arXiv:2004.07353v2 [math.CT] 27 Oct 2020 * upre yNFadAFOSR. and NSF by Supported ehd fmcielann n aaaayi,tencescntuto losest un- to of practices seems daily also and the analysis. behind tasks construction data lurking the nucleus and content by computation the mathematical network driven treatment general analysis, around, dual and data a pure way A and remarkably behind other cover learning applications The monad. machine concrete theory. a of remarkably 2-category of methods uncovers pure nucleus This of the fragment comonads. extracts monad notable for Street and resulting same the idempotent, the that achieves strictly show we is construction, monads nucleus on the Lifting 2-categories. in programming. ads for suggest particularly They toolkits, comonadicity. comonad idem- and and of monadicity monad in role the colimits of central are and extensions the coalgebras limits interesting elucidate the absolute presentations the where new It of side The and coalgebras. the of potents, versa. and on nucleus vice algebras algebras and The of revealing presented, categories both, normally coalgebras. induced for of back the presentations and resolved between new are algebras adjunction provides They of an is categories other. adjunction the one induced an on on the compositions decompositions over and and adjunctions modalities structures dynamics into or algebraic coalgebraic topology, of of from view and saturated operators side, a interior providing the while and logic, closure from the generalize decomposition comonads bases. value and canonical singular their the the displays as displays operators just linear nucleus of adjunction, resulting pair original The adjoint the an sense. in of strict nuclear implicit a a was into in that resolved idempotent be concept is can resolution adjunction its every The when repre- that nuclear to show adjunction. connections, adjunction We category an Galois other. each call each of We from determine denominator projects quantifiers. categories common generalized concept, and the a spectra, are or theories, Adjunctions structure, sentation a back. how and displays other, It the categories. of pair a nvriyo aai oouuHI Honolulu Hawaii, of University nhssmnlerywr,Rs tetdsrbda ducinbtenmnd n comon- and monads between adjunction an described Street Ross work, early seminal his In Monads comonad. a and monad a induce functors of pair adjoint an of composites two The relating and transformations, natural of pair a by related functors of pair a is adjunction An uk Pavlovic Dusko n h tetmndo monads on monad Street the and [email protected] h ulu fa adjunction an of nucleus The * Abstract 1 pl n. uetn CA Cupertino Inc., Apple [email protected] oii .D Hughes D. J. Dominic Contents
1 Introduction 4 1.1Nuclearadjunctionsandtheadjunctionnuclei...... 4 1.1.1 Definition...... 4 1.1.2 Result...... 4 1.1.3 Upshot...... 5 1.1.4 Background ...... 6 1.1.5 Terminology...... 7 1.1.6 Schema...... 7 1.2TheStreetmonad...... 8 1.3Asimplifyingassumption...... 9 1.4Overviewofthepaper...... 9
2 Example 1: Concept lattices and poset bicompletions 10 2.1Fromcontextmatricestoconceptlattices,intuitively...... 10 2.2Formalizingconceptanalysis...... 10 2.3Summary...... 14
3 Example 2: Nuclei in linear algebra 15 3.1Matricesandlinearoperators...... 15 3.2Nucleusasanautomorphismoftherankspaceofalinearoperator...... 16 3.2.1 Hilbertspaceadjoints:Notationandconstruction...... 16 3.2.2 Factorizations...... 18 3.3 Nucleus as matrix diagonalization ...... 18 3.4Summary...... 20
4 Example 3: Nuclear Chu spaces 21 4.1Abstractmatrices...... 21 4.1.1 Posets...... 21 4.1.2 Linearspaces...... 22 4.1.3 Categories...... 22 4.2Representabilityandcompletions...... 22 4.3Abstractadjunctions...... 22 4.3.1 TheChu-construction...... 23 4.3.2 Representingmatricesasadjunctions...... 23 4.3.3 Separatedandextensionaladjunctions...... 24 4.4 What does the separated-extensional nucleus capture in examples 4.1? ...... 25 4.4.1 Posets...... 25 4.4.2 Linearspaces...... 25 4.4.3 Categories...... 25 4.5Discussion:Combiningfactorization-basedapproaches...... 25 4.5.1 Hownucleidependonfactorizations?...... 26
2 4.5.2 Exercise...... 27 4.5.3 Workout...... 27 4.6Towardsthecategoricalnucleus...... 30
5 Example ∞: Nuclear adjunctions, monads, comonads 32 5.1Thecategories...... 32 5.2 Assumption: Idempotentscanbesplit...... 33 5.3Tools...... 34 5.3.1 Extendingmatricestoadjunctions...... 34 5.3.2 Comprehendingpresheavesasdiscretefibrations...... 35 5.4Thefunctors...... 36 5.4.1 The functor MA : Mat →− Adj ...... 36 5.4.2 Fromadjunctionstomonadsandcomonads,andback...... 37
6Theorem 38
7Propositions 39
8 Simple nucleus 51
9 Little nucleus 58
10 Example 0: The Kan adjunction 63 10.1Simplicesandthesimplexcategory...... 64 10.2Kanadjunctionsandextensions...... 64 10.3 Troubles with localizations ...... 66
11 What? 69 11.1Whatwedid...... 69 11.2Whatwedidnotdo...... 69 11.3 What needs to be done ...... 70 11.4Whatarecategoriesandwhataretheirmodelstructures?...... 70
A Appendix: Factorizations 79
B Appendix: Adjunctions, monads, comonads 81 B.1 Matrices (a.k.a. distributors, profunctors, bimodules) ...... 81 B.2Adjunctions...... 82 B.3Monads...... 82 B.4Comonads...... 82 B.5 The initial (Kleisli) resolutions KM : Mnd →− Adj and KC : Cmn →− Adj ...... 82 B.6 The final (Eilenberg-Moore) resolutions EM : Mnd →− Adj and EC : Cmn →− Adj .. 86
C Appendix: Split equalizers 86
3 1 Introduction
We begin with an informal overview of the main result here in Sec. 1, and motivate it through the general examples in Sections 2–5. A categorically minded reader may prefer to start from the categorical definitions in Sec. 5, proceed to the general constructions in Sections 6–9, and come back for examples and explanations. Some general definitions can be found in the Appendix.
1.1 Nuclear adjunctions and the adjunction nuclei 1.1.1 Definition. ∗ We say that an adjunction F = (F F∗ : B →− A)isnuclear when the right adjoint F∗ is monadic and the left adjoint F∗ is comonadic. This means that the categories A and B determine one another, and can be reconstructed from each other: ←− ←− F ∗ • F∗ is monadic when B is equivalent to the category A of algebras for the monad F = F∗F : A →− A, whereas →− • F∗ is comonadic when A is equivalent to the category BF of coalgebras for the comonad →− ∗ F = F F∗ : B →− B. The situation is reminiscent of Maurits Escher’s “Drawing hands” in Fig.1.
←− F →− AB F
∗ F F F∗ F
←− BA F →− F
→− ←− ∗ F F Figure 1: An adjunction (F F∗) is nuclear when A B and B A .
1.1.2 Result ←− ←− The nucleus construction N extracts from any adjunction F its nucleus NF ∗ F = (F F∗ : B →− A) ←− ←− →− (1) F F NF = F F : A →− B
4 ←− F The functor F is formed by composing the forgetful functor A →− A with the comparison functor →− →− A →− BF ,whereasF is the composite of the forgetful functor BF →− B with the comparison ←− F B →− A . Hence the left-hand square in Fig. 2. We show that the functors F and F are adjoint,
←− ⇐= F F =⇒ →− ←− F ABF AF
∗ F F F F∗ F F