812, vol 1, book 1, section 2, chapter 1) one might call “repulsion”, call “attraction”/“cohesion” and then call this unity of opposites “continuum”. Indeed, by the discussion at cohesive topos, this does quite well capture the geometric notion of continuum geometry.
Infinitesimal Continuuum : infin. repulsion infinit. cohesion
For equipped moreover with differential cohesion, there is the infinitesimal version of shape modality flat modality namely the adjoint modality
infinitesimal shape modality infinitesimal flat modality
The corresponding unity-transformation is the
maps from the coefficients for crystalline cohomology to the de Rham space of types , where all infinitesimal neighbour points are identified.
In view of the above the unity exhibited here is clearly to be called the “infinitesimal continuum”.
The combination of the above two examples of Continuum and Quantity is an adjoint triple of modalities
shape modality flat modality sharp modality
characteristic of a cohesive topos.
Skeleta and Co-Skeleta
simplicial skeleton simplicial coskeleton
Formal completion Torsion approximation
For a commutative ring or more generally an E-∞ ring and a suitable ideal, then -adic completion and -torsion approximation form an adjoint modality on the stable (∞,1)-category of ∞-modules over .
(-adic completion) (-torsion approximation)
Let be an E-∞ ring and let be a finitely generated ideal in its underlying commutative ring.
Then there is an adjoint triple of adjoint (∞,1)-functors
This is effectively the content of (Lurie “Proper morphisms”, section 4):
the existence of is corollary 4.1.16 and remark 4.1.17
the existence of is lemma 4.2.2 there;
the equivalence of sub--categories is proposition 4.2.5 there.
See at fracture theorem for more.
Fermions and supergeometry
On super smooth infinity-groupoids there is an adjoint modality deriving from the adjoint triple relating plain algebra and superalgebra. The right adjoint deserves to be called the bosonic modality (“body”), hence its left adjoint the fermionic modality. This expresses the presence of supergeometry/fermions, hence ultimately the Pauli exclusion principle. Following PN§290 this unity of opposties might hence be called “asunderness”.
Totally distributive categories
For a totally distributive category it induces on its category of presheaves an adjoint modality whose right adjoint is the Yoneda embedding postcomposed with its left adjoint .
See at recollement.
The concept of dialectical reasoning is usually attributed to
Hegel in his History of Philosophy writes that dialectic begins with Zeno (one of the characters in that dialogue).
This is much amplified and expanded in
The origins of its proposed formalization in category theory are recalled in
- Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982)
In terms of adjoint triples of (co-)reflections and localizations
Conceived of in terms of adjoint triples of (co-)reflections and localizations the concept appears in
William Lawvere, Some Thoughts on the Future of Category Theory in A. Carboni, M. Pedicchio, G. Rosolini, Category Theory , Proceedings of the International Conference held in Como, Lecture Notes in Mathematics 1488, Springer (1991)
William Lawvere, Cohesive Toposes and Cantor's "lauter Einsen", Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. (pdf)
William Lawvere, Tools for the advancement of objective logic: closed categories and toposes, in J. Macnamara and Gonzalo Reyes (Eds.), The Logical Foundations of Cognition, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference “Logic and Cognition”), pages 43-56, 1994.
William Lawvere, Unity and Identity of Opposites in Calculus and Physics, Proceedings of ECCT 1994 Tours Conference, Applied Categorical Structures, 4: 167-174 Kluwer Academic Publishers, (1996).
F. W. Lawvere, Adjoint Cylinders, message to catlist November 2000. (link)
In terms of adjoint pairs of modal operators
In terms of adjoint pairs of modal operators and hence of Galois connections, the concept appears in
Gonzalo Reyes, H. Zolfaghari, Topos-theoretic approaches to modality, Lecture Notes in Mathematics 1488 (1991), 359-378.
Gonzalo Reyes, A topos-theoretic approach to reference and modality, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (Euclid)
with further developments in
M. Sadrzadeh, R. Dyckho, Positive logic with adjoint modalities: Proof theory, semantics and reasoning about information, Electronic Notes in Theoretical Computer Science 249, 451-470, 2009, in Proceedings of the 25th Conference on Mathematical Foundations of Programming Semantics (MFPS 2009).
Claudio Hermida, section 3.3. of A categorical outlook on relational modalities and simulations, 2010 (pdf)
Wojciech Dzik, Jouni Järvinen, Michiro Kondo, Characterising intermediate tense logics in terms of Galois connections (arXiv:1401.7646)
Formalization specifically in modal type theory is in
For an overview of the role of adjunctions in modal logic see:
- M. Menni, C. Smith, Modes of Adjointness , J. Philos. Logic 43 no.3-4 (2014) pp.365-391.