Refactor mapped structures

database/data/functors/abelianization.yaml

@@ -6,6 +6,7 @@ codomain: Ab
 description: This functor maps a group $G$ to its abelianization $G^{\ab} \coloneqq G/[G,G]$.
 nlab_link: https://ncatlab.org/nlab/show/abelianization
 left_adjoint: null
+right_adjoint: forget_abelian
 
 tags:
   - algebra

database/data/functors/binary_coproduct_sets.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a pair of sets $(X,Y)$ to their coproduct $X + Y$. It is an example of a right-invertible left adjoint functor which is not a reflector.
 nlab_link: null
 left_adjoint: null
+right_adjoint: diagonal_sets
 
 tags:
   - set theory

database/data/functors/binary_product_sets.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a pair of sets $(X,Y)$ to their product $X \times Y$. It is an example of a right-invertible right adjoint functor which is not a coreflector.
 nlab_link: null
 left_adjoint: diagonal_sets
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/brauer_group.yaml

@@ -6,6 +6,7 @@ codomain: Ab
 description: The Brauer group $\Br(K)$ of a field $K$ consists of equivalence classes of central simple algebras over $K$, where $A \sim B$ iff $A \otimes_K M_n(K) \cong B \otimes_K M_n(K)$ for some $n \geq 0$. The group structure is given by $[A] \cdot [B] \coloneqq [A \otimes_K B]$, $1 \coloneqq [K]$ and $[A]^{-1} \coloneqq [A^{\op}]$. A homomorphism $K \to L$ induces the homomorphism $\Br(K) \to \Br(L)$ defined by $[A] \mapsto [A \otimes_K L]$.
 nlab_link: https://ncatlab.org/nlab/show/Brauer+group
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/continuous-functions.yaml

@@ -6,6 +6,7 @@ codomain: CAlg(R) # TODO: specify that R is IR
 description: 'This functor maps a topological space $X$ to the commutative $\IR$-algebra $C(X)$ of continuous functions $X \to \IR$. A continuous map $f : X \to Y$ is mapped to the algebra homomorphism $f^* : C(Y) \to C(X)$, $u \mapsto u \circ f$.'
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - topology

database/data/functors/countable_copower_sets.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a set $X$ to the product $\IN \times X$, which can also be seen as the copower $\IN \otimes X = \coprod_{n \in \IN} X$. It is an example of a polynomial functor.
 nlab_link: null
 left_adjoint: null
+right_adjoint: sequences_sets
 
 tags:
   - set theory

database/data/functors/diagonal_sets.yaml

@@ -6,6 +6,7 @@ codomain: SetxSet
 description: 'Every category $\C$ has a (binary) diagonal functor $\Delta : \C \to \C^2$, $X \mapsto (X,X)$. Here, we specify that $\C$ is the category of sets.'
 nlab_link: https://ncatlab.org/nlab/show/diagonal+functor
 left_adjoint: binary_coproduct_sets
+right_adjoint: binary_product_sets
 
 tags:
   - set theory

database/data/functors/discrete_topology.yaml

@@ -6,6 +6,7 @@ codomain: Top
 description: This functor maps a set $X$ to the discrete topological space $D(X) \coloneqq (X, P(X))$ in which every subset is open. It is a typical example of a fully faithful functor that preserves finite but does not preserve infinite products.
 nlab_link: https://ncatlab.org/nlab/show/discrete+and+indiscrete+topology
 left_adjoint: null
+right_adjoint: forget_topology
 
 tags:
   - topology

database/data/functors/doubling_sets.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a set $X$ to its double $2 X \coloneqq X + X$. It is a simple example of a polynomial functor.
 nlab_link: null
 left_adjoint: null
+right_adjoint: squaring_sets
 
 tags:
   - set theory

database/data/functors/empty_sets.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: 'Every category $\C$ has a unique functor $!_{\C} : \varnothing \to \C$. Here, we specify $\C = \Set$, but most of the properties do not depend on the choice of $\C$, as long as $\C$ is non-empty. This is the simplest example of a functor to $\Set$ that is both continuous and cocontinuous, but is neither representable nor a left or right adjoint.'
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/enveloping_group.yaml

@@ -6,6 +6,7 @@ codomain: Grp
 description: 'This functor maps a monoid $M$ to the group $F(M)$ that is equipped with a universal homomorphism $i_M : M \to F(M)$. It is called the (universal) enveloping group or the group completion of $M$; in the commutative case, it is known as the Grothendieck group of $M$. As a possible construction of $F(M)$, take the free group on generators $\underline{m}$ for $m \in M$ subject to the relations $\underline{1} = 1$ and $\underline{m \cdot n} = \underline{m} \cdot \underline{n}$.'
 nlab_link: https://ncatlab.org/nlab/show/free+functor
 left_adjoint: null
+right_adjoint: forget_inverses
 
 tags:
   - algebra

database/data/functors/forget_abelian.yaml

@@ -6,6 +6,7 @@ codomain: Grp
 description: This functor maps an abelian group to itself, considered merely as a group.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: abelianization
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_addition.yaml

@@ -6,6 +6,7 @@ codomain: Mon
 description: This functor maps a ring to its underlying multiplicative monoid, which as "forgotten" the addition of the ring.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: monoid_ring
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_commutative.yaml

@@ -6,6 +6,7 @@ codomain: Ring
 description: This is the inclusion functor $\CRing \hookrightarrow \Ring$ that maps a commutative ring to itself, regarded merely as a ring.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null # TODO: add the left adjoint to the database
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_finite.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor is the inclusion functor $\FinSet \hookrightarrow \Set$ mapping a finite set to itself. It can also be regarded as a forgetful functor since it makes finite sets "forget" their finiteness. The functor is a basic example of a representable functor which is not a right adjoint.
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/forget_finite_abelian_group.yaml

@@ -6,6 +6,7 @@ codomain: Ab
 description: 'This is the inclusion functor $\FinAb \hookrightarrow \Ab$ that maps a finite abelian group to itself, regarded as an abelian group that has "forgotten" that it is finite. It provides an example of a fully faithful functor that is neither finitary nor cofinitary.'
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_finite_group.yaml

@@ -6,6 +6,7 @@ codomain: Grp
 description: 'This is the inclusion functor $\FinGrp \hookrightarrow \Grp$. It can also be viewed as a forgetful functor that forgets the property of being finite. Among other things, it provides an example of a fully faithful functor that is neither finitary nor cofinitary.'
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_group.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a group $G$ to its underlying set $U_{\Grp}(G)$.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: free_group
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_group_pointed.yaml

@@ -6,6 +6,7 @@ codomain: Set_*
 description: This functor maps a group $G$ to its underlying pointed set $U_{\Grp,\Set_*}(G)$, whose base point is the identity element of $G$. It is an example of an essentially surjective functor which is not right-invertible.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_hausdorff.yaml

@@ -6,6 +6,7 @@ codomain: Top
 description: This is the inclusion functor $\Haus \hookrightarrow \Top$ that maps a Hausdorff space to itself. It can also be viewed as a forgetful functor, since Hausdorff spaces "forget" that they are Hausdorff.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null # TODO: add the Hausdorff reflection functor
+right_adjoint: null
 
 tags:
   - topology

database/data/functors/forget_inverses.yaml

@@ -6,6 +6,7 @@ codomain: Mon
 description: This functor maps a group to its underlying monoid. We view groups as structured sets $(X,m,e,i)$ (consisting of a set, a multiplication, a neutral element, and an inverse operation), and monoids as structured sets $(X,m,e)$. This forgetful functor precisely maps $(X,m,e,i)$ to $(X,m,e)$. From this point of view, it does <i>not</i> merely forget a property; it forgets an operation. This perspective is useful in contexts where the inverse operation is no longer reducible to a property, for example, the forgetful functor from topological groups to topological monoids.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: enveloping_group
+right_adjoint: group_units
 
 tags:
   - algebra

database/data/functors/forget_ring.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a ring $R$ to its underlying set $U_{\Ring}(R)$.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_topology.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a topological space $X$ to its underlying set $U_{\Top}(X)$.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: discrete_topology
+right_adjoint: indiscrete_topology
 
 tags:
   - topology

database/data/functors/forget_torsion.yaml

@@ -6,6 +6,7 @@ codomain: Ab
 description: 'This is the inclusion functor $\TorsAb \hookrightarrow \Ab$. It can also be viewed as a forgetful functor that forgets the property of being torsion. It is a typical example of a fully faithful functor that preserves finite products but does not preserve infinite products.'
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null # we only have the torsion functor Ab -> Ab in the database, not Ab -> TorsAb
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_torsion_free.yaml

@@ -6,6 +6,7 @@ codomain: Ab
 description: 'This is the inclusion functor $\TorsFreeAb \hookrightarrow \Ab$. It can also be seen as a forgetful functor which forgets the property of being torsion-free. The functor provides a typical example of a fully faithful functor that does not preserve coequalizers and does not preserve epimorphisms.'
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/forget_vector.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a vector space $V$ (over a fixed field $K$) to its underlying set $U_{\Vect}(V)$.
 nlab_link: https://ncatlab.org/nlab/show/forgetful+functor
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/free_group.yaml

@@ -6,6 +6,7 @@ codomain: Grp
 description: This functor maps a set $X$ to the free group $F_{\Grp}(X)$ on that set. In the proofs, we abbreviate $F \coloneqq F_{\Grp}$.
 nlab_link: https://ncatlab.org/nlab/show/free+functor
 left_adjoint: null
+right_adjoint: forget_group
 
 tags:
   - algebra

database/data/functors/group_units.yaml

@@ -6,6 +6,7 @@ codomain: Grp
 description: This functor maps a monoid $M$ to its group of units $M^{\times}$, consisting of pairs $(a,b) \in M^2$ satisfying $ab=ba=1$. Equivalently, it takes the submonoid of invertible elements of $M$, equipped with the inverse operation.
 nlab_link: https://ncatlab.org/nlab/show/group+of+units
 left_adjoint: forget_inverses
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/id_Set.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: 'Every category $\C$ has an identity functor $\id_{\C} : \C \to \C$. Here, we specify that $\C$ is the category of sets.'
 nlab_link: https://ncatlab.org/nlab/show/identity+functor
 left_adjoint: id_Set
+right_adjoint: id_Set
 
 tags:
   - set theory

database/data/functors/inclusion_ordinals.yaml

@@ -6,6 +6,7 @@ codomain: On
 description: 'This is the inclusion map from the partially ordered set $(\IN \cup \{\infty\},\leq)$ (considered as a thin category as usual) into the partially ordered collection $(\On,\leq)$, where we map $\infty$ to the ordinal $\omega$. It is an example of a functor that preserves binary products, but not terminal objects.'
 nlab_link: https://ncatlab.org/nlab/show/identity+functor
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/indiscrete_topology.yaml

@@ -6,6 +6,7 @@ codomain: Top
 description: This functor maps a set $X$ to the indiscrete topological space $I(X) \coloneqq (X, \{\varnothing,X\})$ in which only the empty set and $X$ are open.
 nlab_link: https://ncatlab.org/nlab/show/discrete+and+indiscrete+topology
 left_adjoint: forget_topology
+right_adjoint: null
 
 tags:
   - topology

database/data/functors/modulo-p.yaml

@@ -6,6 +6,7 @@ codomain: Ab
 description: This functor maps an abelian group $A$ to the quotient $T^p(A) \coloneqq A/pA$, where $p$ is a fixed prime number. This group can also be represented as $A \otimes \IZ/p$.
 nlab_link: null
 left_adjoint: null
+right_adjoint: p-torsion
 
 tags:
   - algebra

database/data/functors/monoid_ring.yaml

@@ -6,6 +6,7 @@ codomain: Ring
 description: This functor maps a monoid $M$ to the monoid ring $\IZ[M]$, which consists of finite sums of elements in $M$.
 nlab_link: https://ncatlab.org/nlab/show/group+algebra
 left_adjoint: null
+right_adjoint: forget_addition
 
 tags:
   - algebra

database/data/functors/morphism_endpoints_inclusion.yaml

@@ -6,6 +6,7 @@ codomain: walking_morphism
 description: This is the functor that embeds the discrete category $\{0,1\}$ into the walking morphism $\{0 \to 1\}$. It provides an example of a faithful functor that is full on isomorphisms but not full.
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - category theory

database/data/functors/nerve.yaml

@@ -6,6 +6,7 @@ codomain: sSet
 description: The nerve of a small category $\C$ is the simplicial set $N(\C)$ whose $n$-simplices are chains of morphisms $X_0 \to \cdots \to X_n$. Among other things, it provides an example of a fully faithful functor that does not preserve regular epimorphisms.
 nlab_link: https://ncatlab.org/nlab/show/nerve
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - category theory

database/data/functors/opposite_category.yaml

@@ -6,6 +6,7 @@ codomain: Cat
 description: 'This functor maps a small category $\C$ to its opposite category $\C^{\op}$ and a functor $F : \C \to \D$ to the opposite functor $F^{\op} : \C^{\op} \to \D^{\op}$.'
 nlab_link: https://ncatlab.org/nlab/show/opposite+category
 left_adjoint: opposite_category
+right_adjoint: opposite_category
 
 tags:
   - category theory

database/data/functors/opposite_monoid.yaml

@@ -6,6 +6,7 @@ codomain: Mon
 description: 'This functor maps a monoid $M$ to its opposite monoid $M^{\op}$ which has the multiplication $a *^{\op} b \coloneqq a * b$. A monoid homomorphism $f : M \to N$ is also a monoid homomorphism $f^{\op} : M^{\op} \to N^{\op}$.'
 nlab_link: https://ncatlab.org/nlab/show/opposite+magma
 left_adjoint: opposite_monoid
+right_adjoint: opposite_monoid
 
 tags:
   - algebra

database/data/functors/p-torsion.yaml

@@ -9,6 +9,7 @@ description: >-
   where $p$ is a fixed prime number. This group can also be represented as $\HomInternal(\IZ/p,A)$.
 nlab_link: null
 left_adjoint: modulo-p
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/pi_0.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a topological space $X$ to its set $\pi_0(X)$ of path components. Thus, $\pi_0(X) = U(X) / {\sim}$, where $U(X)$ is the underlying set and $x \sim y$ when there is a path from $x$ to $y$.
 nlab_link: https://ncatlab.org/nlab/show/connected+space
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - topology

database/data/functors/pi_1.yaml

@@ -6,6 +6,7 @@ codomain: Grp
 description: The fundamental group $\pi_1(X,x_0)$ of a pointed topological space $(X,x_0)$ is the group of homotopy classes of loops at $x_0$. The group operation is concatenation of paths. For example, we have $\pi_1(S^1,1) \cong \IZ$ (see Hatcher's <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html" target="_blank">Algebraic Topology</a>, Theorem 1.7).
 nlab_link: https://ncatlab.org/nlab/show/fundamental+group
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - topology

database/data/functors/power_set_contravariant.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: 'This functor $P_{\forall}$ maps a set $X$ to its power set $P(X)$ and a map of sets $f : X \to Y$ to the induced preimage operator $f^* : P(Y) \to P(X)$.'
 nlab_link: https://ncatlab.org/nlab/show/power+set
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/power_set_covariant.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: 'This functor $P_{\exists}$ maps a set $X$ to its power set $P(X)$ and a map of sets $f : X \to Y$ to the induced image operator $f_* : P(X) \to P(Y)$.'
 nlab_link: https://ncatlab.org/nlab/show/power+set
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/rational_product.yaml

@@ -6,6 +6,7 @@ codomain: Top
 description: This functor maps a topological space $X$ to the topological space $X \times \IQ$, where $\IQ \subseteq \IR$ carries the usual topology. It is a typical example of a functor that preserves epimorphisms but not regular epimorphisms.
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - topology

database/data/functors/ring_idempotents.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor sends a ring $R$ to its set $\Id(R)$ of idempotent elements. A ring homomorphism $R \to S$ restricts to a map $\Id(R) \to \Id(S)$. Among other things, it provides an example of a representable functor that does not preserve regular epimorphisms.
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/sequences_sets.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a set $X$ to the countable power $X^{\IN}$, i.e. the set of sequences in $X$. It is an example of a polynomial functor. It is also an example of a monadic functor for which the crude monadicity theorem does not apply.
 nlab_link: null
 left_adjoint: countable_copower_sets
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/simple_group_probing.yaml

@@ -12,6 +12,7 @@ description: >-
   This is the canonical example of a continuous functor $\Grp \to \Set$ that is not representable, and not a right adjoint.
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra

database/data/functors/span_endpoints_inclusion.yaml

@@ -6,6 +6,7 @@ codomain: walking_span
 description: This is the functor that embeds the discrete category $\{1,2\}$ into the walking span $\{1 \leftarrow 0 \rightarrow 2\}$. Among other things, it provides an example of a fully faithful functor which is not left-invertible.
 nlab_link: null
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - category theory

database/data/functors/squaring_sets.yaml

@@ -6,6 +6,7 @@ codomain: Set
 description: This functor maps a set $X$ to its square $X^2$. It is a simple example of a polynomial functor.
 nlab_link: null
 left_adjoint: doubling_sets
+right_adjoint: null
 
 tags:
   - set theory

database/data/functors/stone-cech-compactification.yaml

@@ -9,6 +9,7 @@ description: >-
   Among other things, this functor provides an example of a reflector that does not preserve binary products.
 nlab_link: https://ncatlab.org/nlab/show/Stone-%C4%8Cech+compactification
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - topology

database/data/functors/torsion.yaml

@@ -8,6 +8,7 @@ description: >-
   $$T(A) \coloneqq \{a \in A : \exists n \geq 1 \, (na = 0)\}.$$
 nlab_link: https://ncatlab.org/nlab/show/torsion+subgroup
 left_adjoint: null
+right_adjoint: null
 
 tags:
   - algebra