Validate seed files

database/data/functors/abelianization.yaml

@@ -1,12 +1,13 @@
 id: abelianization
 name: abelianization functor for groups
 notation: $(-)^{\ab}$
-domain: Grp
-codomain: Ab
+associated:
+  domain: Grp
+  codomain: Ab
+  left_adjoint: null
+  right_adjoint: forget_abelian
 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

@@ -1,12 +1,13 @@
 id: binary_coproduct_sets
 name: binary coproduct functor on sets
 notation: $+$
-domain: SetxSet
-codomain: Set
+associated:
+  domain: SetxSet
+  codomain: Set
+  left_adjoint: null
+  right_adjoint: diagonal_sets
 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

@@ -1,12 +1,13 @@
 id: binary_product_sets
 name: binary product functor on sets
 notation: $\times$
-domain: SetxSet
-codomain: Set
+associated:
+  domain: SetxSet
+  codomain: Set
+  left_adjoint: diagonal_sets
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: brauer_group
 name: Brauer group functor
 notation: $\Br$
-domain: Fld
-codomain: Ab
+associated:
+  domain: Fld
+  codomain: Ab
+  left_adjoint: null
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: continuous_functions
 name: functor of continuous functions
 notation: $C$
-domain: Top_op
-codomain: CAlg(R) # TODO: specify that R is IR
+associated:
+  domain: Top_op
+  codomain: CAlg(R) # TODO: specify that R is IR
+  left_adjoint: null
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: countable_copower_sets
 name: countable copower functor on sets
 notation: $\IN \times (-)$
-domain: Set
-codomain: Set
+associated:
+  domain: Set
+  codomain: Set
+  left_adjoint: null
+  right_adjoint: sequences_sets
 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

@@ -1,12 +1,13 @@
 id: diagonal_sets
 name: binary diagonal functor on the category of sets
 notation: $\Delta$
-domain: Set
-codomain: SetxSet
+associated:
+  domain: Set
+  codomain: SetxSet
+  left_adjoint: binary_coproduct_sets
+  right_adjoint: binary_product_sets
 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

@@ -1,12 +1,13 @@
 id: discrete_topology
 name: discrete topology functor
 notation: $D$
-domain: Set
-codomain: Top
+associated:
+  domain: Set
+  codomain: Top
+  left_adjoint: null
+  right_adjoint: forget_topology
 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

@@ -1,12 +1,13 @@
 id: doubling_sets
 name: doubling functor on sets
 notation: $2(-)$
-domain: Set
-codomain: Set
+associated:
+  domain: Set
+  codomain: Set
+  left_adjoint: null
+  right_adjoint: squaring_sets
 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

@@ -1,12 +1,13 @@
 id: empty_sets
 name: empty functor to the category of sets
 notation: $!_{\Set}$
-domain: '0'
-codomain: Set
+associated:
+  domain: '0'
+  codomain: Set
+  left_adjoint: null
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: enveloping_group
 name: enveloping group functor
 notation: $F_{\Mon,\Grp}$
-domain: Mon
-codomain: Grp
+associated:
+  domain: Mon
+  codomain: Grp
+  left_adjoint: null
+  right_adjoint: forget_inverses
 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

@@ -1,12 +1,13 @@
 id: forget_abelian
 name: forgetful functor from abelian groups to groups
 notation: $U_{\Ab,\Grp}$
-domain: Ab
-codomain: Grp
+associated:
+  domain: Ab
+  codomain: Grp
+  left_adjoint: abelianization
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_addition
 name: forgetful functor from rings to monoids
 notation: $U_{\Ring,\Mon}$
-domain: Ring
-codomain: Mon
+associated:
+  domain: Ring
+  codomain: Mon
+  left_adjoint: monoid_ring
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_commutative
 name: forgetful functor from commutative rings to rings
 notation: $U_{\CRing,\Ring}$
-domain: CRing
-codomain: Ring
+associated:
+  domain: CRing
+  codomain: Ring
+  left_adjoint: null # TODO: add the left adjoint to the database
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_finite
 name: forgetful functor from finite sets to sets
 notation: $U_{\FinSet,\Set}$
-domain: FinSet
-codomain: Set
+associated:
+  domain: FinSet
+  codomain: Set
+  left_adjoint: null
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_finite_abelian_group
 name: forgetful functor from finite abelian groups to abelian groups
 notation: $U_{\FinAb, \Ab}$
-domain: FinAb
-codomain: Ab
+associated:
+  domain: FinAb
+  codomain: Ab
+  left_adjoint: null
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_finite_group
 name: forgetful functor from finite groups to groups
 notation: $U_{\FinGrp, \Grp}$
-domain: FinGrp
-codomain: Grp
+associated:
+  domain: FinGrp
+  codomain: Grp
+  left_adjoint: null
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_group
 name: forgetful functor for groups
 notation: $U_{\Grp}$
-domain: Grp
-codomain: Set
+associated:
+  domain: Grp
+  codomain: Set
+  left_adjoint: free_group
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_group_pointed
 name: forgetful functor from groups to pointed sets
 notation: $U_{\Grp,\Set_*}$
-domain: Grp
-codomain: Set_*
+associated:
+  domain: Grp
+  codomain: Set_*
+  left_adjoint: null
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_hausdorff
 name: forgetful functor from Hausdorff spaces to topological spaces
 notation: $U_{\Haus,\Top}$
-domain: Haus
-codomain: Top
+associated:
+  domain: Haus
+  codomain: Top
+  left_adjoint: null # TODO: add the Hausdorff reflection functor
+  right_adjoint: null
 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

@@ -1,12 +1,13 @@
 id: forget_inverses
 name: forgetful functor from groups to monoids
 notation: $U_{\Grp,\Mon}$
-domain: Grp
-codomain: Mon
+associated:
+  domain: Grp
+  codomain: Mon
+  left_adjoint: enveloping_group
+  right_adjoint: group_units
 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

@@ -1,12 +1,13 @@
 id: forget_ring
 name: forgetful functor for rings
 notation: $U_{\Ring}$
-domain: Ring
-codomain: Set
+associated:
+  domain: Ring
+  codomain: Set
+  left_adjoint: null
+  right_adjoint: null
 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