# A note on the non-existence of functors

Emmanuel Dror Farjoun<EMAIL_ADDRESS>,  Sergei O. Ivanov
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Anatolii Zaikovskii
<EMAIL_ADDRESS>
###### Abstract.

We consider several cases of non-existence theorems for functors. For example,
there are no nontrivial functors from the category of groups (or the category
of pointed sets, or vector spaces) to any small category. See Theorem 2.3.

Another kind of nonexistence is that of (co-)augmented functors. For example,
every augmented functor from groups to abelian groups, $AG\to G$ is trivial,
i.e. has a trivial augmentation map. Every surjective co-augmented functor
$G\to PG$ from groups to perfect groups or to free groups is also trivial.

The second named author is supported by Yanqi Lake Beijing Institute of
Mathematical Sciences and Applications (BIMSA)

##  1\. Introduction

It is natural to expect that many large and ”rich” categories should have only
trivial functors to any small category. A typical example proven below is the
statement that any functor from the category of groups to the category of
finite groups is trivial, having a constant value on objects and maps. Here we
show that under a mild connectedness condition, any functor from a category
$\mathpzc{C},$ with (any small) products, to a small category is a constant
functor. Further, see 2.9, any functor from the category ${\sf Gr}^{\sf
count}$ of countable groups, to the category ${\sf Gr}^{\sf f.g.},$ of
finitely generated groups, is constant.

In a slightly different vein, consider the impossibility of certain
(co-)augmented functors, such as the abelianization $G\to G_{\sf ab},$ on the
category of groups, the commutator subgroup $[G,G]\to G$ or universal cover
$\widetilde{X}\to X,$ on the category of pointed topological spaces. It turns
out that the values of (co-)augmented functors cannot lie in certain sub-
categories, in case these are closed (or not closed) under limits or co-
limits.

Speaking suggestively (and empirically), let $\mathcal{V}$ be a given proper
subcategory of groups which is not closed under either limits or co-limits.
For example, abelian groups, free groups, or perfect groups. Then the
slogan—for groups—should be that (under some mild conditions, e.g. excluding
the identity functor) there is a non-trivial injective augmented functor
$F:{\sf Gr}\to\mathcal{V},$ namely, $F(G)\hookrightarrow G,$ only in case
there is no surjective co-augmented functor $\Phi:{\sf Gr}\to\mathcal{V},$
namely $G\twoheadrightarrow\Phi(G),$ and vise versa. Thus, since there is a
canonical perfect subgroup $PG\subseteq G,$ that gives a non-trivial augmented
functor, all surjective co-augmented functors from the category of groups to
that of perfect groups: $P:{\sf Gr}\to{\sf PerfGr},$ are trivial, i.e with the
trivial co-augmentation map $G\to PG$. Next, since there is a surjective co-
augmented functor ${\sf Gr}\to{\sf Ab},$ there are only trivial augmented ones
$AG\to G$ to abelian groups. Further, since there is a non-trivial augmented
functor $FG\to G$ where $FG$ is a free group (the free group generated by the
elements of $G,$) any (surjective?) co-augmented functor $G\to\Phi(G)$ with
$\Phi(G)$ a free group should have a trivial co-augmentation. This general
slogan is elaborated upon below with several partial results.

In general, of course, there are many endo-functors
$\mathpzc{C}\to\mathpzc{C},$ both augmented and co-augmented or even from
$\mathpzc{C}$ to a given large subcategory $\mathpzc{D}\subset\mathpzc{C},$
usually given by adjoint functors to the inclusion.

This brief account leaves many problems open since e.g. for any subcategory
$\mathpzc{D}$ of groups, one can ask whether all ( or augmented, co-augmented)
functors ${\sf Gr}\to\mathpzc{D}$ are constant, or otherwise trivial. For
example, $\mathpzc{D}$ can be that of simple groups. In this direction,
according to the results below, if, in contrast to the domain category
$\mathpzc{C},$ the range subcategory $\mathpzc{D},$ does not have ”large
products” or ”large sums” then one might guess that all functors
$\mathpzc{C}\to\mathpzc{D}$ are trivial. See 2.3 and 2.5 below.

It is natural to expect that analog non-existence propositions are valid in
other categories, such as the category of spaces. A small step was taken in
[3] where the impossibility of certain (co-)augmented is considered.

For example, it is natural to expect that in no non-trivial co-augmented
functor $L:Spaces_{*}\to LoopSpaces_{*},$ namely, $FX\to X$ where $FX$ lies in
the category of loop spaces. But we have made only suggestive progress in that
direction.

##  2\. Functors from large to small categories

In preparation for the propositions below, we establish some notations. Let
$\mathpzc{C}$ be a category. For any family of objects ${\bf
c}=(c_{\alpha})_{\alpha\in A}$ such that the product
$\prod_{\alpha}c_{\alpha}$ exists, and any $\beta\in A$ we denote by

(2.1) $p^{\bf c}_{\beta}:\prod_{\alpha}c_{\alpha}\longrightarrow c_{\beta}$

the canonical projection from the product. For any object $c^{\prime}$ we
denote by

(2.2)
$t:\prod_{\alpha}\mathpzc{C}(c^{\prime},c_{\alpha})\longrightarrow\mathpzc{C}\left(c^{\prime},\prod_{\alpha}c_{\alpha}\right)$

the standard bijection, which is uniquely defined by the formula

(2.3) $p^{\bf c}_{\beta}\circ t(\varphi)=\varphi_{\beta}$

for any family $\varphi=(\varphi_{\alpha}:c\to c_{\alpha})_{\alpha\in A}$ and
any $\beta\in A.$

Let $\mathpzc{C},\mathpzc{D}$ be two categories, and let
$F:\mathpzc{C}\to\mathpzc{D}$ be a functor.

Assume that ${\bf c}=(c_{\alpha})_{\alpha\in A}$ is a family of objects of
$\mathpzc{C}$ such that the products $\prod_{\alpha}c_{\alpha}$ and
$\prod_{\alpha}F(c_{\alpha})$ exist. Consider the standard assembly morphism

(2.4) $v^{\bf
c}:F\left(\prod_{\alpha}c_{\alpha}\right)\longrightarrow\prod_{\alpha}F(c_{\alpha})$

such that $p^{F({\bf c})}_{\beta}\circ v^{{\bf c}}=F(p^{{\bf c}}_{\beta})$ for
any $\beta\in A.$ Here we denote by $F({\bf c})$ the family
$(F(c_{\alpha}))_{\alpha\in A}.$

The following is a key observation, saying that for any
$F:\mathpzc{C}\to\mathpzc{D}$ as above, certain natural maps between ”large
products” factors through a mapping set inside $\mathpzc{D}:$

######  Lemma 2.1.

For any object $c^{\prime}$ of $\mathpzc{C}$ and any family ${\bf
c}=(c_{\alpha})_{\alpha\in A}$ of objects of $\mathpzc{C}$ such that the
products $\prod_{\alpha}c_{\alpha}$ and $\prod_{\alpha}F(c_{\alpha})$ exist
the diagram

${\prod\limits_{\alpha}\mathpzc{C}(c^{\prime},c_{\alpha})}$${\prod\limits_{\alpha}\mathpzc{D}(F(c^{\prime}),F(c_{\alpha}))}$${\mathpzc{C}\left(c^{\prime},\prod\limits_{\alpha}c_{\alpha}\right)}$${\mathpzc{D}\left(F(c^{\prime}),F\left(\prod\limits_{\alpha}c_{\alpha}\right)\right)}$${\mathpzc{D}(F(c^{\prime}),\prod\limits_{\alpha}F(c_{\alpha}))}$$\leftarrow$$\rightarrow$$\scriptstyle{\prod
F}$$\leftarrow$$\rightarrow$$\scriptstyle{t}$$\scriptstyle{\cong}$$\leftarrow$$\rightarrow$$\scriptstyle{t}$$\scriptstyle{\cong}$$\leftarrow$$\rightarrow$$\scriptstyle{F}$$\leftarrow$$\rightarrow$$\scriptstyle{v^{\bf
c}_{*}}$

is commutative. In other words, for any family of morphisms
$\varphi=(\varphi_{\alpha}:c^{\prime}\to c_{\alpha})_{\alpha\in A}$ we have

$v^{\bf c}\circ F(t(\varphi))=t({F(\varphi)}).$

###### Proof.

It follows from the defining property of $t(F(\varphi))$ (2.3) and the formula

(2.5) $p^{F({\bf c})}_{\beta}\circ v^{\bf c}\circ F(t(\varphi))=F(p^{\bf
c}_{\beta}\circ t(\varphi))=F(\varphi_{\beta}).$

∎

The following non-existence results are quick implications of the above lemma:
A category $\mathpzc{C}$ is called _strongly connected_ if for any two objects
$c,c^{\prime}\in\mathpzc{C}$ the hom-set $\mathpzc{C}(c,c^{\prime})$ is not
empty.

######  Theorem 2.2.

Let $\kappa$ be a cardinal, $\mathpzc{C},\mathpzc{D}$ be categories with
products indexed by sets of cardinality $\kappa,$ and let
$F:\mathpzc{C}\to\mathpzc{D}$ be a functor. Assume that
$|\mathpzc{D}(F(c^{\prime}),F(c))|\leq\kappa$ for any two objects
$c^{\prime},c$ of $\mathpzc{C}.$ Then the following holds

  1. (1)

for any two parallel morphisms $f,g:c^{\prime}\to c$ of $\mathpzc{C}$ we have
$F(f)=F(g);$

  2. (2)

if $\mathpzc{C}$ is strongly connected, then $F$ is isomorphic to a constant
functor.

###### Proof.

(1) Assume the contrary that there exists morphisms $f,g:c^{\prime}\to c$ such
that $F(f)\neq F(g).$ Take a set $A$ such that $|A|=\kappa$ and consider a
constant family ${\bf c}=(c_{\alpha})_{\alpha\in A},$ where $c_{\alpha}=c.$
Since the image of the map
$\mathpzc{C}(c^{\prime},c_{\alpha})\to\mathpzc{D}(F(c^{\prime}),F(c_{\alpha}))$
has at least two elements, the cardinality of the image of the map
$\prod\mathpzc{C}(c^{\prime},c_{\alpha})\to\prod\mathpzc{D}(F(c^{\prime}),F(c_{\alpha}))$
has cardinality at least $2^{\kappa}>\kappa.$ On the other hand, by Lemma 2.1
the map
$\prod\mathpzc{C}(c^{\prime},c_{\alpha})\to\prod\mathpzc{D}(F(c^{\prime}),F(c_{\alpha}))$
factors through the set $\mathpzc{D}(F(c^{\prime}),F\left(\prod
c_{\alpha}\right))$ whose cardinality $\leq\kappa$ by the assumption, which is
impossible.

(2) Since $\mathpzc{C}$ is strongly connected, it is sufficient to prove that
for any morphism $f:c^{\prime}\to c$ the morphism $F(f)$ is an isomorphism.
Take a morphism $g:c\to c^{\prime}.$ By (1) we obtain $F(g)F(f)=F(gf)=F({\sf
id}_{c^{\prime}})={\sf id}_{F(c^{\prime})}$ and $F(f)F(g)=F(fg)=F({\sf
id}_{c})={\sf id}_{F(c)}.$ Then $F(f)$ is an isomorphism. ∎

######  Theorem 2.3.

Let $\mathpzc{C},\mathpzc{D}$ be categories and $F:\mathpzc{C}\to\mathpzc{D}$
be a functor. Assume that $\mathpzc{D}$ is equivalent to a small category and
there exists a cardinal $\kappa$ such that $\mathpzc{C}$ has products indexed
any set of cardinality $\kappa$ while $|\mathpzc{D}(d^{\prime},d)|\leq\kappa$
for all objects $d^{\prime},d$ of $\mathpzc{D}.$ Then the following holds

  1. (1)

for any two parallel arrows $f,g:c^{\prime}\to c$ we have $F(f)=F(g);$

  2. (2)

if $\mathpzc{C}$ is strongly connected, then $F$ is isomorphic to a constant
functor.

###### Proof.

Without loss of generality we can assume that $\mathpzc{D}$ is small. Consider
the category of presheaves $\hat{\mathpzc}{D}={\sf
Funct}(\mathpzc{D}^{op},{\sf Sets})$ and the Yoneda embedding
$Y:\mathpzc{D}\to\hat{\mathpzc}{D}.$ By the Yoneda lemma we obtain that

$|\hat{\mathpzc}{D}(Y(d^{\prime}),Y(d))|=|\mathpzc{D}(d^{\prime},d)|\leq\kappa$

for any objects $d^{\prime},d$ of $\mathpzc{D}.$ We also know that
$\hat{\mathpzc}{D}$ has all small products. Then we can apply Theorem 2.2 to
the functor $YF:\mathpzc{C}\to\hat{\mathpzc}{D}.$ The assertion follows from
the fact that $Y$ is fully faithful. ∎

######  Corollary 2.4 (cf. [8, Th.2.1]).

Any small category with small products is a preorder.

###### Proof.

Take $\mathpzc{C}=\mathpzc{D},$ $F={\sf Id}$ and $\kappa=|{\sf
Mor}(\mathpzc{C})|,$ and apply Theorem 2.3(1). ∎

######  Corollary 2.5.

Let $\mathpzc{C}$ be a strongly connected category with small products and
$\mathpzc{D}$ be a category that is equivalent to a small category. Then any
functor $\mathpzc{C}\to\mathpzc{D}$ is isomorphic to a constant functor.

######  Remark 2.6.

There are dual versions of all of the above statements in this section, where
products are replaced by coproducts.

######  Corollary 2.7.

Let $\mathpzc{C}$ be one of the following categories:

  * •

the category of non-empty sets;

  * •

the category of pointed sets;

  * •

the category of groups;

  * •

the category of modules over some ring;

and let $\mathpzc{D}$ be one of the following categories

  * •

finite sets;

  * •

finite groups;

  * •

countable groups;

  * •

finitely generated modules over a ring $R.$

Then any functor from $\mathpzc{C}$ to $\mathpzc{D}$ is isomorphic to a
constant functor.

######  Example 2.8.

Denote by $O$ the category, whose objects are the ordinals and morphisms are
canonical embeddings. So $O$ is a “big poset,” in fact a “big well ordered
set.” Then $O$ has small products defined by the infimum and small coproducts
defined by the supremum. However, $O$ is not strongly connected, and for any
ordinal $\alpha>0$ there is a non-constant functor to the small sub-category
$\alpha+1$ of $O:$

(2.6) $F:O\to\alpha+1,\hskip 28.45274ptF(\beta):={\sf min}(\beta,\alpha).$

Therefore, the assumption of being strongly connected in Corollary 2.5 is
essential.

######  Proposition 2.9.

Let ${\sf Gr}^{\sf count}$ be the category of countable groups and ${\sf
Gr}^{\sf f.g.}$ be the category of finitely generated groups. Any functor
${\sf Gr}^{\sf count}\to{\sf Gr}^{\sf f.g.}$ is isomorphic to a constant
functor.

###### Proof.

Countable free product of countable groups is countable. Hence ${\sf Gr}^{\sf
count}$ has countable coproducts. The hom-sets between finitely generated
groups are countable, because any homomorphism is uniquely defined by its
values of generators. Therefore, if we take $\kappa=\aleph_{0}$ and
$\mathpzc{C}={\sf Gr}^{\sf count},$ and $\mathpzc{D}={\sf Gr}^{\sf f.g.}$ and
use the dual version of the Theorem 2.2, we obtain that any functor ${\sf
Gr}^{\sf count}\to{\sf Gr}^{\sf f.g.}$ is isomorphic to a constant functor. ∎

For a cardinal $\kappa$ we denote by ${\sf Sets}^{\kappa}_{\neq\emptyset}$ the
category of non-empty sets of cardinality $\leq\kappa$ and by ${\sf
Sets}^{\kappa}_{*}$ the category of pointed sets of cardinality $\leq\kappa.$
Note that the categories ${\sf Sets}^{\kappa}_{\neq\emptyset}$ and ${\sf
Sets}^{\kappa}_{*}$ are strongly connected, while the category of all sets
${\sf Sets}$ is not strongly connected, because there is no a map from non-
empty set to the empty set.

######  Proposition 2.10.

Let $\kappa,\kappa^{\prime}$ be two cardinals such that $\kappa\geq
2^{\kappa^{\prime}}.$ Assume that $\kappa$ is infinite. Then all functors

${\sf Sets}^{\kappa}_{\neq\emptyset}\longrightarrow{\sf
Sets}^{\kappa^{\prime}}_{\neq\emptyset},\hskip 28.45274pt{\sf
Sets}^{\kappa}_{*}\longrightarrow{\sf Sets}^{\kappa^{\prime}}_{*}$

are isomorphic to constant functors.

###### Proof.

We prove the statement for the category of non-empty sets, the case of pointed
sets is similar. Since $\kappa$ is infinite, we have
$\kappa\times\kappa=\kappa,$ and hence, the disjoint union of non-empty sets
of cardinality $\leq\kappa$ indexed by a set of cardinality $\kappa$ has
cardinality $\leq\kappa.$ Hence ${\sf Sets}^{\kappa}_{\neq\emptyset}$ is
strongly connected and has coproducts indexed by sets of cardinality $\kappa.$
We claim that for any sets $X,Y$ of cardinality $\leq\kappa^{\prime}$ the set
of functions $X^{Y}$ has cardinality $\leq\kappa.$ If $\kappa^{\prime}$ is
finite, it is obvious. If $\kappa^{\prime}$ is infinite, then it is well known
that $(\kappa^{\prime})^{\kappa^{\prime}}=2^{\kappa^{\prime}}$ (see [9, XV.2],
[6, Ch.2, exersice 4, p.47]) and hence $|X^{Y}|\leq
2^{\kappa^{\prime}}\leq\kappa.$ The assertion follows from Theorem 2.2. ∎

Note, however, that not all functors from ”larger sets” to ”smaller sets” are
constant.

######  Example 2.11 (Functors from sets).

The category of all sets ${\sf Sets}$ is not strongly connected, because there
is no a map from a non-empty set to the empty set. So we can’t use Theorems
2.2, 2.3 for this category. Indeed, for any category $\mathpzc{D}$ and any its
morphism $\varphi:d^{\prime}\to d$ there is a functor

$F_{\varphi}:{\sf Sets}\longrightarrow\mathpzc{D}$

such that

$F_{\varphi}(X)=\begin{cases}d^{\prime},&X=\emptyset\\\
d,&X\neq\emptyset\end{cases},\hskip 28.45274ptF_{\varphi}(f:X\to
Y)=\begin{cases}\varphi,&X=\emptyset,\ Y\neq\emptyset\\\
1_{d^{\prime}},&X=Y=\emptyset\\\ 1_{d},&X\neq\emptyset,\
Y\neq\emptyset\end{cases}.$

If $\mathpzc{D}$ is small, using the fact that all functors ${\sf
Sets}_{\neq\emptyset}\to\mathpzc{D}$ are isomorphic to constant functors, it
is easy to check that all functors ${\sf Sets}\to\mathpzc{D}$ are isomorphic
to functors of the form $F_{\varphi}.$

######  Example 2.12.

Here for a natural number $n\geq 2$ we give an example of a functor

(2.7) $F:{\sf Sets}^{n}_{\neq\emptyset}\longrightarrow{\sf
Sets}^{2}_{\neq\emptyset}$

which is not isomorphic to a constant functor. There are two types of
morphisms in ${\sf Sets}^{n}_{\neq\emptyset}$: (0) functions with the image of
cardinality $<n;$ (1) bijections between sets of cardinality $n.$ A
composition of a morphism of type (0) with any other morphism from any side is
a morphism of type (0). The composition of two morphisms of type (1) is a
morphism of type (1). Denote by $f_{0}:\\{0,1\\}\to\\{0,1\\}$ the map sending
all elements to zero $f_{0}(x)=0.$ We define $F$ so that $F(X)=\\{0,1\\}$ for
any set $X;$ for any morphism $f$ of type (0) we set $F(f)=f_{0};$ for any
morphism $f$ of type (1) we set $F(f)={\sf id}_{\\{0,1\\}}.$ It is easy to
check that $F$ is a well-defined functor non-isomorphic to a constant functor.

##  3\. Subfunctors and quotient functors of the identity functor

In this section, we consider the nonexistence of nontrivial (co)-augmented
functors in the category of groups. In particular, it is shown here that for
any augmented functor $A,$ from the category of groups to that of abelian
groups, its augmentation $A(G)\to G,$ is the trivial homomorphism. This is a
special case of a proposition about functors to a proper subvariety of groups.
But for the variety of abelian groups, the proof is rather elementary.

A subgroup $H$ of a group $G$ is called _characteristic_ (denoted by
$H\triangleleft_{\sf ch}G$) if for any automorphism $\varphi:G\to G$ we have
$\varphi(H)\subseteq H.$ Any characteristic subgroup is normal because it is
invariant with respect to inner automorphisms. An advantage of characteristic
subgroups over normal subgroups is that $K\triangleleft_{\sf ch}H$ and
$H\triangleleft_{\sf ch}G$ implies $K\triangleleft_{\sf ch}G.$

Throughout this section, we denote by $F$ a given sub-functor of the identity
functor on the category of groups

(3.1) $F\subseteq{\sf Id}_{\sf Gr}.$

######  Lemma 3.1.

  1. (1)

$F(G)\triangleleft_{\sf ch}G$ for any $G;$

  2. (2)

for any $H\triangleleft_{\sf ch}F(G)$ we have $F(G)/H\subseteq F(G/H);$

  3. (3)

for any family of groups $(G_{\alpha})_{\alpha\in A}$ the standard assembly
map is injective:
$\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{$\ast$}}}_{\alpha\in
A}F(G_{\alpha})\subseteq
F(\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{$\ast$}}}_{\alpha\in
A}G_{\alpha}).$

  4. (4)

for any family of groups $(G_{\alpha})_{\alpha\in A}$ the standard assembly
map is injective: $F(\prod_{\alpha\in I}G_{\alpha})\subseteq\prod_{\alpha\in
A}F(G_{\alpha}).$

###### Proof.

(1) For any automorphism $\varphi$ of $G$ the map $F(\varphi)$ is an
automorphism of $F(G).$ Since $F(\varphi)$ is a restriction of $\varphi$ to a
subgroup, we obtain $\varphi(F(G))\subseteq F(G).$

(2) Since $H\triangleleft_{\sf ch}F(G)$ and $F(G)\triangleleft_{\sf ch}G,$ we
obtain that $H$ is normal in $G.$ Hence we can consider the projection $p:G\to
G/H.$ By the naturality of $F$ we obtain that $F(G/H)$ contains
$p(F(G))=F(G)/H.$

(3) The group
$\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{$\ast$}}}_{\alpha\in
A}F(G_{\alpha})$ is the subgroup of
$\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{$\ast$}}}_{\alpha\in
A}G_{\alpha}$ generated by the images of $F(G_{\alpha}).$ The assertion
follows.

(4) For any projection $p_{\beta}:\prod_{\alpha\in A}G_{\alpha}\to G_{\beta}$
we have $p_{\beta}(F(\prod_{\alpha\in A}G_{\alpha}))\subseteq F(G_{\beta}).$
The assertion follows. ∎

######  Corollary 3.2.

[cf. [5, Prop. 3.1, Prop 3.1*]]

  1. (1)

The class of groups $G$ such that $F(G)=G$ is closed with respect free
products and taking quotients.

  2. (2)

The class of groups $G$ such that $F(G)=0$ is closed with respect products and
taking subgroups.

######  Corollary 3.3.

If $F(\mathbb{Z})=\mathbb{Z},$ then $F={\sf Id}.$

###### Proof.

This follows from Corollary 3.2 (1) above. Any free group is a free product of
copies of $\mathbb{Z},$ and any group is a quotient of a free group. ∎

######  Theorem 3.4.

Let $\kappa$ be a cardinal number and $\mathcal{V}$ be a class of groups that
does not contain any simple group of cardinality $\geq\kappa.$ Then any sub-
functor of the identity functor $F\subseteq{\sf Id}$ with values in
$\mathcal{V}$ is trivial.

###### Proof.

By Lemma 3.1 $F(G)$ must be normal in $G$ for any group $G.$ By a theorem of
Noam Elkis [2] every group can be embedded into a simple group. It is easy to
see that we can strengthen this statement by saying that for any group $G$
there is a simple group $S$ such that $G\subseteq S$ and $|S|\geq\kappa.$
Since $F(S)$ is normal in $S$ and $S\notin\mathcal{V},$ we get $F(S)=1.$
Combining this with the inclusion $F(G)\subseteq F(S),$ we obtain $F(G)=1.$ ∎

######  Corollary 3.5.

If $F\subseteq{\sf Id}_{\sf Gr}$ is a subfunctor of the identity functor such
that $F(G)$ is abelian (or free, or nilpotent, or solvable) for every $G,$
then $F$ is trivial.

Next, consider augmented functors from the category of groups to any proper
variety of groups. A class of all groups satisfying a fixed system of identity
relations is called a _variety of groups_. It is also well known that a
variety of groups can be defined as a full subcategory of the category of
groups $\mathcal{V}\subseteq{\sf Gr}$ that is closed under small products,
taking images and taking subgroups. A variety is called proper if
$\mathcal{V}\neq{\sf Gr}.$ Examples of proper varieties of groups are abelian
groups, nilpotent groups of class $n$, metabelian groups, $n$-Burnside groups
(i.e. groups $G$ satisfying $g^{n}=1$ for all elements $g\in G$). It is clear
that the free group with two generators ${\sf Free}(x,y)$ is not an element of
any proper variety of groups because it contains a free subgroup with any
finite number of generators, and these generators don’t satisfy any identity
relation.

For any groups $G,H$ we denote by $G*H$ the free product and set

(3.2) $G\square H:={\rm Ker}(G*H\to G\times H).$

The following statement is well-known and the standard reference is [7] (see
also [4]).

######  Proposition 3.6 (Levi [7]).

For any groups $G,H$ the group $G\square H$ is a free group freely generated
by the commutators $[g,h]$ for $g\in G\setminus\\{1\\},h\in H\setminus\\{1\\}$

$G\square H\cong{\sf Free}((G\setminus\\{1\\})\times(H\setminus\\{1\\})).$

######  Theorem 3.7.

Let $\mathcal{V}$ be a proper variety of groups and $F\subseteq{\sf Id}_{\sf
Gr}$ be a sub-functor of the identity functor on the category of groups.
Assume that for every group $G,$ one has $F(G)\in\mathcal{V}.$ Then $F(G)$ is
the trivial group for every group $G.$

###### Proof.

Assume the contrary that $F(G)\neq 1$ for some $G.$ Set $G^{\prime}=G*G$ and
$G^{\prime\prime}=G^{\prime}*G^{\prime}.$ By Lemma 3.1(3) $F(G)*F(G)\subseteq
F(G^{\prime})$. Therefore $F(G^{\prime})$ is infinite. Using Lemma 3.1(3)
again we obtain that $F(G^{\prime})\square F(G^{\prime})\subseteq
F(G^{\prime})*F(G^{\prime})\subseteq F(G^{\prime\prime}).$ By Proposition 3.6,
we obtain that $F(G^{\prime\prime})$ contains an infinitely generated free
group. Hence $F(G^{\prime\prime})$ is not in $\mathcal{V},$ since the latter
is a proper variety, contrary to the assumption on $F.$ ∎

######  Theorem 3.8.

Let $\mathpzc{N}$ be a class of groups that do not contain groups isomorphic
to non-trivial finite cyclic groups $\mathbb{Z}/n$ and there exists a natural
$m\geq 2$ such that $\mathpzc{N}$ does not contain a group isomorphic to
$\mathbb{Z}^{m}$. Then there is no non-trivial quotient of the identity
functor taking values in $\mathpzc{N}.$

###### Proof.

We denote by $F$ a subfunctor of the identity functor and set $P(G)=G/F(G).$
We assume that $P(G)\in\mathpzc{N}.$ If $F(\mathbb{Z})=\mathbb{Z},$ then by
Corollary 3.3, $P$ is trivial. Assume that $F(\mathbb{Z})\neq\mathbb{Z}.$
Since, by the assumption, non-trivial finite cyclic groups are not in
$\mathpzc{N},$ we obtain $F(\mathbb{Z})=0.$ We denote by $m$ a natural number
$m\geq 2$ such that $\mathpzc{N}$ does not contain a group isomorphic to
$\mathbb{Z}^{m}.$ Since $F(\mathbb{Z})=0,$ Corollary 3.2 (2) implies that
$F(\mathbb{Z}^{m})=0$ and $P(\mathbb{Z}^{m})\cong\mathbb{Z}^{m},$ which makes
a contradiction. ∎

######  Corollary 3.9.

There is no non-trivial quotient of the identity functor on the category of
groups taking values in the subcategory of perfect groups (or free groups, or
non-abelian groups).

The above general statements are far from covering all cases of impossible
(co-)augmented functors into a subcategory of groups, modules etc. For
example, in the category of abelian groups while there is a co-augmented
functor $A\to A\otimes\mathbb{Q}$ to the subcategory of uniquely divisible
groups there is no such non-trivial injective augmented one $F(A)\subseteq A.$

While, when dropping the condition of injection, there is an augmented functor
to the category of uniquely divisible group:

${\sf Hom}(\mathbb{Q},A)\to A,\hskip 28.45274ptf\mapsto f(1)$

when $A$ is an abelian group in addition to the none-surjective co-augmented
one: $A\to A\otimes\mathbb{Q}.$

On the other hand, in the category of abelian groups, while there is an
augmented injective functor to the subcategory of (abelian) torsion groups,
there is no surjective co-augmented one, to this subcategory, etc.

Another simple example (leaving the proof to the reader): In the category of
sets, while there is a co-augmented composition functor $X\to F(X),$ where $F$
decomposes:

$Sets\longrightarrow Groups\overset{underlying}{\longrightarrow}Sets,$

e.g. the free group functor; there is no augmented one: $G(X)\to X$ such that
the functor $G$ factors, as above, through the category of groups.

##  4\. Impossible (co-)augmented functors in infinity categories

Several of the statements above have quite natural analogs-extensions to
statements about impossible functors on infinity categories such as the
category of spaces or spectra. Clearly, the above statements imply that there
are no functors from the (infinity) category of (pointed) spaces $S_{\star}$
to its full subcategory of spaces of finite type, or spaces $X$ with
$\pi_{i}(X)$ a finite group for all $i\leq 0$ –in the appropriate sense of
taking components and base points into account.

Looking at the statements about abelian groups it makes sense to expect that
for every augmented (composition) functor

$A:PointedSpaces\to LoopSpaces\overset{forget}{\longrightarrow}PointedSpaces$

the augmentation map $AX\to X$ is (coherently) null-homotopic. The same seems
to be true when one replaces ”loop spaces” with ”H-spaces.” Similarly, one
expects that there are no none-trivial co-augmented endo-functors $X\to F(X),$
on (pointed) spaces $F:S_{\star}\to S_{\star}$ with $F(X)$ an acyclic space
for all $X.$ One expects that given the existence of an augmented functor on
pointed ($CW$-) spaces (the universal cover) whose image lies in 1-connected
ones, there can be no such co-augmented one. Similarly, one strongly expects
that there is no non-trivial co-augmented functor $FX\to X$ on pointed spaces
whose image $FX$ is an $n-$truncated space for $n\geq 1.$

Let us consider connected pointed spaces $\mathcal{S_{*}}$. An augmented
functor $LX\to X,$ where $L$ takes values in the category of loop spaces,
gives, in particular, an augmented functor from groups to abelian groups. By
corollary 3.5 above, the map on the fundamental groups $\pi_{1}LX\to\pi_{1}X$
is the trivial map. Thus the augmentation map factors through the universal
cover $\widetilde{X}\to X.$

Using the vanishing of whitehead products on loop spaces one may hope to
proceed by induction on the connectivity of $X,$ and show that the map $LX\to
X$ is coherently null-homotopic, by showing that it canonically factors
through an arbitrarily high connected cover of $X.$

For this, we may now assume that $X$ is simply connected, and show that the
map $i1$ below induces zero map on the second homotopy group $\pi_{2}.$ Let
$0\neq\alpha 1\in\pi_{2}X$ a no trivial element in the image of $\pi_{2}(i1)$,
in the diagram below. For spaces with finitely generate homotopy groups, the
Whitehead product $\pi_{2}\otimes\pi_{2}\to\pi_{3}$ is non-trivial i.e.
$[\alpha 1,\alpha 2]\neq 0$ on the corresponding elements in the wedge sum
i.e. we get $\pi_{2}(X\bigvee X).$ But since the Whitehead product in the
homotopy groups of any connected loop space is trivial we get a contradiction
by naturality since we assume that $LX$ is a loop space. [1]

$\textstyle{LX\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i1}$$\textstyle{L(X\bigvee
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i2}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\bigvee
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$

The above argument is quite far from proving that such augmented functors must
have trivial augmentation.

It is interesting to note, however, that while in the homotopy category
$hoS_{*}$ of pointed spaces one can construct an augmented endo-functor
$W(X)\to X$ with $\pi_{i}W(X)=0$ for all $i\geq 1,$ namely:

$W(X):=\bigvee_{[S^{1}\to X]}S^{1}\to X$

summing up over all elements in the fundamental group $\pi_{1}X,$ such a
functor does not  exists non-trivially in the infinity category of spaces.
This is a similar phenomenon to the non-existence of a natural section of the
canonical map $X\to\pi_{0}X$ in the (infinity) category of pointed (or
unpointed) spaces.

## References

  * [1] Anthony  “nontrivial whitehead product”  In _mathoverflow.net/questions/391001/whitehead-product-and-a-homotopy-group-of-a-wedge-sum_ , 2021
  * [2] Noam Elkis  “embedding in simple groups”  In _mathoverflow.net/users/14830/noam-d-elkies_ , 2016
  * [3] Emmanuel Dror Farjoun  “Higher homotopies of natural constructions”  In _Journal of Pure and Applied Algebra_ 108, 1996
  * [4] ND Gilbert and Philip J Higgins  “The non-abelian tensor product of groups and related constructions”  In _Glasgow Mathematical Journal_ 31.1 Cambridge University Press, 1989, pp. 17–29
  * [5] SA Huq  “Semivarieties and subfunctors of the identity functor”  In _Pacific Journal of Mathematics_ 29.2 Mathematical Sciences Publishers, 1969, pp. 303–309
  * [6] Irving Kaplansky  “Set theory and metric spaces, Chelsea Publ”  In _Co., New York_ 18, 1977, pp. 152
  * [7] Friedrich Levi  “The commutator group of a free product”  Verlag nicht ermittelbar, 1941
  * [8] Michael A Shulman  “Set theory for category theory”  In _arXiv preprint arXiv:0810.1279_ , 2008
  * [9] Wacław Sierpiński  “Cardinal and ordinal numbers”  In _Warszawa_ 4, 1958


11institutetext:  Instituto de Astronomía, Universidad Católica del Norte, Av.
Angamos 0610, Antofagasta, Chile 22institutetext: Department of Physics and
JINA Center for the Evolution of the Elements, University of Notre Dame, Notre
Dame, IN 46556, USA 33institutetext: Universidade de São Paulo, IAG, Rua do
Matão 1226, Cidade Universitária, São Paulo 05508-900, Brazil 44institutetext:
Depto. de Cs. Físicas, Facultad de Ciencias Exactas, Universidad Andrés Bello,
Av. Fernández Concha 700, Las Condes, Santiago, Chile 55institutetext: Vatican
Observatory, V00120 Vatican City State, Italy 66institutetext: Leibniz-
Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam,
Germany 77institutetext: Laboratório Interinstitucional de e-Astronomia -
LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
88institutetext: School of Physics and Astronomy, Sun Yat-sen University,
Zhuhai 519082, China 99institutetext: Universidade Federal do Rio Grande do
Sul, Instituto de Física, Av. Bento Gonçalves 9500, Porto Alegre, RS, Brazil
1010institutetext: Departamento de Astronomía, Casilla 160-C, Universidad de
Concepción, Concepción, Chile 1111institutetext: Department of Astronomy -
Universidad de La Serena - Av. Juan Cisternas, 1200 North, La Serena, Chile
1212institutetext: Instituto de Investigación Multidisciplinario en Ciencia y
Tecnología, Universidad de La Serena. Benavente 980, La Serena, Chile
1313institutetext: Centro de Investigación en Astronomía, Universidad Bernardo
O’Higgins, Avenida Viel 1497, Santiago, Chile

# Galactic ArchaeoLogIcaL ExcavatiOns (GALILEO)

I. An updated census of APOGEE N-rich giants across the Milky Way

José G. Fernández-Trincado Correspondence should be addressed to:
<EMAIL_ADDRESS>and/or<EMAIL_ADDRESS>Timothy C. Beers 22
Beatriz Barbuy 33 Dante Minniti 4455 Cristina Chiappini 6677 Elisa R. Garro 44
Baitian Tang 88 Alan Alves-Brito 99 Sandro Villanova 1010 Doug Geisler
101011111212 Richard R. Lane 1313 Danilo G. Diaz 11

(Received …; Accepted …)

We use the 17th data release of the second phase of the Apache Point
Observatory Galactic Evolution Experiment (APOGEE-2) to provide a homogenous
census of N-rich red giant stars across the Milky Way (MW). We report a total
of 149 newly identified N-rich field giants toward the bulge, metal-poor disk,
and halo of our Galaxy. They exhibit significant enrichment in their nitrogen
abundance ratios ([N/Fe] $\gtrsim+0.5$), along with simultaneous depletions in
their [C/Fe] abundance ratios ([C/Fe] $<+0.15$), and they cover a wide range
of metallicities ($-1.8<$ [Fe/H] $<-0.7$). The final sample of candidate
N-rich red giant stars with globular-cluster-like (GC-like) abundance patterns
from the APOGEE survey includes a grand total of $\sim$ 412 unique objects.
These strongly N-enhanced stars are speculated to have been stripped from GCs
based on their chemical similarities with these systems. Even though we have
not found any strong evidence for binary companions or signatures of pulsating
variability yet, we cannot rule out the possibility that some of these objects
were members of binary systems in the past and/or are currently part of a
variable system. In particular, the fact that we identify such stars among the
field stars in our Galaxy provides strong evidence that the nucleosynthetic
process(es) producing the anomalous [N/Fe] abundance ratios occurs over a wide
range of metallicities. This may provide evidence either for or against the
uniqueness of the progenitor stars to GCs and/or the existence of chemical
anomalies associated with likely tidally shredded clusters in massive dwarf
galaxies such as “Kraken/Koala,” Gaia-Enceladus-Sausage, among others, before
or during their accretion by the MW. A dynamical analysis reveals that the
newly identified N-rich stars exhibit a wide range of dynamical
characteristics throughout the MW, indicating that they were produced in a
variety of Galactic environments.

###### Key Words.:

stars: abundances – stars: chemically peculiar – globular clusters:general –
techniques: spectroscopic

##  1 Introduction

The formation and assembly history of the Milky Way (MW) reveal that during
early epochs it was an inexhaustible galactic cannibal, which has completely
devoured a number of other large and massive dwarf galaxies some 9 to 11 Gyr
ago (see e.g., Kruijssen et al., 2020; Helmi, 2020). Some of these now defunct
dwarf galaxies have been identified as stellar structures or debris that
populate the present-day Galactic field (e.g., the bulge, disk, and/or halo of
the MW), and they are very phase mixed. Among the most populated progenitors
associated to these systems are the Helmi Stream (Helmi et al., 1999; Chiba &
Beers, 2000), the Gaia-Sausage (Belokurov et al., 2018) also referred to as
Gaia-Enceladus (Helmi et al., 2018), Sequoia (Myeong et al., 2019; Barbá et
al., 2019), Thamnos (Koppelman et al., 2019), Kraken/Koala (Kruijssen et al.,
2020; Forbes, 2020), and Pontus (Malhan et al., 2022), which serve as the
fossil records of the events that created them, acting as time capsules that
preserve the dynamical and chemical memory of the MW assembly history.

Besides the ancient accretion events, there is a wealth of observational
evidence for a more recent ($<$ 7 Gyr ago; see e.g., Kruijssen et al., 2020)
event which corresponds to the Sagittarius dwarf spheroidal galaxy (Ibata et
al., 1994; Hasselquist et al., 2017, 2019) and its dispersed streams
(Hasselquist et al., 2017, 2019), which also preserve traces of tidal
stripping and disruption of globular clusters (GCs) (see e.g., Fernández-
Trincado et al., 2021c), followed by the unusual Omega Centauri (NGC 5139)
system (Villanova et al., 2014), together with its hypothesized “Fimbulthul”
stream (Ibata et al., 2019) as well as its controversial tidal tails and
stellar debris (Da Costa & Coleman, 2008; Majewski et al., 2012; Fernández-
Trincado et al., 2015a, b; Marconi et al., 2014; Sollima, 2020), and a number
of stellar streams with variate origins and over a wide range of metallicities
(Wan et al., 2020; Ibata et al., 2021; Martin et al., 2022). These claims are
bolstered by the detection of extratidal stars with unusual GC-like abundance
patterns, similar to that seen toward the Large and Small Magellanic systems
today (Fernández-Trincado et al., 2020b), the stellar halo (Hanke et al.,
2020), and the Galactic bulge (Minniti et al., 2018; Fernández-Trincado et
al., 2021b).

In addition to the major accretion events, the Galactic field is also polluted
by a plethora of minor substructures (Koppelman et al., 2019), some of which
have been named Aleph, Arjuna, I’itoi, LMS-1 (Wukong), among other (see e.g.,
Naidu et al., 2020; Malhan et al., 2022). These substructures are in turn
accompanied by a number of chemically atypical stars associated with dissolved
GCs, such as the genuine population of giants with GC second-generation-like
chemical patterns (Fernández-Trincado et al., 2016a; Fernández-Trincado et
al., 2017) identified toward the bulge, disk, and inner halo (Fernández-
Trincado et al., 2020d). The recently discovered “Jurassic” substructure
(Fernández-Trincado et al., 2019a, 2020a) is accompanied by a prominent
population of aluminum-enhanced stars buried in the inner Galaxy (Fernández-
Trincado et al., 2020c; Lucey et al., 2022). There is also a recently
discovered population of high-metallicity ([Fe/H]$>-0.7$) giants with
atmospheres strongly enhanced in nitrogen and aluminum (Fernández-Trincado et
al., 2021d) well above typical Galactic levels.

All of these chemically unusual populations were likely formed after the
infall of major satellites and/or stellar clusters during different epochs of
MW evolution, as envisioned by many studies (see e.g., Leon et al., 2000;
Majewski et al., 2012; Kunder et al., 2014; Anguiano et al., 2015; Fernández-
Trincado et al., 2015b, a, 2016b, 2016a; Lind et al., 2015; Recio-Blanco et
al., 2017; Minniti et al., 2018; Kundu et al., 2019b, a; Hanke et al., 2020;
Sollima, 2020; Kundu et al., 2021, and references therein). Based on low- and
high-resolution spectroscopic surveys, several pioneering works have searched
for traces of a unique and exclusive collection of field stars that contrast
with the typical chemical composition of the majority of Galactic stars, and
they are uniquely distinctive in some of the canonical abundance planes and
speculated to belong to the population of GCs’ escapee. Among these field
stars are the nitrogen-enhanced metal-poor stars (NEMP; Johnson et al., 2007)
with [C/N] $<-0.5$ and [N/Fe] $>+0.5$, which have also been exclusively
identified in GC environments (see e.g., Simpson & Martell, 2019).

Previous studies along these lines include a unique collection of carbon-
deficient ([C/Fe] $<+0.15$) nitrogen-enhanced field stars (hereafter N-rich
stars) identified toward the inner Galaxy by Fernández-Trincado et al. (2016a)
and Martell et al. (2016); toward the bulge field by Schiavon et al. (2017),
Fernández-Trincado et al. (2017), and Fernández-Trincado et al. (2019b); and
an extra-tidal N-rich star from the bulge GC NGC 6723 was identified by
Fernández-Trincado et al. (2021b). A number of more recently added examples
are from Kisku et al. (2021) in the inner Galaxy, toward the disk field, and
beyond the tidal radius of Palomar 5 by Fernández-Trincado et al. (2019b), and
in the field of the Large and Small Magellanic Clouds (LMC and SMC,
respectively) and the Sagittarius Spheroidal galaxy by Fernández-Trincado et
al. (2020b) and Fernández-Trincado et al. (2021c), respectively. Additionally,
the similarity in orbital properties between GCs and N-rich field stars (or
previously called “CN-strong field stars”) was demonstrated in Carollo et al.
(2013) and Tang et al. (2020). While the wide gamut of elemental abundances
recently examined by Yu et al. (2021a) with optical spectra for some of these
N-rich stars strengthened their exclusive link to GCs.

Figure 1: Distribution of [N/Fe] vs. [Fe/H], [Al/Fe] vs. [Mg/Fe], and [Mg/Mn]
vs. [Al/Fe] for the newly identified N-rich stars (navy) compared to GC stars
(black dashed contours) from Mészáros et al. (2020), LMC (solid black
contours), SMC (cyan dashed contours), and Sgr (solid red contours) stars from
Helmi et al. (2018) with ASPCAP abundance information. The black dashed lines
roughly define the criterion to “distinguish” in situ from accreted
populations, similar to that as defined in Das et al. (2020) and Horta et al.
(2021). Also included are kernel density estimation (KDE) models showing the
density of objects belonging to the MW. The continuous black line in the first
column represents the fourth order polynomial fit at the 4$\sigma$ level
selection criterion for N-rich stars as described in the text. Typical error
bars are shown as black plus symbols in each panel.

Besides the carbon-deficient N-rich field stars, there is a significant
population of carbon-deficient aluminum-enriched field stars (hereafter Al-
rich stars) that exhibit low and high levels of nitrogen abundance ratios
(Fernández-Trincado et al., 2020c), suggesting that there are at least two
groups of N-rich field stars that have likely been formed from two different
nucleosynthetic pathways over a wide range of metallicities. Such anomalous
abundance patterns are found in many GCs over a wide range of metallicities
(Fernández-Trincado et al., 2019d, 2020e, 2021a; Mészáros et al., 2020, 2021;
Geisler et al., 2021; Romero-Colmenares et al., 2021; Fernández-Trincado et
al., 2022a, b; Masseron et al., 2019), and even into the most metal-poor MW GC
(see e.g., Fernández-Trincado et al., 2021e). Thus, the escape of stars from
GCs could be a plausible explanation for the presence of these chemically
anomalous stars in the Galactic field.

Although the recognized number of such chemically anomalous field stars has
expanded greatly during the last decades, the origin of their unusual
abundance signatures remains under discussion. For instance, recently, Bekki
(2019) proposed an alternative way to produce these N-rich stars, which could
be formed in high-density environments. More recently, Fernández-Trincado et
al. (2019c) has confirmed the existence of a N-rich star in a genuine binary
system, which owes its nitrogen enrichment to binary mass transfer of an
evolved asymptotic giant branch (AGB) star that is likely now a white dwarf.
There still remains the possibility that a high fraction of the N-rich and Al-
rich field stars could be evolved objects, possibly an ”early-AGB” or an AGB
star, rather than being formed through GC self-enrichment (see e.g., Pereira
et al., 2017). The presence of such a young and moderately metal-poor star in
the Galactic field would have important implications. However, its nature
remains uncertain.

As emphasized by Tang et al. (2019), some of the CH-normal field stars and the
N-rich field stars could have a very similar origin, and likely they share the
same nucleosynthetic pathways. The wide range of elemental abundances recently
examined by Yu et al. (2021b) for some of these N-rich stars has strengthened
the proposed link to GCs. Thus, future dedicated long-term radial-velocity
monitoring of all of these stars would naturally be the best course of action
in order to offer clues about the rare astrophysical events behind these
unique and atypical objects in the MW.

The zoo of chemically anomalous stars in the Galactic field is very diverse.
In addition to the previously mentioned N-rich and Al-rich stars, substantial
evidence has been offered for the existence of Si-rich stars (Fernández-
Trincado et al., 2019a, 2020a), K-rich stars (Kemp et al., 2018), r-process
enriched stars (Holmbeck et al., 2020; Gudin et al., 2021; Shank et al.,
2022), s-process enriched stars (Anders et al., 2018; Pereira et al., 2019a),
Na-rich stars (Pereira et al., 2019b; Koo et al., 2022), and P-rich stars
(Masseron et al., 2020a, c) over a wide range of metallicities, and they are
thought to form through different nucleosynthetic channels.

Here, we report the results of the analysis of the final and complete data
release of the Apache Point Observatory Galactic Evolution Experiment II
survey (APOGEE-2; Majewski et al., 2017) through Galactic ArchaeoLogIcaL
ExcavatiOns (GALILEO). This project dedicated to searching for and detecting
chemically anomalous giants – including the stars that show anomalously high
levels of [N/Fe] and [Al/Fe] – for long-term radial-velocity monitoring to
better understand the nature of these stars with unusual elemental abundances,
as well as its nucleosynthetic pathways.

##  2 Observations and data

We employed high-resolution ($R\sim 22,500$) near-infrared (NIR) spectra taken
from the final APOGEE-2 seventeenth data release (DR 17; Abdurro’uf et al.,
2021), which is one of the internal programs of the Sloan Digital Sky Survey-
IV (Blanton et al., 2017). These programs have been developed to provide
precise radial velocities (RVs) $<$1 km s-1 and elemental abundances for over
700,000 unique stars across the MW and nearby systems.

APOGEE-2 observations were carried out through two identical spectrographs
(Wilson et al., 2019) from the Northern Hemisphere on the 2.5m telescope at
Apache Point Observatory (APO, APOGEE-2N; Gunn et al., 2006) and from the
Southern Hemisphere on the Irénée du Pont 2.5m telescope (Bowen & Vaughan,
1973) at Las Campanas Observatory (LCO, APOGEE-2S). Each instrument records
most of the H band (1.51$\mu$m – 1.69$\mu$m) on three detectors, with coverage
gaps between $\sim$1.58–1.59$\mu$m and $\sim$1.64–1.65$\mu$m, and with each
fiber subtending a $\sim$2” diameter on-sky field of view for the northern
instrument and 1.3” for the southern one. The targeting strategy and plan for
the updates of the APOGEE-2 survey is fully described in Zasowski et al.
(2013), Zasowski et al. (2017), and Beaton et al. (2021) for Northern
Hemisphere observations, and in Santana et al. (2021) for Southern Hemisphere
observations.

The APOGEE-2 spectra have been reduced as described in Nidever et al. (2015),
Holtzman et al. (2018), and Jönsson et al. (2018), and analyzed using the
APOGEE Stellar Parameters and Chemical Abundance Pipeline (ASPCAP; García
Pérez et al., 2016), as well as the libraries of synthetic spectra described
in Zamora et al. (2015). The customized H-band line lists are fully described
in Shetrone et al. (2015), Hasselquist et al. (2016) (neodymium lines, Nd II),
Cunha et al. (2017) (cerium lines, Ce II), Masseron et al. (2020b) (phosphorus
lines), and have been recently updated and reviewed in Smith et al. (2021).

##  3 Data selection

Our goal was to search for the ambiguous population of N-rich stars that
likely detached from GCs across the MW by making use of the latest data
release of the APOGEE-2 survey. Before selecting potential N-rich stars, we
were required to apply a number of criteria to certify the quality of the data
for our analysis. We first cleaned the sample from sources with unreliable
parameters by removing all stars with STARFLAG and ASPCAPFLAG different from
zero (see e.g., Holtzman et al., 2015). That is to say, ASPCAPFLAG $==0$
ensures that there are not major flagged issues, such as a low signal-to-noise
ratio (S/N), poor synthetic spectral fit, stellar parameters near grid
boundaries, among others; whilst STARFLAG $==0$ minimizes potentially
problematic object spectra. We then limited the data to red giant stars with
spectral S/Ns larger than 70 pixel-1, stellar effective temperatures in the
range 3200 K – 5500 K, surface gravities $\log g<$ 3.6, and metallicities
([Fe/H]) between $-1.8$ and $-0.7$, so as to minimize the presence of sources
from the thin and thick disk at the metal-rich end, and to include giants with
reliable carbon and nitrogen abundances at the metal-poor end. We also removed
sources with C_FE_FLAG $==$ BAD, N_FE_FLAG $==$ BAD, O_FE_FLAG $==$ BAD, and
FE_H_FLAG $==$ BAD.

Following the conventions in the literature (see e.g., Fernández-Trincado et
al., 2016a, 2019b), we required noncarbon-enhanced stars with [C/Fe] $<+$0.15,
which are typically found in GC environments. These criteria allowed us to
reduce the presence of possible CH stars (Karinkuzhi & Goswami, 2015).

Next, we removed a total of 1826 stars from the sample which are deemed to be
potential members of GCs, as listed in the the catalogs of Masseron et al.
(2019), Mészáros et al. (2020), and Mészáros et al. (2021), and in the more
recent compilation from Baumgardt’s web
service111https://people.smp.uq.edu.au/HolgerBaumgardt/globular/ which is
fully described in Vasiliev & Baumgardt (2021) and Baumgardt & Vasiliev
(2021). We also removed N-rich red giant stars previously reported in the
literature (see e.g., Fernández-Trincado et al., 2016a; Fernández-Trincado et
al., 2017; Fernández-Trincado et al., 2019b, a, c, 2020c, 2020b, 2020a, 2021b,
2021c; Martell et al., 2016; Schiavon et al., 2017; Kisku et al., 2021).

The application of the above filters and removing duplicate entries left us
with a sample of 8812 unique sources with high-quality spectral information.
The data for the objects of interest were extracted from this final sample as
described in the following sections.

##  4 Newly identified N-rich stars

We started with a nitrogen- and metallicity-based selection criterion in the
[Fe/H] versus [N/Fe] plane, which is presented in Figure 1 (top-left panel),
and similar to the one used in Martell et al. (2016) and Fernández-Trincado et
al. (2019b). We fit a fourth-order polynomial to the distribution in the
[N/Fe]–[Fe/H] plane which captures the mean behavior of our data set, and we
selected all stars more than 4$\sigma$ above that curve as N-rich. As can be
seen in the top-left panel in Figure 1, the newly identified N-rich stars
exhibit typical nitrogen abundance ratios of [N/Fe] $>+0.5$. This returns a
large sample of 149 newly identified unique red giant stars that are strongly
nitrogen-enhanced relative to the final data set. The main kinematic
properties, orbital elements, and elemental abundances of these newly
identified N-rich stars are listed in Table 1.

Table 1 also lists five newly identified N-rich star candidates toward the SMC
(three stars: 2M01055438$-$6833068, 2M01021954$-$7147493, and
2M00592789$-$6943291) and LMC (two stars: 2M05023164$-$7157181 and
2M05481230$-$7937052), which display similar physical properties to that seen
in the stars of these systems. We do not provide orbital calculations for
these stars because of the reasons provided in Section 6. These stars have
been cataloged as Orbital_Sense $==$ “Unknow/LMC-Candidate” and Orbital_Sense
$==$ “Unknow/SMC-Candidate”.

N-rich stars in Table 1 with Orbital_Sense$==$“prograde/HVNS-Candidate,” have
been classified as hypervelocity N-rich star (HVNS) candidates. While N-rich
stars in our sample with unreliable orbital solutions have been tagged as
Orbital_Sense$==$“Unknow,” stars in different Galactic orbital configurations
have been classified as Orbital_Sense$==$“P--R” to refer to the orbits
changing their sense of motion from prograde to retrograde (P–R) during the
integration time and vice versa, as Orbital_Sense$==$“Prograde” to highlight
those stars in a prograde orbital configuration, and as
Orbital_Sense$==$“Retrograde” to refer to those that lie in retrograde orbits.
We characterize this newly identified N-rich population in Section 6.

Table 1: Description of the columns of the catalog containing the 149 newly
identified N-rich stars. This table is published in its entirety as supporting
information with the electronic version of the article.

ID | Column name | Units | Column Description
---|---|---|---
1 | APOGEE_ID |  | APOGEE id
2 | RA | degrees |  $\alpha$ (J2000)
3 | DEC | degrees |  $\delta$ (J2000)
4 | J_2MASS |  | 2MASS J band
5 | H_2MASS |  | 2MASS H band
6 | K_2MASS |  | 2MASS K band
7 | BP_GaiaEDR3 |  |  Gaia EDR3 BP band
8 | RP_GaiaEDR3 |  |  Gaia EDR3 RP band
9 | G_GaiaEDR3 |  |  Gaia EDR3 G band
10 | SNR |  pixel-1 | spectral signal-to-noise
11 | RV |  km s-1 | APOGEE-2 radial velocity
12 | VSCATTER |  km s-1 | APOGEE-2 radial velocity scatter
13 | TEFF | K | effective temperature
14 | ERROR_TEFF | K | uncertainty in TEFF
15 | LOGG | [cgs] | surface gravity
16 | ERROR_LOGG | [cgs] | uncertainty in LOGG
17 | C_FE |  |  [C/Fe] from ASPCAP
18 | ERROR_C_FE |  | uncertainty in [C/Fe]
19 | N_FE |  |  [N/Fe] from ASPCAP
20 | ERROR_N_FE |  | uncertainty in [N/Fe]
21 | MG_FE |  |  [Mg/Fe] from ASPCAP
22 | ERROR_MG_FE |  | uncertainty in [Mg/Fe]
23 | AL_FE |  |  [Al/Fe] from ASPCAP
24 | ERROR_AL_FE |  | uncertainty in [Al/Fe]
25 | MN_FE |  |  [Mn/Fe] from ASPCAP
26 | ERROR_MN_FE |  | uncertainty in [Mn/Fe]
27 | FE_H |  |  [Fe/H] from ASPCAP
28 | ERROR_FE_H |  | uncertainty in [Fe/H]
29 | VR |  km s-1 | Galactocentric radial
|  |  |  velocity in VR
30 | ERROR_VR |  km s-1 |  uncertainty in VR
31 | VPHI |  km s-1 | Galactocentric azimuthal
|  |  |  velocity in Vϕ
32 | ERROR_VPHI |  km s-1 |  uncertainty in Vϕ
33 | PERIGALACTICON | kpc | Perigalactocentric distance
34 | ERROR_PERIGALACTICON | kpc | uncertainty in PERIGALACTICON
35 | APOGALACTICON | kpc | Apogalactocentric distance
36 | ERROR_APOGALACTICON | kpc | uncertainty in APOGALACTICON
37 | Eccentricity |  | Orbital eccentricity
38 | ERROR_Eccentricity |  | uncertainty in Eccentricity
39 | Zmax | kpc | Maximum vertical excursion
|  |  | from the Galactic plane
40 | ERROR_Zmax | kpc | uncertainty in Zmax
41 | Orbital_Sense |  | Orbital sense
42 | ruwe |  |  Gaia EDR3 renormalized
|  |  | unit weight error
43 |  d_STARHORSE† | kpc |  Bayesian StarHorse distance, 50th percentile
44 | ERROR_d_STARHORSE | kpc | uncertainty in d_STARHORSE ,
|  |  |  ERROR $=$ (84th - 16th)/2 percentile
45 | pmRA_GaiaEDR3 |  mas yr-1 |  $\mu_{\alpha}\cos{}(\delta)$ from Gaia EDR3
46 | ERROR_pmRA_GaiaEDR3 |  mas yr-1 |  uncertainty in $\mu_{\alpha}\cos{}(\delta)$
|  |  |  from Gaia EDR3
47 | pmDEC_GaiaEDR3 |  mas yr-1 |  $\mu_{\delta}$ from Gaia EDR3
48 | ERROR_pmDEC_GaiaEDR3 |  mas yr-1 |  uncertainty in $\mu_{\delta}$ from Gaia EDR3

Note: †For the stars 2M17025992$-$3537464 and 2M19175783$-$1343049, we provide
the estimated Bayesian StarHorse and Bailer-Jones distances and their
resulting parameters in two different entries of the catalog.

It is also important to note that none of the 149 N-rich candidates without
strong carbon enhancement have a particularly strong variability in their
radial velocity over the period of the APOGEE-2 observations, thus limiting
the possible orbital properties than can be detected in our data. Those
sources with available multi-epoch observations exhibit a typical V${}_{\rm
scatter}<0.9$ km s-1, with the exception of three sources, which show a
slightly high Vscatter between 1.64 – 2.6 km s-1. We also found no matches
with the ASAS-SN Catalog of Variable Stars (Jayasinghe et al., 2021).

Although we have not found strong evidence for a binary companion or a
pulsating variable star as of yet, we cannot rule out the possibility that
some of these stars were members of a binary system in the past (see e.g.,
Fernández-Trincado et al., 2019c), or they could likely be part of the semi-
regular (SR) variable populations (see e.g., Fernández-Trincado et al.,
2020b). Therefore, long-term radial-velocity monitoring for all the identified
N-rich stars is currently being carried out with the SDSS-V panoptic
spectroscopic survey (Kollmeier et al., 2017) which, with a baseline that is
better ($\lesssim$ 6 months) than that of the APOGEE-2 DR 17 survey, would
naturally be the best course to establish the number of such objects that
formed through the binary channel or that currently belong to a kind of
pulsating variable.

Table 2 summarizes the nearly complete census in APOGEE of N-rich field stars,
for a grand total of $\sim$412 unique carbon-depleted N-rich candidates across
the MW. It is expected that this number will increase further once the results
from the new Milky Way Mapper (MWM) program of SDSS-V (Kollmeier et al., 2017)
are published. Table 2 does not include the recent exploration by Phillips et
al. (2022), which reports on the detection of two N-rich stars along the
Palomar 5 Stream. These two stars were already identified previously by
Martell et al. (2016) for 2M15204588$+$0055032, suggested as potential extra-
tidal material of Palomar 5 in Fernández-Trincado et al. (2019b) for
2M15183589$+$0027100, and confirmed as members of the Palomar 5 Stream by the
STREAMFINDER algorithm in Ibata et al. (2021). Table 3 lists the APOGEE ids
and origin on the literature of the 412 compiled APOGEE N-rich stars.

Finally, we found that there are no known GCs within an angular separation of
$1.5\times{}{\rm r_{tidal,GC}}$ arcmin out of the 149 newly reported N-rich
stars in our sample, and there is not an association with the newly discovered
stellar streams in the Gaia universe Ibata et al. (2021). However, some of
them could be made up of dissipated GCs, which is investigated in detail by
assessing other chemical species not accessible from the APOGEE-2 survey.

Table 2: Census updated of $\sim$412 unique field carbon-depleted N-rich
([C/Fe]$<+0.15$; [N/Fe]$\gtrsim+0.5$) stars in the APOGEE survey

Number of | Galactic component | Data release (DR) | Code | Literature
---|---|---|---|---
N-rich stars |  |  |  |
5 | Halo: Migrants | DR 12 | ASPCAP | Martell et al. (2016)
1 | Halo | DR 12 |  iSpec(1) and MOOG(2) | Fernández-Trincado et al. (2016a)
58 | Bulge | DR 12 | ASPCAP | Schiavon et al. (2017)
11 | Bulge, Disk and Halo | DR 13 | Turbospectrum(3) | Fernández-Trincado et al. (2017)
31 | Bulge, Disk, Halo, and | DR 14 | Payne | Fernández-Trincado et al. (2019b)
| Palomar 5: extra-tidal candidate |  |  |
1 | Halo: Binary system | DR 14 | BACCHUS | Fernández-Trincado et al. (2019c)
7 | Bulge, Disk and Halo: N-/Si-rich | DR 16 | BACCHUS | Fernández-Trincado et al. (2020a)
| (with simultaneously C,N,O determinations) |  |  |
44 | Magellanic Clouds: LMC & SMC | DR 16 | BACCHUS | Fernández-Trincado et al. (2020b)
16 | Bulge and inner Halo: N-/Al-rich | DR 16 | BACCHUS | Fernández-Trincado et al. (2020c)
47 out of 83 | Located in the inner 4 kpc of the MW; | DR 16 | ASPCAP | Kisku et al. (2021)
| excluding previously identified N-rich stars, and |  |  |
| stars positioned inside the tidal radius of known GCs |  |  |
2 |  Sgr$+$M 54 field | DR 16 | BACCHUS | Fernández-Trincado et al. (2021c)
40 out of 42 |  Bulge, Disk and Halo ([Fe/H]$>-0.7$) | DR 17 | ASPCAP | Fernández-Trincado et al. (2021d)
| excluding two previously identified N-rich stars |  |  |
149 |  LMC, SMC, Bulge, Disk and Halo ([Fe/H]$<-0.7$) | DR 17 | ASPCAP | This work

Note: (1)Blanco-Cuaresma et al. (2014); (2)Sneden (1973); and (3)Alvarez &
Plez (1998).

Table 3: Compilation of the 412 N-rich stars indentified thus far in the APOGEE survey. The APOGEE id and literature source (doi) is listed. The newly identified N-rich stars are listed in this table as “NEW.” (This table is published in its entirety as supporting information with the electronic version of the article. A portion is shown here for guidance regarding its form and content). APOGEE_ID | doi
---|---
2M16011638$-$1201525 | 10.3847/1538-4357/833/2/132
2M16051144$-$2330557 | 10.1093/mnras/stab525
2M16100035$-$1933027 | 10.1093/mnras/stab525
2M16161605$-$2838569 | 10.1093/mnras/stab525
2M16180906$-$2442217 | 10.1093/mnras/stab525
2M16304650$-$2949522 | 10.1093/mnras/stab525
2M16314726$-$2945273 | 10.1093/mnras/stab525
2M16333703$-$3028333 | 10.1093/mnras/stab525
2M16335569$-$1344044 | 10.1093/mnras/stab525
2M16493016$-$2557027 | 10.1093/mnras/stab525
2M17043840$-$2608052 | 10.1093/mnras/stab525
2M17132833$-$1820552 | 10.1093/mnras/stab525
2M17281699$-$3024573 | 10.1093/mnras/stab525
2M17285196$-$2013080 | 10.1093/mnras/stab525
2M17292082$-$2126433 | 10.1093/mnras/stab525
2M17293012$-$3006008 | 10.1093/mnras/stab525
2M17293730$-$2725594 | 10.1093/mnras/stab525
2M17305645$-$3030155 | 10.1093/mnras/stab525
… | …
2M19175783$-$1343049 | NEW
… | …

##  5 Elemental abundance analysis

In this section we analyze the chemical properties of the newly identified
N-rich stars alongside the MW halo and disk, the LMC and SMC members, the
Sagittarius spheroidal dwarf galaxy (Sgr), and GC members, in the chemical
diagnostic plans of [N/Fe] versus [Fe/H], [Al/Fe] versus [Mg/Fe], and [Mg/Mn]
versus [Al/Fe]. Our sample selection of LMC, SMC, and Sgr members was obtained
by cross-matching the catalogs from Helmi’s web
service222https://www.astro.rug.nl/~ahelmi/research/dr2-dggc/ (Gaia
Collaboration et al., 2018) against the APOGEE-2 DR 17 catalog (Abdurro’uf et
al., 2021), and adopting the quality cuts described in Section 3. GC sources
were adopted from Mészáros et al. (2020) and Mészáros et al. (2021), but we
adopted the APOGEE-2/ASPCAP DR 17 abundance determinations in order to avoid
the systematic elemental abundances produced between BACCHUS (Masseron et al.,
2016) and ASPCAP (García Pérez et al., 2016), as our sample relies on
abundance ratios determined from the ASPCAP pipeline.

Figure 1 summarizes the elemental properties of the 149 newly identified
N-rich red giant stars. The first column of the panels in this figure shows
that the [N/Fe] abundance ratio of the N-rich stars is more compatible with
the nitrogen-enriched GC population, rather than the LMC, SMC, and Sgr
populations.

The second column of panels in Figure 1 reveals that there appears to be at
least two main groups in the [Mg/Fe] versus [Al/Fe] plane, where [Al/Fe] is
anticorrelated with [Mg/Fe], similar to what is seen in GC populations,
primarily for [Al/Fe] $\gtrsim$ 0, and where it also overlaps with the main
bulk of ME stars. While the second group of N-rich stars with subsolar
[Al/Fe]$<0$ fall within the group dominated by the dwarf members plus GC
populations with low-aluminum enrichment more likely the first-generation of
GC stars as envisioned by Mészáros et al. (2020). The third column of panels
in Figure 1 shows [Mg/Mn] versus [Al/Fe], as originally envisioned by Das et
al. (2020). The newly identified N-rich populations expand the same locus of
that chemical plane deemed to contain GC populations, even (in)outside the
“accreted” population area defined by Das et al. (2020) which runs roughly
above [Mg/Mn] $>+0.25$ and [Al/Fe] $\lesssim+0.2$.

While Das et al. (2020) and Horta et al. (2021) suggest that the
[Al/Fe]–[Mg/Mn] plane could be used to identify stellar populations that
formed ex situ, some caution is recommended since metal-poor ([Fe/H]
$\lesssim-0.7$) nitrogen-enriched populations that likely formed ex situ share
the same locus as the in situ MW population in this plane. In particular, for
subsolar [Al/Fe], there is a mix of stellar populations belonging to GCs and
dwarf populations.

##  6 Dynamical properties of selected N-rich stars

We examined the kinematic and dynamical properties of the newly identified
N-rich stars by making use of the GravPot16333https://gravpot.utinam.cnrs.fr
model (Fernández-Trincado, 2017), and computing the ensemble of orbits
integrated over a 3 Gyr time span. For the orbit calculations, we assumed a
bar pattern speed of 41$\pm$10 km s-1 kpc-1 (Sanders et al., 2019; Bovy et
al., 2019). We note that our model has some limitations in the processes
considered; for example, secular changes in the adopted MW potential as well
as dynamical friction are not included.

The orbit calculations were performed by adopting a simple Monte Carlo
approach that considers the errors of the observables. The resulting values
and their errors were taken as the 16th, 50th, and 84th percentiles from the
generated distributions. Heliocentric distances were estimated with the
Bayesian StarHorse code (see e.g., Anders et al., 2019; Queiroz et al., 2018,
2020, 2021) as part of the APOGEE_STARHORSE Value Added
Catalog444https://data.sdss.org/sas/dr17/env/APOGEE_STARHORSE/, proper motions
are from Gaia EDR 3 (Gaia Collaboration et al., 2021), and radial velocities
are from the APOGEE-2 DR 17 database.

Figure 2 shows the resulting kinematic properties and orbital elements for 124
out of 149 stars in our sample which have a Gaia re-normalized unit weight
error (RUWE) of less than 1.4 (indicating the high quality of their
astrometric solutions), and uncertainties on the heliocentric distances of
less than 20%. It is important to note that five potential N-rich SMC and LMC
members were excluded from our dynamical analysis; as our MW model does not
take the Magellanic Clouds’ contribution to the global gravitational potential
into account, their orbital elements will be more uncertain.

###  6.1 Kinematical properties

Figure 2 reveals that a significant fraction (65%; 80 stars) lie within the
fraction dominated by Gaia-Enceladus-Sausage-like (GES-like; Belokurov et al.,
2018; Helmi et al., 2018) stars and a halo-like population with
$V_{\rm\phi}<100$ km s-1, which exhibit a typical azimuthal velocity
uncertainty below 40 km s-1, with the exception of 2M16053258$-$4502140
exhibiting a relatively large azimuthal velocity uncertainty of
$V_{\rm\phi}=-264\pm 163$ km s-1, but it is clearly well separated
kinematically from the GES-like component and disk populations (see Figure 2).
While 34% (42 stars) of the N-rich stars exhibit disk-like kinematics, with an
azimuthal velocity of $V_{\rm\phi}>100$ km s-1 with typical uncertainties
$<35$ km s-1 – with the exception of 2M06243113$+$4213097 exhibiting a large
uncertainty in $V_{\rm\phi}\sim 237\pm 73$ km s-1 – they are still related to
the disk-like kinematic population. However, as is described below, the newly
identified N-rich stars cover a wide range of dynamical characteristics with
overlapping Galactocentric radial and azimuthal velocities.

###  6.2 Dynamical properties

A more elaborate inspection of their orbital elements reveal that there are at
least four main groups of N-rich stars in our sample, whose kinematic
distributions are shown in Figure 3. The first group (Group #1) is dominated
by a significant fraction (72 out of 124 stars) of N-rich stars which lie on
pure prograde orbits with an azimuthal velocity of $V_{\rm\phi}\gtrsim-50$ km
s-1.

A second group (Group #2) is made up of 25 out of 124 N-rich stars which
exhibit $V_{\rm\phi}<100$ km s-1. They also have a retrograde orbital
configuration with respect to the rotation of the Galactic bar.

A third group (Group #3) composed of 25 out of 124 N-rich stars with Galactic
azimuthal velocities ranging from $-150$ km s-1 $\lesssim V_{\rm\phi}\lesssim
150$ km s-1 exhibit an unusual dynamical behavior, that is to say they are in
the so-called P–R orbital configuration which changes their sense of motion
from prograde to retrograde during the integration time and vice versa, which
could also be related to chaotic behavior. Further, for these three groups, we
find no strong correlation between their orbital elements and [Fe/H], which
could be indicative of an early chaotic phase of the evolution of the MW, as
well.

In addition, the N-rich stars occupying the GES-like component in Figure 2 are
dominated by a wide range of orbital configurations (including P–R, prograde
and retrograde orbits), and they are almost completely confined to
eccentricities larger than 0.65 peaking at $\sim$ 0.95, similar to that of the
GES-like kinematic substructure identified by Naidu et al. (2020).

A fourth group (Group #4) is made up of two N-rich outliers in the Vϕ versus
VR plane at $V_{\rm\phi}>600$ km s-1. We refer to these stars as HVNS, and
they likely form as part of the intriguing subgroup of the high-velocity stars
(Hawkins et al., 2015, and references therein); readers are encouraged to
refer to the discussion in Section 7.

Figure 2: Galactocentric radial (VR) and azimuthal (Vϕ) velocity, and orbital-
element properties for the newly identified N-rich stars. The color-coding of
the symbols in the panels is based on their [Fe/H], while the gray background
shading represents the KDE model of the overlaid points, excluding the HVNS
candidates (see text). The distribution of the velocity component VR vs. Vϕ is
shown in panel (a). The blue dashed line represents the approximate region for
sources with the Gaia-Enceladus-Sausage-like kinematics in VR vs. Vϕ, based on
Belokurov et al. (2018). The area occupied by disk-like stars is also
highlighted with the empty black circle. The perigalactic (b), apogalactic
distances (c), and the maximum vertical excursion (d) from the Galactic plane
as a function of the orbital eccentricity are shown. Black up-tick symbols
mark the stars with likely chaotic behavior, i.e., those with orbits that
change their direction of motion from prograde to retrograde (and vice versa)
with respect to the direction of Galactic rotation, while the small, empty
black concentric symbols indicate the stars with retrograde orbits.
Uncertainties are shown as gray plus symbols in each panel. The large star
symbols mark the two HVNS candidates assuming the Bayesian StarHorse
heliocentric distance estimations, while the large hexagonal markers refer to
the same stars with Bailer-Jones heliocentric distance estimations.

Figure 2 also reveals that most of the metal-poor N-rich stars lie on high-
eccentricity orbits with large vertical excursions above the Galactic plane,
with some noticeable evidence of N-rich stars that likely formed in the outer
halo, while the most metal-rich N-rich stars lie on prograde, low-
eccentricity, inner halo- and disk-like orbital configurations. Interestingly,
two (2M06243113$+$4213097 and 2M08534023$-$2554252) of the most metal-rich
N-rich stars exhibit near circular orbital configurations in the outer disk of
the MW, suggesting that they might have been formed in situ from material
chemically self-enriched by a first generation of stars in the outer disk, or
they could have formed from a GC disruption in the inner Galaxy and radially
migrated to their current location and/or are trapped by disk ripples (Xu et
al., 2015).

Figure 3: Same as Figure 2(a), but here showing the KDE models of the newly
identified N-rich stars confined to four main groups (see text). HVNS
candidates (Grupo #4; 2 stars) are shown as empty black star symbols for the
assumed Bayesian StarHorse distances and as empty black hexagonal symbols by
adopting the Bailer-Jones distances, while N-rich stars in prograde (Grupo #1;
72 stars), retrograde (Grupo #2; 25 stars), and P–R (Grupo #3; 25 stars)
orbital configurations are represented by the gray, green, and red contours,
respectively.

##  7 Hypervelocity N-rich star (HVNS) candidates

Strikingly, we also identified two N-rich stars with a high azimuthal velocity
of $V_{\rm\phi}>600$ km s-1. Thus, these could be classified as atypical fast-
moving N-rich stars likely escaping the MW potential, with the exception of
the N-rich star 2M17025992$-$3537464 which exhibits a larger uncertainty in
$V_{\rm\phi}\sim 540$ km s-1 overlapping with the kinematic of typical disk
stars. We have classified these two stars as HVNS candidates.

Figure 4 reveals that the predicted orbital configurations of these HVNS
candidates are racing through space at above the escape speed of the MW,
having a perigalactic distance of $r_{\rm peri}\sim 11.7\pm 0.87$ kpc
(2M19175783$-$1343049) and $r_{\rm peri}\sim 17.09\pm 2.98$ kpc
(2M17025992$-$3537464), clearly indicating that they did not originate in the
Galactic Center. However, their atypical chemistry suggests that these HVNS
candidates likely have a GC formation site.

These HVNS stars could have been produced via different physical processes,
including the three-body encounters at the center of a GC; which through a
process known as resonance scattering a binary star in the stellar cluster can
capture a star in the cluster (thus forming a bound triple system), and after
some time one of the stars (likely the HVNS) is ejected, or via flyby where a
captured HVNS is ejected in the interaction (see e.g., Hut & Bahcall, 1983;
Pichardo et al., 2012; Fernández-Trincado et al., 2016b). This finding could
aid in providing useful information about the environments from which the
high-velocity halo stars are produced.

It is also important to note that Bailer-Jones et al. (hereafter Bailer-Jones;
2021) provides estimated heliocentric distances for these two stars which put
them closer to the Sun (d${}_{\odot}<6.5$ kpc) than those estimated with the
Bayesian StarHorse code (d${}_{\odot}>$19 kpc). Figures 2–4 reveal that
adopting closer heliocentric distances for 2M19175783$-$1343049 and
2M17025992$-$3537464 puts these stars in an orbital configuration which
inhabits the inner regions of the Galactic bulge in prograde and high
eccentric ($\gtrsim$0.6) orbits, overlapping with the kinematic properties of
the GES-like component ($V_{\rm\phi}<100$ km s-1).

However, the estimated distances from Bailer-Jones et al. (2021) for these two
stars should be more uncertain as they should be dominated by a weak prior for
a density distribution of the Galaxy, thus the priors would make us expect for
these stars to be much closer than they truly are, producing unrealistic
distances. Thus, it is likely that the distances from the Bailer-Jones catalog
for 2M19175783$-$1343049 and 2M17025992$-$3537464 should just be following the
priors. In addition, the Bayesian StarHorse code provides smaller
uncertainties at larger ranges compared to the Bailer-Jones catalog.
Therefore, we decided to use the Bayesian StarHorse distances derived by
Queiroz et al. (2021) as the primary distance set for this work, also based on
their better extinction treatment. However, with the upcoming Gaia DR3, we
will be able to constrain the distances and proper motions better for these
two HVNS candidates.

##  8 Conclusions

We report on the discovery of 149 newly identified carbon-depleted ([C/Fe]
$<+0.15$) N-rich stars over a wide range of metallicities ($-1.8<$ [Fe/H]
$<-0.7$) for a grand total of $\sim$412 N-rich stars across the MW. Similar to
the previously identified nitrogen-enhanced population, the new one exhibits
elemental abundances comparable to those exclusively seen in GC environments.

We also examined the position of the newly identified N-rich stars in the
[Al/Fe] versus [Mg/Mn] plane as envisioned by Das et al. (2020), finding that
N-rich stars occupy a wide range at and beyond the locus deemed to contain an
“accreted” population ([Al/Fe] $\lesssim 0$ and [Mg/Mn] $\gtrsim+0.25$) and to
that dominated by an in situ population of the MW. This suggests that the
region occupied by the in situ population of the MW has a non-negligible
presence of stars with unusual elemental abundances, likely associated to
tidal stripping and/or the disruption of GCs over different epochs of
evolution of the MW. The diverse dynamical and kinematical characteristics of
the newly identified N-rich stars reveal that they do not share the same
birthplaces in the MW, and that some of them could be the debris of GCs that
were fully or partially destroyed during previous major merger accretion
events such as the GES.

Figure 4: Ensemble of ten thousand orbits for the two HVNS candidates
2M17025992$-$3537464 and 2M19175783$-$1343049, projected on the equatorial
(top) and meridional (bottom) Galactic planes in the inertial reference frame
with a bar pattern speed of 41$\pm$10 km s-1 kpc-1, and time-integrated
forward and backward over 1 Gyr. The yellow and orange colors correspond to
more probable regions of the space, which are most frequently crossed by the
simulated orbits. The black dashed and solid black lines show the 50th
percentile for the forward and backward orbital path, respectively. The top
panels show, for guidance, a MW bulge (solid white circle) and disk (white
dashed circle) radius of $\sim 3.5$ kpc (Barbuy et al., 2018) and $\sim$15
kpc, respectively. While the MW disk extension until $\sim$15 kpc is also
denoted by a thick white line in the bottom panels. Columns 1 and 3 show the
orbital configurations by assuming the Bayesian StarHorse distances, whilst
columns 2 and 4 display the predicted orbits by adopting the Bailer-Jones
geometric distances.

###### Acknowledgements.

The author is grateful for the enlightening feedback from the anonymous
referee.
J.G.F-T gratefully acknowledges the grant support provided by Proyecto
Fondecyt Iniciación No. 11220340, and also from ANID Concurso de Fomento a la
Vinculación Internacional para Instituciones de Investigación Regionales
(Modalidad corta duración) Proyecto No. FOVI210020.
J.G.F-T and D.G.D. gratefully acknowledges the grant support from the Joint
Committee ESO-Government of Chile 2021 (ORP 023/2021).
T.C.B. acknowledges partial support for this work from grant PHY 14-30152:
Physics Frontier Center / JINA Center for the Evolution of the Elements (JINA-
CEE), awarded by the US National Science Foundation.
B.B. acknowledges grants from FAPESP, CNPq and CAPES - Financial code 001.
D.M. gratefully acknowledges support by the ANID BASAL projects ACE210002 and
FB210003, and Fondecyt Project No. 1220724.
E.R.G acknowledges support from ANID PhD scholarship No. 21210330.
S.V. gratefully acknowledges the support provided by Fondecyt regular n.
1220264, and by the ANID BASAL projects ACE210002 and FB210003.
D.G. gratefully acknowledges support from the ANID BASAL project ACE210002.
D.G. also acknowledges financial support from the Dirección de Investigación y
Desarrollo de la Universidad de La Serena through the Programa de Incentivo a
la Investigación de Académicos (PIA-DIDULS).
This work has made use of data from the European Space Agency (ESA) mission
Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS- IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution for
Science, Carnegie Mellon University, the Chilean Participation Group, the
French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrofìsica de Canarias, The Johns Hopkins University, Kavli
Institute for the Physics and Mathematics of the Universe (IPMU) / University
of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für
Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatory of China, New Mexico State University, New York University,
University of Notre Dame, Observatório Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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