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Classify the kinds of adjoint strings.

Goals

• Show how the Yoneda Lemma expresses four levels of knowledge - Whether, What, How, Why.
• Understand how the Yoneda embedding and the Yoneda lemma relate to adjunction, for example, how the "do nothing" action appears.
• Understand how Grothendieck's six operations express these four levels of knowledge (through f) and increasing and decreasing slack (through functors based on an object X).
• Classify adjoint strings. Understand how adjoint strings carve out contexts.
• Understand how they manifest that information stays the same but context changes.
• Classify the kinds of trivial functors.
• Understand isomorphism and equivalence as adjunctions.
• Explain the qualitative difference between left and right adjoints.
• Understand how the qualitative difference between left and right adjoints arises from the asymmetry of how their arrows are related, as shown by the universal mapping property, pointing in parallel or at each other.
• Intuit how the left adjoint exprėsses a local solution and the right adjoint express a global solution. Note that a division of everything relates a particular local perspective within the global overview.
• Understand how the various definitions of adjunction are related
• Consider if there are four definitions which together express the formula for perspective (požiūrio lygtis)
• Understand how to go from one definiton of adjunction to another and gain intuition on what this means in practice for various examples.
• Understand in what sense self-adjoint operators on Hilbert spaces are related to adjunctions.

Write up

• Why the two definitions of equivalences (in terms of natural isomorphisms and in terms of full, faithful, essentially surjective functor) are the same and what that means in the case of two object categories.

Steps to take

• Understand the three different definitions of adjunction.
• Correct and improve my template for understanding adjunction in terms of the universal mapping property.
• Use the adjunction template to write out the logic by which functors have or have not adjoint functors.
• Explain why there are three or more definitions of adjunction.
• Understand which definition is most closely related to the Yoneda lemma and what do the other definitions say about the Yoneda lemma.
• Understand how to go between the three different definitions of adjunction.
• Classify adjoint strings.
• Collect examples of equivalences, especially those which are not isomorphisms.
• Classify the examples of equivalences.
• Relate the kinds of equivalences to the kinds of adjunctions.
• Correct and improve my diagram relating the definitions of isomorphism, equivalence and adjunction.
• Classify the kinds of trivial functors.
• Why and how do adjoint functors arise from trivial functors?
• Adjunctions are with regard to various "do nothing" functors. Catalogue the possibilities.
• Linearity, multilinearity, tensors, etc. are triviality - how does that relate to adjunctions?
• What is the difference between trivial actions (defined by adjunctions) and the null action (defined by the Yoneda lemma)?
• Think of the kinds of triviality as the ways of the abuses of notation.
• Classify the kinds of illustrative examples.
• Adjunctions should relate to developing a theory of most illustrative examples. Develop such a theory of examples for category theory. Perhaps there may be several examples needed in more sophisticated cases. The set of examples would relate to the perspectives in the corresponding division of everything.

Adjoints, perspectives and divisions of everything

• In an adjoint string, what are the end points, the "zeroes"?
• Does a sequence of adjoint functors express a division of everything?
• What is a perspective? Is it a morphism between adjoint functors?
• What do adjunctions say about knowledge and the foursome?
• Adjunction. If you can't tell the difference between objects, what does knowledge mean? How do levels of knowledge correspond to various abilities to distinguish between objects as such?
• Express composition of "God's point of view" and "Human's point of view" in terms of adjoint functors.
• What is a functor with no adjoints? In what sense is it the twosome?
• Is the fivesome an adjoint quadruple as with cohesive topos?

Free and forgetful

• What exactly is the distinction between including and forgetting?
• What would be the adjoint functors to various forgettings in analysis such as going from Euclidean space to a metric space (retaining simply the distance metric). When does the adjoint functor exist?

Syntax-semantics adjunction

• If a functor takes us from a syntactic category to a semantic category, then what does the adjoint functor mean?

Tableaux and {$K^{-1}K=I$}

• Can "the fundamental unit of information" (a certain tableaux-partition) be thought of as an expression of adjoint functors? Consider the duality of matrices and the symmetric group.
• Consider the equation {$K^{-1}K=I$} as a back and forth process.
• Use the nonexistence of an adjoint functor to show that there is no combinatorial interpretation of an involution for {$K^{-1}K=I$}.
• Think about Wenbo's infinite factorization of adjunction.

Mandelbrot set and Catalan set

• Think of generating the Mandelbrot set in terms of adjoint functors that express {$z\Leftrightarrow z^2+c$}.

Examples of left adjoints and right adjoints

Overview

Organize by trivial functors.

• General principle: left adjoint is local, right adjoint is global.
• Identity functor
• Isomorphism
• Equivalence.
• The unit is a natural isomorphism from {$FG$} to {$I_D$} and likewise the counit is a natural isomorphism from {$GF$} to {$I_C$}. Alternatively, {$F\dashv G$} where both functors are full and faithful.
• Duality.
• Equivalence in opposite directions.
• Compatible adjoint functors
• Greatest lower bound and least upper bound. Galois connection.
• Efficient solution vs. Difficult problem
• Minimal axiomatization (smallest set of keys to open a set of locks) left adjoint to Totality of satisfaction (maximum set of locks that can be opened by a set of keys)
• Concatenation by an object.
• Tensor product (internal structure) is left adjoint to Hom functor (external relations).
• Insertion of copies into a diagram.
• Left adjoint is colimit of diagram. Right adjoint is limit of diagram. (This grounds my intuition about internal structure and external relationships.)
• Ignoring algebraic structure given by operations.
• Free construction is left adjoint to Forgetful functor. Left adjoint to free construction is the enveloping functor. Right adjoint to Forgetful functor is the substructure satisfying the additional structure.
• Inclusion (of Z into R), extending the domain.
• Left adjoint is ceiling function, inverse from above. Right adjoint is floor function, inverse from below.
• Ignoring topological structure given by grouping.
• Discrete topology is left adjoint to this, and indiscrete topology is right adjoint to this.
• Defining a constant function.
• Left adjoint is "there exists" value, right adjoint is "for all" value.
• Aggregation of elements.
• Left adjoint of inverse functor is direct image functor, and right adjoint of inverse functor is the proper image functor. And the right adjoint functor of the proper image functor leads to the infinite three-cycle.
• Unclear
• Dual functors. Vector space mapped to dual vector space and back.

One functor (Self-adjoint)

Identity functor

 identity functor {$\textrm{id}_\textbf{C}:\textbf{C}\rightarrow\textbf{C}$} identity functor {$\textrm{id}_\textbf{C}:\textbf{C}\rightarrow\textbf{C}$}

Equivalences

An equivalence is a translation. A translation is sucessful if the corresponding terms (the translated term and the compared term) have analogous structure and if the terms are explicitly connected. If the terms are different in structure or if they have no connecting link then the translation is not achieved.

An equivalence can be defined as follows:

• An equivalence is a pair of functors {$F$}, {$G$} with natural isomorphisms {$\eta:\textbf{1}_\textbf{C}\rightarrow {G∘F}$}, {$\epsilon:F∘G\rightarrow {1}_\textbf{D}$}. The natural isomorphisms {$\eta_{C} \eta^{-1}_{C}={1}_{G∘F(C)}$}, {$\eta^{-1}_{C} \eta_C ={1}_{C}$}, {$\epsilon_{D} \epsilon^{-1}_{D}={1}_{D}$}, {$\epsilon^{-1}_{D}\epsilon_{D}={1}_{F∘G(D)}$}.
• This is similar to the definition of adjunction in terms of unit and counit.
• A functor is an equivalence if and only if it is full, faithful and essentially surjective on objects.
• A functor is full if the function {$\textbf{C}(C,C')\rightarrow \textbf{D}(F(C),F(C'))$} is surjective for all {$C$}, {$C'$} in {$\textbf{C}$}.
• A functor is faithful if the function {$\textbf{C}(C,C')\rightarrow \textbf{D}(F(C),F(C'))$} is injective for all {$C$}, {$C'$} in {$\textbf{C}$}.
• A functor is essentially surjective on objects if for all {$D\in \textbf{D}$}, there exists {$C\in\textbf{C}$} such that {$F(C)\cong D$}.
• This is similar to the definition of adjunction in terms of homsets, and also in terms of the universal mapping property.

• Leinster 36. "let A be the category whose objects are groups and whose maps are all functions between them, not necessarily homomorphisms. Let {$Set_{\neq ∅}$} be the category of nonempty sets. The forgetful functor U : A → {$Set_{\neq ∅}$} is full and faithful. It is a (not profound) fact that every nonempty set can be given at least one group structure, so U is essentially surjective on objects. Hence U is an equivalence. This implies that the category A , although defined in terms of groups, is really just the category of nonempty sets. Andrius: This example shows that the faithfulness and fullness are with regard to the homsets, not the objects. Thus we may have two different groups on the same elements and which thus get mapped to the same set but their homsets remain in bijection between the category of groups and the category of nonempty sets.
• Wikipedia: Consider the category {$C$} having a single object {$c$} and a single morphism {$1_{c}$}, and the category {$D$} with two objects {$d_{1}$}, {$d_{2}$} and four morphisms: two identity morphisms {$1_{d_{1}}$}, {$1_{d_{2}}$} and two isomorphisms {$\alpha \colon d_{1} \to d_{2}$} and {$\beta \colon d_{2} \to d_{1}$}. The categories {$C$} and {$D$} are equivalent; we can (for example) have {$F$} map {$c$} to {$d_{1}$} and {$G$} map both objects of {$D$} to {$c$} and all morphisms to {$1_{c}$}.
• By contrast, the category {$C$} with a single object and a single morphism is not equivalent to the category {$E$} with two objects and only two identity morphisms. The two objects in {$E$} are not isomorphic in that there are no morphisms between them. Thus any functor from {$C$} to {$E$} will not be essentially surjective. This example shows that isomorphism is defined not absolutely as to the form but with regard to the morphisms in the category itself.
• Leinster p.35 Let A be any category, and let B be any full subcategory containing at least one object from each isomorphism class of A . Then the inclusion functor {$B ,→ A$} is faithful (like any inclusion of subcategories), full, and essentially surjective on objects. Hence B is equivalent to A. So if we take a category and remove some (but not all) of the objects in each isomorphism class, the slimmed-down version is equivalent to the original. Conversely, if we take a category and throw in some more objects, [along with the relevant morphisms to and from all isomorphic objects, and other objects], each of them isomorphic to one of the existing objects, it makes no difference: the new, bigger, category is equivalent to the old one.
• Leinster p.35 Let FinSet be the category of finite sets and functions between them. For each natural number n, choose a set n with n elements, and let B be the full subcategory of FinSet with objects 0, 1, . . . . Then B is equivalent to FinSet, even though B is in some sense much smaller than FinSet.
• Leinster p.35 Let C be the full subcategory of CAT whose objects are the one-object categories. Let Mon be the category of monoids. Then C ' Mon. To see this, first note that given any object A of any category, the maps A → A form a monoid under composition (at least, subject to some set-theoretic restrictions). There is, therefore, a canonical functor F : C → Mon sending a one-object category to the monoid of maps from the single object to itself. This functor F is full and faithful (by Example 1.2.7) and essentially surjective on objects. Hence F is an equivalence.
• If f is the inclusion of a closed subspace X ⊆ Y then the direct image functor {$f_*$} is exact. Actually, in this case f∗ is an equivalence between sheaves on X and sheaves on Y supported on X. It follows from the fact that the stalk of ( f ∗ F ) y {\displaystyle (f_{*}{\mathcal {F}})_{y}} (f_* \mathcal F)_y is F y {\displaystyle {\mathcal {F}}_{y}} \mathcal F_y if y ∈ X {\displaystyle y\in X} y \in X and zero otherwise (here the closedness of X in Y is used).

Dualities

Leinster p.35: An equivalence of the form {$A^{op}\simeq B$} is sometimes called a duality between A and B. One says that A is dual to B. There are many famous dualities in which A is a category of algebras and B is a category of spaces.

Leinster: For certain classes of space, the passage from X to C(X) loses no information: there is a way of reconstructing the space X from the ring C(X). For this and related reasons, it is sometimes said that ‘algebra is dual to geometry’. Given a topological space X, let C(X) be the ring of continuous real-valued functions on X. The ring operations are defined ‘pointwise’: for instance, if {$p_1 , p_2 : X → R$} are continuous maps then the map {$p_1 + p_2 : X → R$} is defined by {$(p_1 + p_2 )(x) = p_1 (x) + p_2 (x)(x ∈ X)$}. A continuous map {$f : X → Y$} induces a ring homomorphism {$C( f ) : C(Y) → C(X)$}, defined at {$q ∈ C(Y)$} by taking {$(C( f ))(q)$} to be the composite map {X \overset{f}{\rightarrow} Y \overset{q}{\rightarrow} R.$} Note that {$C( f )$} goes in the opposite direction from f . After checking some axioms (Exercise 1.2.26), we conclude that C is a contravariant functor from Top to Ring. • Stone duality: the category of Boolean algebras is dual to the category of totally disconnected compact Hausdorff spaces. • Gelfand–Naimark duality: the category of commutative unital C ∗ -algebras is dual to the category of compact Hausdorff spaces. (C ∗ -algebras are certain algebraic structures important in functional analysis.) • Algebraic geometers have several notions of ‘space’, one of which is ‘affine variety’. Let k be an algebraically closed field. Then the category of affine varieties over k is dual to the category of finitely generated k-algebras with no nontrivial nilpotents. • Pontryagin duality: the category of locally compact abelian topological groups is dual to itself. As the words ‘topological group’ suggest, both sides of the duality are algebraic and geometric. Pontryagin duality is an abstraction of the properties of the Fourier transform. Isbell duality. Preorders. from nLab... Isbell duality In the simplest case, namely for an ordinary category {$\mathcal{C}$}, the adjunction between presheaves and copresheaves arises as follows... • The presheaf category {$[\mathcal{C}^{op}, \mathrm{Set}]$} has all limits, so we can extend the Yoneda embedding to a continuous functor {$Y \colon [\mathcal{C}, \mathrm{Set}]^{op} \to [\mathcal{C}^{op}, \mathrm{Set}]$} from copresheaves to presheaves. • Dually, the copresheaf category {$[\mathcal{C}, \mathrm{Set}]^{op}$} has all colimits, so we can extend the co-Yoneda embedding to a cocontinuous functor {$Z \colon [\mathcal{C}^{op}, \mathrm{Set}] \to [\mathcal{C}, \mathrm{Set}]^{op}$} from presheaves to copresheaves. • Isbell duality says that these are adjoint functors: Y is right adjoint to Z. The more general case deals with enrichment. nLab: Spec is the left Kan extension of the Yoneda embedding along the contravariant Yoneda embedding, while {$\mathcal{O}$} is the left Kan extension of the contravariant Yoneda embedding along the Yoneda embedding. nLab: The codensity monad of the Yoneda embedding is isomorphic to the monad induced by the Isbell adjunction, {$Spec \mathcal{O}$} (Di Liberti 19, Thrm 2.7). Preorders. Greatest lower bound and least upper bound. Galois connection. Lower adjoint vs. Upper adjoint • Let (A, ≤) and (B, ≤) be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone functions: F : A → B and G : B → A, such that for all a in A and b in B, we have F(a) ≤ b if and only if a ≤ G(b). Galois connection An essential property of a Galois connection is that an upper/lower adjoint of a Galois connection uniquely determines the other: • F(a) is the least element ~b with a ≤ G(~b), and • G(b) is the largest element ~a with F(~a) ≤ b. A consequence of this is that if F or G is invertible, then each is the inverse of the other, i.e. {$F = G^{-1}$}.  Lower adjoint Upper adjoint Notes {$F$} {$G$} Galois connection {$Fa≤b\Leftrightarrow a≤Gb$} for preorders. An antitone Galois connection between A and B is just a monotone Galois connection between A and the order dual {$B^{op}$} of B. • the connection between fields and groups in Galois theory • the connection between covering spaces and fundamental groups in topology • the connection between polynomials and their roots in algebraic geometry • Is this a Galois connection? The ideal quotient is adjoint to the multiplication by ring ideals. Any relation R on X×Y defines a Galois connection. • the theory of polynomial equations (Lagrange, Galois), • the modern Galois theory (Dedekind, Artin), • the origins of lattice theory (Dedekind, Schröder), • the polarities and lattice-theoretical aspects (Birkhoff), • the order-theoretical Galois connections (Ore), • the logical calculus (Boole, Peirce, Schröder), • the residuation theory (Krull, Ward, Dilworth) • {$B\subseteq fA$} (each feature in B is possessed by every element in A) iff {$A\subseteq gB$} (each element in A has all the features in B). Preorders. Defining a constant function There is a trivial functor that takes us from a constant to a function with that constant value. The left functor will be "there exists" and the right functor will be "for all".  {$\exists_X:\Omega^X\rightarrow\Omega$}, {$\exists_XP=\top\Leftrightarrow\exists x\in X(P(x)=\top)$} dummy variable constant functor {$\Delta_X:\Omega\rightarrow\Omega^X$} which sends {$\top$} to {$\top(x)=\top$} and sends {$\bot$} to {$\bot(x)=\bot$} {$\forall_X:\Omega^X\rightarrow\Omega$}, {$\forall_XP=\top\Leftrightarrow\forall x\in X(P(x)=\top)$} {$\bot\leq\top,\Omega=\{\bot,\top\},\Omega^X$} is the set of propositional functions {$P:X\rightarrow\Omega$}. {$P\leq Q\Leftrightarrow\forall X(P\Rightarrow Q)$}. {$\exists_f:\textrm{Sub}(X)\rightarrow\textrm{Sub}(Y)$} {$f^*:\textrm{Sub}(Y)\rightarrow \textrm{Sub}(X)$} {$\forall_f:\textrm{Sub}(X)\rightarrow\textrm{Sub}(Y)$} Given morphism {$f:X\rightarrow Y$} in a category with pullbacks. {$\textrm{Sub}(X)$} is the category that is the preorder of subobjects. Preorders. Based on function f.  direct image functor {$f_*:PA\rightarrow PB$} maps subset {$A′⊂A$} to subset {$f(A′)⊂B$}. Then for {$A′⊂A$} and {$B′⊂B$}, {$f(A′)⊂B′$} if and only if {$A′⊂f^{−1}(B′)$} inverse image functor {$f^{-1}:PB\rightarrow PA$} {$f_!:PA\rightarrow PB$} maps {$A′⊂A$} to the subset of elements of {$B$} whose fibers (inverse images) lie entirely in {$A'$}. Thus {$B′⊂f_!(A′)\Leftrightarrow f^{−1}(B′)⊂A′$} given set function {$f:A\rightarrow B$}, the subsets of {$A$} and subsets of {$B$} for posets, {$PA$} and {$PB$}, ordered by inclusion. left Kan extension {$f_!$} precomposition functor {$f^∗:Sets^{D^{op}}→Sets^{C^{op}}$} given by {$f^∗(Q)(C)=Q(fC)$} right Kan extension {$f_∗$} Awodey 9.17  {$f^{-1}$} inverse image (or pullback functor on sheaves) {$f^*:\textrm{Sh}(Y)\rightarrow \textrm{Sh}(X)$} direct image (or pushforward functor on sheaves) {$f_*:\textrm{Sh}(X)\rightarrow \textrm{Sh}(Y)$} (understand the case for sheaves) Flatness See: Flatness and Inverse image functor. {$f^{*}$} is (in general) only right exact. If {$f^{*}$} is exact, f is called flat. Adjoint quadruple  {$f_!$} {$f^*$} {$f_*$} {$f^!$} {$Π_0$} Disc Γ Codisc expresses cohesion of geometry For cohesive topos by definition the terminal geometric morphism extends to an adjoint quadruple. Preservation of limits and colimits  Left adjoint Right adjoint Notes  F preserves small colimits F is a functor between locally presentable categories G preserves small limits and is an accessible functor G is a functor between locally presentable categories • Left adjoint exists for functor that preserves limits. Right adjoint exists for functor that preserves colimits. • Adjoint functors: Limit preservation Every functor that has a left adjoint (and therefore is a right adjoint) is continuous (i.e. commutes with limits in the category theoretical sense); every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (i.e. commutes with colimits). • Exact functor If the functor F is left adjoint to G, then F is right exact and G is left exact. Dual functors  Left adjoint Right adjoint Notes {$(-)^*:\mathbf{Vect}_k\rightarrow \mathbf{Vect}^{op}_k$} {$(-)^*:\mathbf{Vect}^{op}_k\rightarrow \mathbf{Vect}_k$} each functor send a vector space {$V$} to its dual vector space {$V^*:=\textrm{Hom}(V,k)$} See: Double dualization monad which sends a vector space to its double dual Riehl Ex.5.1.14 similarly discusses the double power set monad Merged object vs. arrows from one to the other  {$- \bigotimes X$} {$\textrm{Hom}(X,-)$} Note that the direction of the arrow switches. There is a natural isomorphism {$\textrm{Hom}_S(Y\bigotimes_R X, Z) \cong \textrm{Hom}_R(Y,\textrm{Hom}_S(X,Z))$}Tensor-Hom adjunction internal tensor product internal Hom functor smash product smash-hom adjunction topological spaces with chosen base points, Tai Danae Bradley suspension functor loop functor case of circle {$S^1$}, Tai Danae Bradley, proof that fundamental group of circle is Z currying f(a,b) to create two functions in sequence x=h(b) where h=g(a) uncurrying functor {$B ↦ B × C$} functor {$A ↦ A^C$} Currying: Set theory for every fixed set C {$L:=X\times –:\mathbf{Set}\rightarrow\mathbf{Set}$} {$R:=\mathbf{Set}(X,–):\mathbf{Set}\rightarrow\mathbf{Set}$} Product-hom adjunction, see Topology: A Categorical Approach. Page 92. Note that {$\mathbf{Set}(LZ,Y)=Y^{X\times Z}\cong (Y^X)^Z = \mathbf{Set}(Z,RY)$} Currying: Logic Under the Curry–Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem ( A ∧ B ) → C ⇔ A → ( B → C ) {\displaystyle (A\land B)\to C\Leftrightarrow A\to (B\to C)} {\displaystyle (A\land B)\to C\Leftrightarrow A\to (B\to C)}, as tuples (product type) corresponds to conjunction in logic, and function type corresponds to implication. Currying: Category Theory In closed monoidal categories: Currying is the statement that the tensor product and the internal Hom are adjoint functors; that is, for every object B {\displaystyle B} B there is a natural isomorphism: H o m ( A ⊗ B , C ) ≅ H o m ( A , B ⇒ C ) . {\displaystyle \mathrm {Hom} (A\otimes B,C)\cong \mathrm {Hom} (A,B\Rightarrow C).} {\displaystyle \mathrm {Hom} (A\otimes B,C)\cong \mathrm {Hom} (A,B\Rightarrow C).} Here, Hom denotes the (external) Hom-functor of all morphisms in the category, while B ⇒ C {\displaystyle B\Rightarrow C} {\displaystyle B\Rightarrow C} denotes the internal hom functor in the closed monoidal category. For the category of sets, the two are the same. When the product is the cartesian product, then the internal hom B ⇒ C {\displaystyle B\Rightarrow C} {\displaystyle B\Rightarrow C} becomes the exponential object C B {\displaystyle C^{B}} {\displaystyle C^{B}}. Currying can break down in one of two ways. One is if a category is not closed, and thus lacks an internal hom functor (possibly because there is more than one choice for such a functor). Another ways is if it is not monoidal, and thus lacks a product (that is, lacks a way of writing down pairs of objects). Categories that do have both products and internal homs are exactly the closed monoidal categories. The setting of cartesian closed categories is sufficient for the discussion of classical logic; the more general setting of closed monoidal categories is suitable for quantum computation.[18] The difference between these two is that the product for cartesian categories (such as the category of sets, complete partial orders or Heyting algebras) is just the Cartesian product; it is interpreted as an ordered pair of items (or a list). Simply typed lambda calculus is the internal language of cartesian closed categories; and it is for this reason that pairs and lists are the primary types in the type theory of LISP, scheme and many functional programming languages. By contrast, the product for monoidal categories (such as Hilbert space and the vector spaces of functional analysis) is the tensor product. The internal language of such categories is linear logic, a form of quantum logic; the corresponding type system is the linear type system. Such categories are suitable for describing entangled quantum states, and, more generally, allow a vast generalization of the Curry–Howard correspondence to quantum mechanics, to cobordisms in algebraic topology, and to string theory.[1] The linear type system, and linear logic are useful for describing synchronization primitives, such as mutual exclusion locks, and the operation of vending machines. Commuting with both limits and colimits. See: Exact functor and consider the difference between left exact functors and right exact functors. Akhil Mathew Having an adjoint tells you that the functor commutes with (either) limits or colimits. If a functor has a left adjoint, then it commutes with colimits, while if it has a right adjoint, it commutes with limits. For nice categories, one can sometimes conclude the converse. One example of this is in an abelian category. In the case of R-modules, for instance, the adjunction between Hom and the tensor product shows that the tensor product is right-exact (a.k.a. commutes with finite colimits). This is a somewhat more conceptual argument than the usual one. Efficient solution vs. Difficult problem  Left adjoint Right adjoint Notes give the most efficient solution to the problem posed pose the most difficult problem that the solution solves See: Adjoint Functors: Introduction and Motivation Free construction vs. Forgetfulness  Free construction Forgetful functor Notes free functor from complete semi-lattice to set MSE: yields the powerset monad free functor from free semigroup to set MSE: yields the list monad free vector space functor {$\mathbb{k}[−]:\textbf{Set}\rightarrow\textbf{Vect}_k$} underlying set functor {$U:\textbf{Vect}_k\rightarrow\textbf{Set}$} Tai-Danae Bradley map set to K-algebra {$X\rightarrow K[X]$} map K-algebra to its underlying set polynomial rings are free commutative algebras turn a graph into a category by concatenating paths - free category from category to underlying graph Tai Danae Bradley also Varela, Kaufmann Stone–Čech construction Inclusion functor U from CHaus into Top Stone-Čech compactification Category of compact Hausdorff spaces CHaus, category of topological spaces Top. The Axiom of Choice is used to prove the existence of the compactification. functor from monoids to rings from ring to underlying monoid integral monoid ring construction take semigroup, add identity, get monoid from monoid to underlying semigroup take a commutative monoid, add inverses, get abelian group take abelian group, forget inverses Grothendieck group construction abelianization: {$G\rightarrow G^{ab}=G/[G,G]$} from abelian group to group induction from a representation of a subgroup to representation of a group which extends it most generally restriction of representation of group to a representation of its subgroup Frobenius reciprocity take rng {$R$} to a ring with identity {$R\times \mathbb{Z}$} from ring to underlying rng assigns to every ring {$R$} the pair {$(R[x],x)$} forgetful functor {$G:\textbf{Ring}_*\rightarrow\textbf{Ring}$} {$\textbf{Ring}_*$} pointed commutative rings with unity (pairs (A,a) where A is a ring, a ∈ A) take integral domain to its field of fractions forgetful functor {$\textbf{Field}\rightarrow\textbf{Dom}_m$} category {$\textbf{Dom}_m$} of integral domains with injective morphisms tensor algebra from algebra to vector space tensor product with {$S$} yields {$F:R\textbf{-Mod}\rightarrow S\textbf{-Mod}$} forgetful functor {$G:S\textbf{-Mod}\rightarrow R\textbf{-Mod}$} given {$\rho:R\rightarrow S$} Riehl page 115: There are many instances of free constructions in mathematics. One can define the free (abelian) group on a set, the free ring on a set or on an abelian group, the free module on an abelian group, the free category on a directed graph, and so on. Sometimes there are competing notions of universal constructions: Does the free graph on a set have a single edge between every pair of vertices or none? Does the free topological space on a set have as many open sets as possible or as few? Other times there are none: For instance, there is no “free field.” Riehl page 115: The universal property of the free vector space functor {$\mathbb{k}[−]:\textbf{Set}\rightarrow\textbf{Vect}_k$} is expressed by saying that it is left adjoint to the underlying set functor {$U:\textbf{Vect}_k\rightarrow\textbf{Set}$}: linear maps {$\mathbb{k}[S]\rightarrow\mathbf{V}$} correspond naturally to functions {$S\rightarrow U(V)$}, which specify the image of the basis vectors {$S⊂\mathbb{k}[S]$}. By the Yoneda lemma, this universal property can be used to define the action of the free vector space functor {$\mathbb{k}[−]$} on maps. The forgetful functor {$U:\textbf{Vect}_k\rightarrow\textbf{Set}$} has no right adjoint, so in this setting there are no competing notions of free construction. Kategoriją išreiškus matrica išryškėja ryšys tarp laisvumo (freely generated) ir užmarštumo (forgetfulness functor), mat išrašus visus kategorijos slinkstis, galima "matricų daugyba" išreikšti kaip jos įvairiai komponuojasi, bet tada tenka išreikšti, kaip jos įvairiai sutampa. Forgetful without left adjoint The forgetful functor Field → Set does not have a left adjoint. (Leinster, Example 6.3.5.) The theory of fields is unlike the theories of groups, rings, and so on, because the operation {$x \mapsto \x^{-1}$} is not defined for all x (only for {$x\neq 0$}). Forgetful without right adjoint If a functor does not commute with colimits, then it does not have a right adjoint. The abelianization functor does not have a right adjoint because the forgetful functor does not commute with colimits. Forgetful with right adjoint  Left adjoint endows the set with the discrete topology (all subsets are open). The left adjoint constructs a topological space from a set {$S$} in such a way that continuous maps from this space to another space {$T$} correspond naturally and bijectively to functions {$S\rightarrow U(T)$}. from topological space to set Right adjoint endows the set with the indiscrete topology (only open sets are the whole set and the empty set). The right adjoint constructs a topological space from a set {$S$} in such a way that continuous maps from {$T$} to this space correspond naturally and bijectively to functions {$U(T)\rightarrow S$}. F(M) is the group obtained from monoid M by throwing in an inverse to every element. from group to monoid R(M) is the group which is the submonoid of M consisting of all invertible elements of M. Leinster, page 45, example 2.13 d Arturo Magidin Ever wondered why the underlying set of a product of topological spaces is the product of the underlying sets, and the underlying set of a coproduct of topological spaces is also the coproduct/disjoint union of the underlying sets of the topological spaces? Why the constructions in topological spaces always seem to start by doing the corresponding thing to underlying sets, but in other categories like Group, R−Mod, only some of the constructions do that? (I know I did) It's because while in Group the underlying set functor has a left adjoint but not a right adjoint, in Top, the underlying set functor has both a left and a right adjoint (given by endowing the set with the discrete and indiscrete topologies). Having a right adjoint means that we have a geometry (a space) because it means that we have a global construction and thus we have a "choice framework", in other words, homogeneity. Having a left adjoint means that we have algebra, we have step-by-step constructions. Free construction with left adjoint  Left adjoint Free construction Forgetful functor Notes The universal enveloping functor U which constructs the most general algebra containing all representations of a Lie algebra Construct a Lie algebra from an algebra by taking the Lie bracket to be the commutator. Forget the Lie bracket and simply have an algebra without it. Universal enveloping algebra Given an underlying field, {$\mathbf{Alg}$} is the category of algebras (unital associative algebras?), {$\mathbf{LieAlg}$} is the category of Lie algebras. Free construction with left adjoint, forgetful with right adjoint Awodey, 2006. 9.6: There is a string of four adjoints between Cat and Sets, {$V\dashv F\dashv U\dashv R$} where {$U:Cat→Sets$} is the forgetful functor to the set of objects {$U(C)=C_0$}. For V, consider the “connected components” of a category. Figure this out as an exercise. Free constructions • Free module: the forgetful functor from M o d ( R ) {\displaystyle \mathbf {Mod} (R)} {\mathbf {Mod}}(R) (the category of R {\displaystyle R} R-modules) to S e t {\displaystyle \mathbf {Set} } \mathbf {Set} has left adjoint Free R {\displaystyle \operatorname {Free} _{R}} \operatorname {Free}_{R}, with X ↦ Free R ⁡ ( X ) {\displaystyle X\mapsto \operatorname {Free} _{R}(X)} X\mapsto \operatorname {Free}_{R}(X), the free R {\displaystyle R} R-module with basis X {\displaystyle X} X. • Free group • Free lattice • Universal enveloping algebra Minimal axiomatization vs. Totality of satisfaction  Left adjoint Right adjoint Notes "syntax functor" F(S) is the minimal axiomatization of S "semantics functor" G(T) is the set of all structures that satisfy the axioms of a theory T. C is the set of all logical theories (axiomatizations), D is the power set of the set of all mathematical structures, S is a subset of G(T) if and only if F(S) logically implies T Diagonal functor The diagonal functor preserves both limits and colimits, thus has both a left adjoint and a right adjoint.  Left adjoint Diagonal functor Right adjoint Notes sum functor diagonal functor {$\Delta:\textbf{C}\rightarrow \textbf{C}\times \textbf{C}$} product functor Accordingly as to whether {$\textbf{C}$} has sums and products colimit functor {$\textrm{lim}:\textbf{C}^J\rightarrow \textbf{C}$} diagonal functor {$\Delta:C\rightarrow C^J$} limit functor {$\textrm{lim}:\textbf{C}^J\rightarrow \textbf{C}$} Accordingly as to whether every diagram of shape J has a colimit and limit. {$\textrm{Hom}(\textrm{colim} F,N)\cong \textrm{Cocone}(F,N)$} {$\textrm{Hom}(N,\textrm{lim} F)\cong \textrm{Cone}(N,F)$} {$F:\mathbb{Ab}^2\rightarrow \mathbb{Ab}$} assigns to every pair {$(X_1, X_2)$} of abelian groups their direct sum {$X_1+X_2$} {$G:\mathbb{Ab}\rightarrow \mathbb{Ab}^2$} is the functor which assigns to every abelian group {$Y$} the pair {$(Y, Y)$} Richard Southwell. Category Theory for Beginners: Kan Extensions. • {$\Sigma$} is left adjoint to {$\Delta$} (defined before 35:00), which is left adjoint to {$\Pi$} (46:00). {$\Sigma$} goes from Graphs to Sets and takes a graph to a set with one element for each weakly connected component in the graph. And diagonalization {$\Delta$} goes from Sets to Graphs by converting each element to an object with an identity arrow. {$\Pi$} sends a graph to its set of points (self-loops). Data migration and Kan extensions Kan extension of adjoint pair is adjoint quadruple. Inclusion (inclusion?) functor with both a left adjoint (from - lean on the inner structure - requalification - best post-approximation) and a right adjoint (to - lean on the external whole - best pre-approximation) What is the difference between inclusion and forgetting? In the case of the preorders Z and R, the morphisms in R for objects of Z all carry over to Z. Whereas that is not the case when we include groups as sets.  Left adjoint Inclusion functor Right adjoint Notes ceiling function {$\mathbb{R}\rightarrow \mathbb{Z}$} {$i\in \mathbb{Z} | x\leq i \wedge$}{$(\forall j\in \mathbb{Z})(x\leq j \Rightarrow i\leq j)$} inclusion {$i:\mathbb{Z}\rightarrow\mathbb{R}$}, {$x_\mathbb{Z}\equiv x_\mathbb{R}$} floor function {$\mathbb{R}\rightarrow \mathbb{Z}$} {$i\in \mathbb{Z} | i\leq x$}{$ \wedge (\forall j\in \mathbb{Z})(j\leq x \Rightarrow j\leq i)$} morphism {$x\rightarrow y$} whenever {$x\leq y$}, think of counting as moving upward, thus defining "pre-approximation" and "post-approximation" Dualizing object Fausk, Hu, May: There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate "dualizing object". Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. Keys and Locks  minKeys() gives the smallest set of keys to open a set of locks maxLocks() gives the maximum set of locks that can be opened by a set of keys Runar Bjarnason "Adjunctions in Everyday Life" multiplication by integer y division by integer y x,y, z,: BigInt (when y >0); (z * y <= x) == (z <= x / y); (f(z) <= x) == (z <= g(x)) Adjoint triple • Go through the examples... • If one of the two adjoint pairs induced from an adjoint triple involving identities, then the other exhibits an adjoint cylinder / unity of opposites. • An adjoint triple {$F\dashv G\dashv H$} is Frobenius if F is naturally isomorphic to H. See Frobenius functor. • An affine morphism is an adjoint triple of functors in which the middle term is conservative. For example, any affine morphism of schemes induce an affine triples of functors among the categories of quasicoherent modules. • An adjoint triple of functors among A ∞A_\infty- or triangulated functors with certain additional structure is called spherical . See e.g. (Anno). The main examples come from Serre functors in a Calabi-Yau category context. • An adjoint triple {$F \dashv G \dashv H$} is called an ambidextrous adjunction (or sometimes ambijunction, for short) if the left adjoint F and the right adjoint H of G are equivalent {$F \simeq H$}. • A context of six operations {$(f_! \dashv f^!)$}, {$(f^\ast \dashv f_\ast)$} induces an adjoint triple when either {$f^! \simeq f^\ast$} or {$f_! = f_\ast$}. This is called a Wirthmüller context or a Grothendieck context, respectively. • {$f_! \dashv f^! \simeq f^\ast \dashv f_\ast$} Wirthmüller context • {$f^\ast \dashv f_\ast = f_! \dashv f^!$} Grothendieck context Yoneda embedding Yoneda embedding is the embedding of a category {$\mathcal{C}$} into {$\mathbf{Set}$} as follows: {$\mathcal{C}(A,B) \simeq \mathbf{Set}(\mathcal{C}(-,B),\mathcal{C}(-,A))$} So we can think of the Yoneda embedding as a fully faithful functor {$Z:\mathcal{C}\rightarrow \mathbf{Set}^{\mathcal{C}}$} which sends {$X$} to the functor {$\mathcal{C}(-,X)$} and which sends {$f:X\rightarrow Y$} to the natural transformation which sends functor {$\mathcal{C}(-,Y)$} to {$\mathcal{C}(-,X)$} by prepending {$f$}. nLab: Adjoint string There is an adjoint 5-tuple between {$[Set^{op}, Set]$} and {$Set$}. Indeed, given a locally small category B, and the Yoneda embedding, {$Y: B \to [B^{op}, Set]$}, then Y being the rightmost functor of an adjoint 5-tuple entails that B is equivalent to Set; see Rosebrugh-Wood.  Left-Left Left Center Right Right-right Notes {$\Lambda$} {$\Pi$} {$\Delta:\mathbf{Set}\rightarrow [\mathscr{O}(X)^{\textrm{op}},\mathbf{Set}]$} assigns a set {$A$} to the presheaf {$\Delta A$} with constant value {$A$}. {$\Gamma$} {$\nabla$} Leinster, page 50 For any category C, there is a functor {$ids: C\to Ar(C)$} from C to its arrow category that assigns the identity morphism of each object. This functor always has both a left and a right adjoint which assign the codomain and domain of an arrow respectively; thus we have an adjoint triple {$cod \dashv ids \dashv dom$}. If C has an initial object 0, then cod has a further left adjoint I assigning to each object x the morphism {$0\to x$}; and dually if C has a terminal object 1 then dom has a further right adjoint T assigning to x the morphism {$x\to 1$}. Thus if C has an initial and terminal object, we have an adjoint 5-tuple. Continuing from the last example, if C is moreover a pointed category with pullbacks and pushouts, then I has a further left adjoint that constructs the cokernel of a morphism {$x\to y$}, i.e. the pushout of {$y \leftarrow x \to 0$}; and T has a further right adjoint that constructs the kernel of a morphism {$x \to y$}, namely the pullback of {$x\to y \leftarrow 0$}. Thus we have an adjoint 7-tuple. In fact, the existence of such an adjoint 7-tuple characterizes pointed categories among categories with finite limits and colimits. The previous two examples apply also to derivators, and the extension of the analogous adjoint 5-tuple to a 7-tuple again characterizes the pointed derivators. Moreover, the stable derivators are characterized by the extension of this 7-tuple to a doubly-infinite adjoint string with period 6 (GrothShul17). • We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of "stability relative to a class of functors", which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein. Long adjoint strings Let [n] denote the totally ordered (n+1)-element set, regarded as a category. For each positive integer n, we have n+1 order-preserving injections from [n-1] to [n], and nn order-preserving surjections from [n] to [n-1]. Regarded as functors, these injections and surjections interleave to form an adjoint chain of length 2n + 1. These categories, functors, and adjunctions form the simplex category regarded as a locally posetal 2-category?; see below. Let C be a category with a terminal object but no initial object. Then there are functors {$δ i:[n+1,C]→[n,C] 0≤i≤n; σ i:[n,C]→[n+1,C] 0≤i≤n \array{ \delta_i \colon [n+1,C] \to [n,C] & 0\leq i \leq n; \\ \sigma_i\colon [n,C] \to [n+1,C] & 0\leq i \leq n }$} such that {$\delta_0 \dashv \sigma_0 \dashv \cdots \dashv \delta_n \dashv \sigma_n$} is a maximal string of adjoint functors (all but {$\sigma_n$} are obtained by applying {$[−,C][-, C]$} to the simplex category example, and {$\sigma_n$} exploits the presence of the terminal object of C). Generalizing the simplex category example: if P is a lax idempotent monad with unit {$u: 1 \to P$} and multiplication {$m: P P \to P$} (so that {$m \dashv u P$}), then there is an adjoint string {$P^{n-1} m \dashv P^{n-1} u P \dashv P^{n-2}m P \dashv \ldots \dashv m P^{n-1} \dashv u P^n$} of length {$2 n + 1$}, back and forth between {$P^{n+1}$} and {$P^n$}. The example of [n][n] and [n+1][n+1] above is based on the fact that the simplex category Δ\Delta, regarded as a locally posetal bicategory, is the walking lax idempotent monoid. Given an ambidextrous adjunction, {$F \dashv G$} and {$G \dashv F$}, we of course get an infinite adjoint string {$\ldots \dashv F \dashv G \dashv F \dashv \ldots$} of period 2. Possibilities vs. Generator  Left adjoint Right adjoint Notes functor from reachable automata to behaviors functor which gives the minimal realization of a behavior minimal realization behavior Goguen 1971 per Maclane's book page 89  Left adjoint Right adjoint Notes Meet (disjunction) Implication Propositional logic, Heyting algebra Unfolding vs. Void  Left adjoint Right adjoint Notes Eternal life God What about good? and life? Composition Identity morphism Unclear  the proper (or extraordinary) direct image {$f_!:Sh(X) → Sh(Y)$} the proper (or extraordinary) inverse image {$f^!:D(Sh(Y)) → D(Sh(X))$} Notes • Cartesian closed categories have binary products and a right adjoint to each functor sending A to AxB, which is essentially the typed lambda-calculus. More examples of adjoint functors • Expanding (replacing Identities with L*R) and collapsing (replacing L*R with Identity). • See Wikipedia: Currying {$B \mapsto B\times C$} is left adjoint to {$A \mapsto A^C$}. This grounds the equation {$A^{B\times C}\cong (A^C)^B$}. Akhil Mathew Let X be a locally compact Hausdorff space. Then the functor Z↦Z×X has an adjoint (namely, the functor Y↦YX). It follows that taking products with X preserves push-out diagrams, and more generally all colimits. This is useful sometimes in algebraic topology. For instance, if you have a push-out A∪BC and homotopies A×I→Z and C×I→Z that agree on B×I, you get a homotopy (A∪BC)×I→Z, the continuity of which might not be immediately obvious otherwise. Jeremy Gibbons: Every pair of adjoint functors gives rise to a monad. The converse holds too: every monad arises in that way. In fact, it does so in two canonical ways. One is the Kleisli construction Petr describes; the other is the Eilenberg-Moore construction. Indeed, Kleisli is the initial such way and E-M the terminal one, in a suitable category of pairs of adjoint functors. They were discovered independently in 1965. If you want the details, I highly recommend the Catsters videos. Tom Ellis regarding monads: • Maybe comes from the free functor into the category of pointed sets and the forgetful functor back • [] comes from the free functor into the category of monoids and the forgetful functor back • Sjoerd Visscher: Cont r comes from the adjunction of the contravariant functor Op r : Hask^op --> Hask with itself, with Op r a = a -> r. • [[Vitus: State monad can be "decomposed" as the pair of adjoint functors F and G: • data F b a = F (a,b) • data G b a = G (b -> a) • instance Functor (F b) where fmap f (F (a,b)) = F (f a, b) • instance Functor (G b) where fmap f (G g) = G (f . g) Bartosz Milewski: Unit lets us introduce the composition R ∘ L anywhere we could insert an identity functor on D; and counit lets us eliminate the composition L ∘ R, replacing it with the identity on C. That leads to some “obvious” consistency conditions, which make sure that introduction followed by elimination doesn’t change anything. • In Haskell, unit is known as "return" (or "pure") and counit is known as "extract". • The unit (or return) is a polymorphic function that creates a default box around a value of arbitrary type. The counit (or extract) does the reverse: it retrieves or produces a single value from a container. nLab: Geometry of Physics: Categories and Toposes: Example 1.61 There is a reflective subcategory-inclusion (Def. 1.60) {$$\textrm{Set} \overset {\overset{\pi_0}{\longleftarrow}} {\underset {\hookrightarrow}{\bot}} \textrm{Grp}$$} of the category of sets (Example 1.2) into the category Grpd (Example 1.16) of small groupoids (Example 1.10) where • the right adjoint full subcategory inclusion (Def. 1.19) sends a set S to the groupoid with set of objects being S, and the only morphisms being the identity morphisms on these objects (also called the discrete groupoid on S, but this terminology is ambiguous) • the left adjoint reflector sends a small groupoid {$𝒢\mathcal{G}$} to its set of connected components, namely to the set of equivalence classes under the equivalence relation on the set of objects, which regards two objects as equivalent, if there is any morphism between them.  Geometric realization Nerve operation nLab: nerve and realization Examples • Adjunction example: initial collapsing terminal 2.1.9 • Relational algebra by way of adjunctions, paper, video • Database queries: Selections, projections, equijoins. • Free functor from sets to commutative monoids generates bags (multisets). Adjoint functor collapses bags to sets. • Elias Zafiris. Category-theoretic analysis of the notion of complementarity for quantum systems. Adjunction between the category of quantum event algebras and the category of presheaves on Boolean event algebras. Establishes local or partial structural congruences between the quantum and Boolean kinds of event structure. If we consider a Boolean modeling functor {$M:B → L$} there exists precisely one corresponding uniquely defined, up to isomorphism, colimit-preserving functor {$\hat{M}:\textrm{Sets}^{B^{op}}\rightarrow \mathfrak{L}$} such that the following diagram commutes... The functor {$\hat{M}:=\textbf{L}$} is the left adjoint of the categorical adjunction between the categories {$\textrm{Sets}^{B^{op}}$} and {$\textbf{L}:\textrm{Sets}^{B^{op}}\rightarrow \mathfrak{L}$} whilst the right adjoint functor {$\textbf{R}:\mathfrak{L}\rightarrow\textrm{Sets}^{B^{op}}$} is physically interpreted as the Boolean realization functor of {$\mathfrak{L}$} in terms of variable local probing frames, functioning as natural contexts for measurement of observables. The existence of the functorial relations designate the fact that a quantum event algebra {$\mathfrak{L}$} in {$L$} can be expressed in terms of structured multitudes of interlocking local Boolean frames capable of carrying all the information encoded in the former. • Kleisli category associated to a monad T is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to the question Does every monad arise from an adjunction? The other extremal solution is the Eilenberg–Moore category. If, for fixed {$K:C\rightarrow D$} and {$E$}, the left {$\textrm{Lan}_K$} and right {$\textrm{Ran}_K$} Kan extensions of any functor {$F: C\rightarrow E$} along {$K$} exist, then these define left and right adjoints to the pre-composition functor {$K^∗: E^D\rightarrow E^C\$}. Riehl 6.1.5

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