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伴隨函子

读物

• Tom Leinster. Basic Category Theory.
• Lukas Heger. A Brief Introduction to Categories, Part 8: Adjunctions I including Galois connections, Part 9: Adjunctions II With many examples.
• David I. Spivak. Category Theory for Scientists (Old Version)
• MO: What is an intuitive view of adjoints? (category theory)
• Tai-Danae Bradley. What is an Adjunction?, Definition, Examples
• David Ellerman. A Theory of Adjoint Functors--with some Thoughts about their Philosophical Significance. Includes philosophical ideas that seem related to the twosome.
• David Ellerman. Adjoints and emergence: applications of a new theory of adjoint functors.
• Daniel M. Kan. Adjoint Functors
• Bartosz Milewski. Adjunctions.
• Adjunctions are fun. by Kyle Ferendo.
• Keegan Smith. Adjoint Functors in Algebra, Topology and Mathematical Logic.
• Fausk, Hu, May. Isomorphisms Between Left and Right Adjoints
• Tai-Danae Bradley, Tyler Bryson, and John Terilla. Topology: A Categorical Approach dedicates a lot of attention to adjoint functors.
  • Chapter 5: Adjunctions and the Compact-Open Topology
  • Chapter 6: Paths, Loops, Cylinders, Suspensions,...
• John Baez. Lectures on advanced category theory
• David Ellerman. A Theory of Adjoint Functors philosophy of communication
• 科学期刊文章: David Ellerman. Adjoints and Emergence: Applications of a new theory of adjoint functors.
• 科学期刊文章: David Ellerman. Mac Lane, Bourbaki, and Adjoints: A Heteromorphic Retrospective
• Louis Kauffman. Mathematical Work of Francisco Varela.
• Goguen, Varela. Systems and Distinctions. Duality and Complementarity.
• Adjunction and inversion of adjunction. Adjunction formula. We call adjunction the process of inferring statements about a subvariety from some knowledge of the ambient variety, while the inverse and usually more complicated process is called inversion of adjunction.
• https://www.researchgate.net/publication/51978740_Relating_Operator_Spaces_via_Adjunctions
• https://hal.archives-ouvertes.fr/hal-02184208/document Crash Course on Category Theory focusing on adjunctions.
• Chris Henderson. Generalized Abstract Nonsense: Category Theory and Adjunctions Equivalence of two definitions.
• Michael Levin. Agency at the Very Bottom.

维基百科

• Wikipedia: Adjoint functors
• Wikipedia: Funtores adjuntos
• Wikipedia: Adjunktion
• Wikipedia: Foncteur adjoint
• Сопряжённые функторы
• 伴隨函子
• Galois connection
• Monad
• Kuratowski closure axioms
• 维基百科: Adjoint Notions of adjoint.
• Adjunction formula Relates the canonical bundle of a variety and a hypersurface inside that variety.

影片

• 影片: Richard Southwell: Adjoint Functors
• Catsters: Adjunctions from morphisms Motivation for the construction of adjoint functors for bundles over sets.
• Emily Riehl. The Formal Theory of Adjunctions Monads Algebras and Descent
• Chris Rogers. Towards an adjunction between the homotopy theories of dg manifolds and Lie ∞-groupoids
• MathProofsable. Adjunction.

Software

• https://varkor.github.io/blog/2020/11/25/announcing-quiver.html Quiver - for drawing commutative diagrams

事实 (Shìshí)

• Uniqueness Two right adjoints of a functor are naturally isomorphic.

直觉 (Zhíjué)

Intuition about adjunction

Meaning of adjunction: Analogy

• Given two worlds (such as {$C=\mathbf{Set}$} and {$D=\mathbf{VectorSpaces}$}) and a functor {$F:C\rightarrow D$} (metaphor) between the two worlds.
• Then the left adjoint is the functor L which defines an analogy between arrows {$L(d)\rightarrow c$} in the world {$C$} and arrows {$d\rightarrow F(c)$} in the world {$D$}.
• And the right adjoint is the functor R which defines an analogy between arrows {$F(c)\rightarrow d$} in the world {$D$} and arrows {$c\rightarrow R(d)$} in the world {$C$}.
• In either case the definition depends on it holding for all {$c$} in {$C$} and all {$d$} in {$D$}. And if it holds, then it is unique up to isomorphism.
• From the analogy it is clear that the left adjoint acts internally as it prepares (preprocesses) the input, as with colimits, whereas the right adjoint acts externally as it processes (postprocesses) the output, as with limits. Indeed, this distinction between colimits and limits is manifest in the adjunctions that arise from the diagonal functors (copying an object into diagrams).
• For example, given the diagonal functor {$\Delta:c\rightarrow (c,c)$}, the left adjoint has to handle the information of {$X$} and {$Y$} by preprocessing, thus internally establishing the coproduct, whereas the right adjoint handles the information by postprocessing, thus by externally establishing the product.
• A functor F with both a left adjoint and a right adjoint plays both roles - preprocessing (with regard to its right adjoint) and postprocessing (with regard to its left adjoint)
• Michael Levin on adjunctions and active inference: Optimal reconstruction (relates to internal modeling of external environment). Laziest way to do something (relates to least action). Prediction is structure preserving transformations. Math is agentic because it makes prediction.

Meaning of adjunction: Metaphor

• Adjointness = metaphors. Metaphors is layered. There are layers of metaphors.
• Meanings are variously related by adjunctions. They enrich the meaning and extend the context.
• Think of my understanding of my three grandfathers as changing with context.

Information in context

• If you wrote {$U=\textrm{Res}^G_H W$} then you would not go back. Except that the knowledge is not in this representation but in the map that takes you back up. So it is outside knowlege, this implicit knowledge.
• Extending context. Adjunction is like extension of the domain, but in terms of structure: extension of structure. Polymorphism.
• Adjunctions present the same information in different context. Information is given in the form of morphisms, not isolated objects. Morphisms are structure preserving maps, or more generally, validations that there is a way to go from one object to another.
• An adjunction relates contexts (categories) that are connected by patches (images of functors). Which is to say, an adjunction synchronizes two categories along a patch that may be all or part of the category on either side.
• Adjoint functors harmonize the semantics of information (which stays the same) and the syntax of context (which changes).
• A morphism synchronizes the clockwork of one representation with the clockwork of another representation. This synchronization is information. An adjunction matches this synchronization in one world (for one group) with the synchronization in another world (for another group). The matching is a natural isomorphism.

From internal structure to external relationships

• Adjunction relates a question (external relationships, what we don't know) and an answer (internal structure, what we know). External relationships are a universal language of questions, of what we don't know.
• Those things are which show themselves to be. This means that internal structure must get expressed through external relationships, as with category theory. Colimits are internal structures and limits are external relationships. Thus colimits need to get expressed as limits. A topos has arbitrary colimits and finite limits, just as a topology has arbitrary unions and finite intersections. Right adjoint functors preserve limits and left adjoint functors preserve colimits. Thus we should move from left adjoint functors (such as adding perspectives, as with free constructions) to right adjoint functors (removing perspectives, as with forgetting).
• Those things are which show themselves to be. Shulman: By analogy with function extensionality and propositional extensionality, univalence could be called typal extensionality. In particular, like function extensionality and propositional extensionality, univalence is an “extensionality” property, meaning that “types are determined by their behavior.”

Perspective

• A functor is a perspective. An adjoint functor is an inverted perspective.
• David Ellerman. On Adjoint and Brain Functors. A heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of di§erent categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain.
• Adjoint functors relate human and divine perspectives in trivial ways.
• Adjoint functors may relate to chains of perspectives of human's view and God's view. The identity triangles for units and counits are in terms of FGF. This reminds me of my thoughts on bisecting a view. FGF also shows the ambiguity of F, that it can be thought of as acting on the outside, F(GF->I), or on the inside (FG->I)_F

A perspective in a division of everything may be not a functor but an adjunction between functors.

• Wall = nullsome
• Wall - Wall = onesome
• Wall - A - Wall = twosome
• Wall - A - B - Wall = thresome (adjunction)
• Wall - A - B - C - Wall = foursome (adjoint string of length 3)

Forwards view and backwards view

• Adjunction is like a half-inverted tube. There is an inversion point which can be changed, which can be freely chosen as desired, but cannot be either end.
• Adjoint functors relate viewing forward and viewing backward as occurs with shifts of perspective in the foursome, fivesome, sixsome, sevensome.

Pushdown automata

• Adjunction's left and right parentheses are like particle and anti-particle, which can be created out of an identity morphism by way of the triangle identity.
• Adjunction's parenthesis in one world mirrors an identity in the other world.
• Adjunction's parentheses (unit and counit) and their respective worlds express reading and writing.
• Adjunction's parenthesis (unit or counit) is the boundary of a world (into or out of the image of the functor).

Approximate inverse functor from above or below

• An adjoint functor is an approximate inverse, either as a pre-adjustment ("on the left") or as a post-adjustment ("on the right").

Left or right

• In adjoint strings, each functor changes from being left adjoint or right adjoint. So it's not possible to directly compare the different homset isomorphisms because the homsets for left and right are not the same.

Ambiguity

• Adjunction includes shared ambiguity inside of the bijection. For example, in the tensor product - homset adjunction, the functors differ as to how they treat y, but they treat x basically the same and likewise they treat z basically the same. In the induction-restriction adjunction there is a great ambiguity as to how to interpret the action on H but they differ on G, where one is defined and the other is not.

Definition of adjunction

• An adjunction is defined by two different natural transformations that are separated by a functor (before and after) and a related functor (after and before).

Classifying trivial cognition

• Adjunction gives the kinds of dualities and trivialities.
• Adjunctions are trivial nonsymmetries.

Wholeness transformation

• An adjoint functor, notably a trivial one, is a wholeness transformation, as described by Christopher Alexander.

Emphasis of morphisms

• Adjunction is a relationship that lifts above the object and says what matters are the morphisms. But it does that with a pair of counterbalancing functors.

Homotopy equivalences

• How algebraic topology and category theory are related. Adjunctions are homotopy equivalences.

Recalibration

• Adjunction: G(F(x)) is the recalibration of x

Explosion of unfolding

• Mathematical explosion can be based on adjunction explosion as when going back and forth between geometry and algebra, each becoming more refined and powerful

Restructuring

• Adjunction can describe the details of restructuring of one structure by another structure and the abyss between them.

Weakening of identity

• Adjunction example: Lending a dollar bill and not getting back the same dollar bill, but an equivalent dollar bill.
• Equivalence - how identity appears on the inside.
• Isomorphism - how identity appears on the outside.
• C and D are equivalent (there are natural isomorphisms from {$FG$} to {$I_D$} and {$I_C$} to {$GF$}) if and only if {$F\dashv G$} and both F and G are full and faithful. More generally, an adjunction describes the ways in which F and G may not be full and faithful. A functor {$F:C\rightarrow D$} induces a function {$F_{X,Y}:\textrm{Hom}_C(X,Y)\rightarrow\textrm{Hom}_D(F(X),F(Y))$}. {$F$} is faithful if {$F_{X,Y}$} is injective and is full if {$F_{X,Y}$} is surjective. So there are four possibilities for F: It is both injective and surjective, only injective, only surjective, and neither. Likewise there are four possiblities for G. Thus there are {$4\times 4=16$} possibilities in all to explore.

Adjunctions and perspectives and divisions

Divisions

• If you can have two functors that are both left and right adjoints of each other, then they can form adjoint strings of arbitrary length, thus express divisions of everything of arbitrary length. An example is the case of induction - restriction - coinduction. This brings to mind the reflection effect for the twosome, by which the direction can change, if you reflect.

Twosome

• The adjunction of free construction (free will) and forgetful (fate) functors expresses the twosome.
• The twosome is modeled by the adjunction of the tensor product functor (opposites coexist - in parallel - commutative, internal duality A=A) and the homset functor (all is the same - in series - noncommutative, external duality A->B).
• Varela. 10.6 Excursus into Dialects. Examples related to the twosome and threesome.

Adjunction - shifts

• +1 self-adjoint operator
• +2 restriction-induction (implicit information - internal structure vs. explicit information - external relationships)
• +3 three-cycle derived functors

Operation +2

• Adjoint strings express the operation +2, the alternation of Human perspective of God's perspective, and also thus the gap that starts with the foursome, grows with the fivesome, sixsome, sevensome, where from that womb keep inserting new perspectives, flipping the direction, what it means to go beyond oneself.

Six functors formalism. Six representatives.

• {$C\rightarrow C$} 2: tensor & homset
• {$C\rightarrow \textrm{Set}$} 4: sheaves

Left and right derived functors

• Left and right derived functors. A main interest of homological algebra is to study functors between categories of modules (or more generally abelian categories) via the homology of their compositions with other functors. One of the crucial tools for doing so is the concept of derived functors. The categories in question must have projective or injective objects, and enough of them at that, for this idea to work. The functor to be studied is then applied to projective or injective resolutions, chain complexes consisting of projective/injective objects, after which homology is taken. [Wei94, § 2.1–§ 2.5], [Bla11, § 11.2–§ 11.3]
• 21. Derived functors/categories (e.g., Tor, Ext, triangulated categories) The topics no. 17, “Homological algebra”, and no. 21, “Derived functors/categories (e.g., Tor, Ext, triangulated categories)”, are for all intents and purposes identical. Having multiple talks on it seems very much possible. Up to four are outlined below. Chain complexes. Chain complexes, usually of modules over a ring, are the prominent entities treated in homological algebra. Chain complexes form a category with chain maps as morphisms. Actually, they form a 2-category in the sense of higher order category theory with chain homotopies as second-order morphisms. The functor given by the operation of taking the homology of given chain complex is what gives the subject of homological algebra its name. It permits measuring to what degree a given chain complex fails to be exact (a property inspired by that of the same name for short or long sequences). [Wei94, § 1.1–§ 1.5], [Bla11, § 11.1]
• Triangulated categories. The theory of derived functors can alternatively be expressed through that of derived categories, a special case of triangulated categories (or more precisely triangulated categories with chosen t-structures). Generalizing abelian categories, triangulated categories are merely additive categories but come equipped with a generalized notion of “exact sequences”, namely exact triangles. Much of classical homological algebra can be carried out on triangulated categories as well. [Wei94, § 10.1–§ 10.2]

Additional topics

Proof of nonexistence of adjoint

{$K^{-1}K=I$}.

• Partitions (tableaux) are structures that relate to decomposition (factoring) of an item. Thus they relate to factorization of monoids. Thus it may be possible to show using adjunctions that they can't yield a graded involution, one that factors through.
• My thesis about eigenvalues of a matrix may involve an adjunction that relates a substitution right adjoint functor (for the matrix entries) and a left adjoint functor for reworking of nodes into edges.

Adjunctions and Math discovery

• Tensor-hom is like a list
• Galois connections are like least upper bound, greatest lower bound
• Diagonal functor is like substitution (one of the methods of proof)
• In the house of knowledge, the threesome creates a division of everything between local (algebra) and global (analysis) as an adjoint string.

Self-adjointness

• Hermitian adjoint The adjoint of an operator plays the role of the complex conjugate of a complex number.
• Self-adjoint operators play the role of the real numbers (being equal to their own "complex conjugate") and form a real vector space. They serve as the model of real-valued observables in quantum mechanics.
• The relation between self-adjoint (reals) and adjoint (complexes) gets repeated with the complexes and the quaternions, and with the quaternions and the octonions.
• If we think of the scalar product as summing over all elements of the group and thus ensuring group invariance, and if that scalar product is semilinear in the second argument, then inverting all of the group elements is the same as transposing and taking the complex conjugate of the matrix. Thus adjunction expresses this duality.
• If a representation is defined with regard to an orthonormal basis, then it is a unitary matrix. We can think of a unitary matrix as expressing that a representation is defined with regard to an orthonormal basis defined with regard to the scalar product.
• Self-adjoint means that it is "observable" because observability requires being able to approach it from both directions of causality, forwards and backwards, as per the critical point - the ambiguous point - of the fivesome.
• A unitary (self-adjoint) transformation has the same information going forward as going backward (there is an adjunction) but that information may be shifted (as when shifting an infinite dimensional vector space). Equality after a shift is what is possible (expresses the freedom) in a closed system, as modeled by mathematics, where you have self superimposed sequence (an internal, analytic symmetry).

Relating adjunctions in category theory and linear algebra

• In linear/Lie algebra - Hermitian adjoint (Hermitian conjugate) - {$<Ah_1,h_2>H_2=<h_1,A*h_2>H_1$}
• Self-adjoint operator: {$A**=A (A*)-1=(A-1)*$}

Adjunction in physics

• Free construction moves us from kinetic energy to potential energy. Forgetting moves us from potential energy to kinetic energy.
• The change in the free energy is the maximum amount of work that a thermodynamic system can perform in a process at constant temperature, and its sign indicates whether a process is thermodynamically favorable or forbidden. Free energy is subject to irreversible loss in the course of such work.
• A propagator {$<u_n(x'),u_n(x)e^{-iE_nt/ħ}>$} relates a shift from {$x$} to {$x'$} with multiplying by a time factor.

Adjunction and physical operators

• Information states: What nature has to say as a sum of terms.
• Space-time weight function: What questions the physicists ask by selecting an interval.

Correlation and causality

• Adjunction can relate correlation and causality. For they are two functors that are synchronized but they have different contexts. A correlation is like a forgetful functor in that it gives partial information. And causality is like a free construction in that it generates more information.

Emergence

• Mathematically, emergence is described by free construction, thus left adjoints.

Adjunction and entropy

• Information context. Probability. The value of the change in a bit.
• Local information is high entropy (Shannon commmunication). Global informaion is low entropy (Physicists).
• von Neumann entanglement entropy - system (there is a composite state) and subsystem (there is no state).

Wisdom

Unfolding vs. Void

Organization of knowledge

Notes: Lecture 15: Abstraction and adjunction, video by Daniel Murfet

• Seminar: How to use adjoint functors and topoi to organise mathematical knowledge, Announcement
• Based on book: Mac Lane, Moerdijk. “Sheaves in Geometry and Logic: A First Introduction to Topos Theory”.
• See: Classifying topos and Classifying space

Notes

• For the Geometry of Moods video I could have added one more slide about Conformal Geometry, SL(2) and the threesome.
• Monads deal with scopes: none, some, and so on. The logic of the sevensome.
• Eugenio Moggi. Notions of computation and monads
• Anurag Mendhekar. So, You Want To Understand Monads?
• An Intuitive Introduction to Monads in Under 10 Minutes Polymorphic embellished composition. Defined for overloads. Monads are generic containers whose prototype can be overridden by its usage in the program and implicitly declares "exceptional" behavior, including side effects that would otherwise not be possible to have in a purely functional programming language, and hides rules for handling such behavior inside of itself.
• Relate triangulated categories (with squiggles {$X\rightsquigarrow W = X\rightarrow TW$}) to monads with likewise squiggles.
• Adjoint School 2023