Tropicalization of the universal Jacobian

In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -- one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.

, where the diagram of topological spaces comes from the description of J trop g,n as a combinatorial cone stack, see Theorem 2.16(i). Moreover, J trop g,n admits generalized cone subcomplexes J trop g,n,d (resp. J trop, spl g,n , resp. J trop g,n (φ) for any φ ∈ V g,n ) parametrizing pairs (Γ , D) such that deg D = d (resp. G(Γ )) is simple, resp. D is φ-semistable). And similarly for their compactifications inside J trop g,n .
Recall now that in non-archimedean geometry, the tropicalization of a toroidal embedding of Artin stacks (U ⊂ X ) (or more generally a logarithmic algebraic stack) is meant to be a generalized cone complex Σ(X ), and its natural compactification Σ(X ) which is a generalized extended cone complex, together with a surjective proper continuous map (that we baptise functorial analytic tropicalization) trop an X : |X | → Σ(X ) which is functorial with respect to locally toric morphisms, see [ACP15] and [Uli19].
The relation between the above two topological spaces is summarized in the next Theorem (see Theorem 5.9 for a more precise version).

Theorem B.
(1) There are canonical isomorphisms Ψ J g,n : Σ(J g,n ) − → J trop g,n and Ψ J g,n : Σ(J g,n ) − → J trop g,n of, respectively, generalized cone complexes and generalized extended cone complexes, in such a way that the map (that we baptise the modular analytic tropicalization map) Moreover, the diagram (1.3) commutes, via the suitable forgetful-stabilization morphisms, with the analogous diagram for M g,n established in [ACP15]. Moreover, the diagram (1.4) is compatible with the restrictions to J g,n,d , J spl g,n and J g,n (φ) for any universal stability condition φ ∈ V g,n .
Theorem B contains as a special case the main result of [AP20]: when n = 1 and φ ∈ V g,1 is a suitable perturbation of the canonical stability condition (which is then general), J g,1 (φ) is isomorphic (up to G m -rigidification) to the Esteves' compactified universal Jacobian stack J φ,g of loc. cit. (see Remark 3.2 below) and so Theorem B states that the non-Archimedean skeleton of Esteves' compactified universal Jacobian J φ,g can be identified with J trop g,1 (φ) (which can be shown to be isomorphic to the generalized extended cone complex J trop φ,g constructed in loc. cit.) making the natural diagram of tropicalization maps commute, which is exactly [AP20, Theorem 6.9] (1) . We believe that one advantage of our approach, with respect to the one of [AP20], is that the spaces J trop g,n (φ), as φ varies in V g,n , are constructed as generalized extended cone sub-complexes of J trop g,n , which should be useful in order to study tropical wall-crossing phenomena (similar to [KP19]).

Fibers of the forgetful-stabilization morphism
In the last Section of the paper we study the fibers of the forgetful-stabilization morphism of cone stacks Φ trop : J See Theorems 6.2, 6.4, 6.5 in Section 6.2.

Theorem C.
(1) The fiber of the forgetful-stabilization morphism of cone stacks Φ trop : J trop g,n → M trop g,n over a stable tropical curve Γ /σ ∈ M trop g,n (σ ) is the Jacobian cone space Jac Γ /σ that we construct in Definition 6.1. Moreover, a similar result is true for the restriction of Φ trop to the cone substacks J trop g,n,d , J trop, spl g,n and J trop g,n (φ) for any φ ∈ V g,n .
(2) Let Γ be a stable tropical curve of type (g, n) with real edge lengths. (i) The fiber of the forgetful-stabilization morphism of topological stacks Φ trop : J trop g,n → M trop g,n over Γ is the Jacobian topological space Jac Γ that we construct in Definition 6.3. Moreover, a similar result is true for the restriction of Φ trop to the topological substacks J for any general universal stability condition φ ∈ V g,n .
(1) However, it seems to us that [AP20, Theorem 6.9] should work, with small changes in the proof, for any n ≥ 1 and any general universal stability condition φ ∈ V g,n .
It follows from the above Theorem that, for any general universal stability condition φ ∈ V g,n , the fiber of the forgetful-stabilization morphism of generalized cone complexes (see (5.13)) Φ trop : J trop g,n (φ) −→ M trop g,n over a point Γ ∈ M trop g,n is homeomorphic to Pic |φ| (Γ )/ Aut(Γ ) (see Remark 6.8), thus recovering (and slightly extending to our more general setting) the result of Abreu-Pacini [AP20, Theorem 5.14]. The advantage of working with topological stacks as in Theorem C(2), rather than generalized cone complexes, is that we do not have to quotient out by the automorphism group of Γ when describing the fiber of a point Γ .

A sequel on LogPic and TroPic
In [MW18] the authors have shown that in the category of logarithmic schemes (and stacks), there is a unique minimal model of the universal logarithmic Jacobian that is not representable by an algebraic stack. The study of this so-called universal logarithmic Picard variety LogP ic g,n,d can be traced back to Illusie [Ill94] and has subsequently received attention in [Kaj93,Ols04,Bel15,FRTU19]. The authors of [MW18] also introduce the universal tropical Picard variety, which universally over M log g,n is denoted by T roP ic g,n,d . It parametrizes tropical curves together with a torsor over the sheaf of harmonic or linear functions on Γ of degree d and it naturally arises as the tropicalization of LogP ic g,n,d via a natural tropicalization morphism LogP ic g,n,d → T roP ic g,n,d .
In the sequel to this article we carefully study the relationship between this construction and what we have done in this article. The gist of this story is that, for a general universal stability condition φ ∈ V g,n , the cone stack J trop g,n (φ) defines a proper subdivision of the universal tropical Picard variety T roP ic g,n,d . This induces a logarithmic modification J log g,n (φ) G m → LogP ic g,n,d -effectively a blowup -by base change along the tropicalization map. As an application, this approach allows one to readily relate the spaces constructed here with Abel-Jacobi theory (see e.g. [MW20] and [AP21]). The perspective adopted in this paper and the sequel is for instance crucial in ongoing work of Holmes, Pandharipande, Pixton, Schmitt, and the second author regarding the double ramification cycle.

Quasi-stable graphs
We will adopt the graph-theoretic terminology of [CCUW20, Section 3.1], which we now briefly recall. A graph G is a triple (X(G), r G , i G ) such that • X(G) is a finite set; • r G : X(G) → X(G) is an idempotent map (called the root map); • i G : X(G) → X(G) is an involution whose fixed set contains the image of r G . We recover the more familiar definition of graphs in the following way. The image of r G is the vertex set V (G) of G. Its complement F(G) := X(G) \ V (G) is the set of flags of G and the root map restricts to a map r G : F(G) → V (G), which we think of as the map that sends a flag to the root from which it emanates. The involution i G restricts to an involution on F(G): the fixed points of this involution are the legs L(G) of G, its non fixed points are the half-edges H(G) of G. Hence we get that

X(G) = V (G) H(G) L(G),
the root map r G is the identity on V (G) and it restricts to a map the involution i G is the identity on V (G) L(G) and it is fixed-point free on H(G). The quotient E(G) := H(G)/i G is the set of edges of G; explicitly, any edge e of G is equal to e = {h 1 , h 2 } with i G (h 1 ) = h 2 and we say that the half-edges h 1 and h 2 belong to the edge e and that they are conjugate half-edges.
We will be dealing with n-marked vertex-weighted graphs G A morphism π : G 1 = (G 1 , h 1 , m 1 ) → G 2 = (G 2 , h 2 , m 2 ) of (n-marked vertex-weighted) graphs consists of a function π : X(G 1 ) → X(G 2 ) with the property that π • r G 1 = r G 2 • π and π • i G 1 = i G 2 • π, and which moreover satisfies the following additional properties: • For any flag f ∈ F(G 2 ), its inverse image π −1 (f ) has one element which is a flag of G 1 .
Recall that a stable graph of type (g, n) is a n-marked vertex-weighted connected graph G of total genus g such that for all v ∈ V (G): where val(v) is the number of flags emanating from (or incident to) v. We now define a slight generalization of stable graphs, namely quasi-stable graphs.
Definition 2.1. Fix (g, n) an hyperbolic pair, i.e. a pair of integers g, n ≥ 0 such that 2g − 2 + n > 0. A quasi-stable graph of type (g, n) is a n-marked vertex-weighted connected graph G = (G, h, m) of total genus g such that any vertex v ∈ V (G) of genus zero has valence at least two and those vertices of genus zero and valence two, called exceptional vertices, are such that: • every exceptional vertex has exactly two edges incident to it (in particular, there are no legs rooted at exceptional vertices); • two distinct exceptional vertices are not adjacent.
Notice that quasistable graphs with no exceptional vertices correspond exactly to stable graphs.
We will denote the set of exceptional vertices of G by V exc (G) and the set of remaining vertices, called nonexceptional, by V nex (G). For any v ∈ V exc (G), we will denote by h 1 v , h 2 v the two half-edges rooted at v and we fix an order of them. The edges to which h 1 v and h 2 v belong are denoted, respectively, by . Moreover, for any v ∈ V exc (G) and any i = 1, 2, we set v i : The half-edges (resp. edges) of the form h 1 v and h 2 v (resp. e 1 v and e 2 v ), for some v ∈ V exc (G), are called exceptional and the set of all exceptional half-edges (resp. edges) is denoted by H exc (G) (resp. E exc (G)); the remaining half-edges (resp. edges), called non-exceptional, are denoted by H nex (G) (resp. E nex (G)).

Definition 2.2.
A quasi-stable graph G is called simple if the graph obtained from G by removing its exceptional vertices is connected.
The stabilization of G, denote by G st , is the (connected) graph with One can easily check that the graph G st becomes a stable graph of type (g, n) with respect to the marking of legs and the vertex-weight function As a notational advice, given v ∈ V nex (G) (resp. h ∈ H nex (G), resp. l ∈ L(G)), we will denote the corresponding element of G st by v (resp. h, resp. l). The edges of G st come with a natural partition E(G st ) := E nex (G st ) E exc (G st ) into exceptional and non-exceptional ones, that can be described by the following bijections . We will also use this notation: • given a non-exceptional edge e ∈ E nex (G st ) we will denote by e ∈ E nex (G) the unique non-exceptional edge of G such that e = e; • given an exceptional edge e = e v ∈ E exc (G st ) we will denote the corresponding (ordered) exceptional edges of G by e i := e i v for i = 1, 2. Note that, by contracting exactly one among the edges e 1 and e 2 for any e ∈ E exc (G), we get a (non-unique) stabilization morphism σ : Stable (resp. quasi-stable) graphs of type (g, n) with respect to morphisms of graphs form a category, that we will denote by SG g,n (resp. QSG g,n ). The stabilization procedure gives rise to a stabilization functor given by The stabilization π st : G st 1 → G st 2 of a morphism π : G 1 → G 2 is defined as follows: given an exceptional edge e v ∈ E exc (G st 1 ), if both e 1 v and e 2 v are mapped by π to a vertex w ∈ V (G 2 ), then π st (e v ) = w; if instead π maps at least one of e 1 v and e 2 v to an edge of G 2 , then it corresponds to a unique edge e in G st 2 and we set π st (e v ) = e. See Figure 1 for a picture of the stabilization functor.

Stability conditions
Let G be a finite graph. A divisor on G is a finite formal sum over the vertices of G with a v ∈ Z. Divisors on G form an abelian group, which we denote by Div(G). The degree of a divisor D = a v v is defined to be deg(D) = v a v . We write Div d (G) for the divisors of degree d. The subset Div 0 (G) is a subgroup of Div(G) and, for any d ∈ Z, the subset Div d (G) is naturally a torsor over Div 0 (G).
We now define what it means for a divisor on a quasi-stable graphs to be admissible and (semi)stable with respect to a (universal) stability condition. Recall from [KP19]: Definition 2.3. A universal stability condition φ of type (g, n) is an assignment of a function for any G ∈ SG g,n (called the stability condition induced by φ on G) such that for any morphism π : G 1 → G 2 in SG g,n we have that We will often denote φ G by φ if there is no danger of confusion. From the definitions, together with the fact that SG g,n admits a final object (namely the graph with one vertex of genus g and no edges), it follows that the integral number |φ G | is independent from G ∈ SG g,n ; it is therefore denoted by |φ| and called the degree of φ.
Remark 2.4. The space of universal stability conditions of type (g, n), denoted by V g,n , is an abelian group with respect to the sum for any G ∈ SG g,n and any v ∈ V (G).
The subset of V g,n formed by the universal stability conditions of degree d is denoted by V d g,n . Note that V 0 g,n is a subgroup of V g,n and V d g,n is a torsor with respect to V 0 g,n . Moreover, V 0 g,n is also a finite dimensional real vector space with respect to the scalar multiplication (λ · φ) G (v) := λ · φ G (v) for any G ∈ SG g,n and any v ∈ V (G).
Hence V d g,n is a real affine space with respect to the vector space V 0 g,n .
Given φ ∈ V g,n , we can define a stability condition φ G : V (G) → R on any graph G that is quasi-stable of type (g, n) by Note that also for a quasi-stable graph G we have that |φ G | = |φ|.
Definition 2.5. Fix an hyperbolic pair (g, n) and let G be a quasi-stable graph of type (g, n).
(2) Let φ ∈ V g,n be a universal stability condition.
is admissible of degree deg D equal to |φ| and the following inequalities hold for any subset S ⊆ V (G): where E(S, S c ) is the set of edges joining a vertex in S with a vertex in the complementary subset S c .
strict unless S or S c is a union (possibly empty) of exceptional vertices. (3) A universal stability condition φ ∈ V g,n is called general (2) if for any quasi-stable graph of type (g, n) we have that for any subset S ⊆ V (G) Some remarks on the above definition are in order.
Remark 2.6. Let us keep the notation of the above Definition 2.5.
(i) The two inequalities in (2.5) for S are equivalent to the two analogous inequalities (but in reverse order) for S c . Hence it is enough to require one of the two inequalities in (2.5) for any S ⊆ V (G). (ii) If S (resp. S c ) is a union of exceptional vertices then the second (resp. the first) inequality in (2.5) is always an equality for any admissible divisor D ∈ Div(G) and for any universal stability condition φ.
Hence the definition of φ-stability in (2) is sharp.
(iii) If S or S c is a union of exceptional vertices, the quantity φ G (S) + |E(S, S c )| 2 is always an integer for any universal stability condition φ (equal to |S| or to d + |S c |, respectively). Hence the definition of general universal stability condition in (3) is sharp. (iv) It follows from the proof of [MV12, Theorem 6.1(i), Proposition 7.3] that if a quasi-stable graph G of type (g, n) admits a φ-stable divisor for some φ ∈ V g,n then G has no separating exceptional vertices, i.e., G is simple. In particular, there are no exceptional vertices on bridges of G st on the support of a φ-stable divisor for a general stability condition. (v) It follows from the proofs of [MV12, Theorem 6.1(iii) and Proposition 7.3] that φ ∈ V g,n is general if and only if for any quasi-stable graph G of type (g, n) every φ-semistable divisor on G is φ-stable. This fact, combined again with [MV12, Theorem 6.1(iii)], shows that our definition of general universal stability conditions coincides with the one of [KP19, Definition 4.1].
(2) This is called non-degenerate in [KP19]. We prefer to call it general according to the terminology used in [MV12], [MRV17], [MRV19a], [MRV19b], [MSV21]. Note that in the first two papers, the term non-degenerate is used for a slightly weaker condition.

The category QDiv g,n
We now can define the category of quasi-stable graphs together with admissible divisors, and some variants of it, that will play a special role in what follows.
(1) Let QDiv g,n be the category whose objects are pairs (G, D) consisting of a quasi-stable graph G of type (g, n) and an admissible divisor D ∈ Div(G), and whose morphisms π : (G, D) → (G , D ) are the morphisms of the underlying graphs π : (2) We will consider the following full subcategories of QDiv g,n : (i) QDiv g,n,d is the full subcategory of QDiv g,n whose objects are pairs (G, D) ∈ QDiv g,n such that deg D = d. (ii) QDiv spl g,n (resp. QDiv spl g,n,d ) is the full subcategory of QDiv g,n (resp. QDiv g,n,d ) whose objects are pairs (G, D) ∈ QDiv such that G is simple. (iii) For any stability condition φ ∈ V g,n , QDiv g,n (φ) is the full subcategory of QDiv g,n whose objects are pairs (G, D) ∈ QDiv g,n such that D is φ-semistable. Note that, if φ is general, QDiv g,n (φ) is a full subcategory of QDiv spl g,n,d where |φ| = d. Remark 2.8. It is easy to check that: • QDiv spl g,n and QDiv g,n (φ) are "under-closed" subcategories of QDiv g,n in the following sense: if π : (G, D) → (G , D ) is a morphism in QDiv g,n and (G, D) ∈ QDiv spl g,n (resp. QDiv g,n (φ)), then also (G , D ) ∈ QDiv spl g,n (resp. QDiv g,n (φ)).
• QDiv g,n,d is a "morphism-closed" subcategory of QDiv g,n in the following sense: consider a morphism π : (G, D) → (G , D ) in QDiv g,n . If either (G, D) or (G , D ) belongs to QDiv g,n,d then both of them belong to QDiv g,n,d . Similarly, QDiv spl g,n,d is a morphism-closed subcategory of QDiv spl g,n . There is a forgetful-stabilization functor (2.6) F : QDiv g,n −→ QSG g,n st − −→ SG g,n that sends an object (G, D) ∈ QDiv g,n into the stabilization G st ∈ SG g,n of G and a morphism π : (G, D) → (G , D ) into the stabilization π st : G st → G st of π : G → G . The restrictions of F to the full subcategories QDiv g,n,d , QDiv spl g,n,(d) and QDiv g,n (φ) will be denoted again by F. Throughout the text, and in order to shorten the statements, when we write (d) in the context of any notation we mean that the sentence applies to the object in discussion either indicating d or not. For instance, when we write QDiv spl g,n,(d) we mean that it works for both QDiv spl g,n and for QDiv spl g,n,d . Given a stable graph G ∈ SG g,n , we will now define the essential fiber of F : QDiv g,n → SG g,n over G (and of its subcategories QDiv g,n,d , QDiv spl g,n,(d) and QDiv g,n (φ)). Definition 2.9 (The category QDiv G ). For any G ∈ SG g,n , let QDiv G be the category such that • the objects of QDiv G are triples (G, D, ρ) such that (G, D) ∈ QDiv g,n and ρ : G − → G st is an isomorphism in SG g,n ; • the morphisms π : (G, D, ρ) → (G , D , ρ ) are the morphisms π : (G, D) → (G , D ) in QDiv g,n such that π st • ρ 1 = ρ 2 .
Proof. Let π : (G, D, ρ) → (G, D, ρ) be an automorphism in QDiv G . The equality π st • ρ = ρ forces π st : G st → G st to be the identity. By the definition of the stabilization graph G st (see (2.1)), we get that the restriction of π V to V nex (G) is the identity and that the restriction of π H to H nex (G) is the identity. Since any exceptional-half edge of G, i.e. an half-edge of the form h i v for i = 1, 2 and v ∈ V exc (G), is conjugate to a non-exceptional half-edge, i.e. i G (h i v ) is non-exceptional, and using that π H commutes with i G , we deduce that π H = id. Finally, using that π : X(G) → X(G) commutes with the root map r G and that π H = id, we get for any exceptional vertex v ∈ V exc (G) and any i = 1, 2: = v, and we deduce that π V = id. Since π is uniquely determined by π V and π H , we conclude that π is the identity.
For later use, we will also need a quotient of the above categories in which two morphisms of (quasi-)stable graphs (or pairs formed by a quasi-stable graph and an admissible divisor) are identified if they induce the same map on the edge sets. These categories are important because they will turn out to be isomorphic to the category of strata of the corresponding toroidal stacks. We refer the reader to Figure 2 for the two types of automorphisms that do not permute the edges.
Definition 2.11 (The categories SG E g,n , QSG E g,n , QDiv E g,n ). Fix an hyperbolic pair (g, n). (i) Let QSG E g,n (resp. SG E g,n ) be the category whose objects are quasi-stable (resp. stable) graphs of type (g, n) and the morphisms from G to G are the equivalence classes of morphisms π : G → G in QSG g,n (resp. in SG g,n ) with respect to the equivalence relation The equivalence class of a morphism π : G → G in QSG g,n (resp. in SG g,n ) will be denoted by [π] : G → G . (3) (ii) Let QDiv E g,n be the category whose objects are pairs (G, D) consisting of a quasi-stable graph G of type (g, n) and an admissible divisor D ∈ Div(G), and whose morphisms from (G, D) to (G , D ) are the equivalence classes of morphisms π : (G, D) → (G , D ) in QDiv g,n with respect to the equivalence relation π ∼ π ⇔ π * E = π * E : E(G ) → E(G).
(3) In general, two equivalent morphisms of graphs map each pair of adjacent vertices to the same set. From this observation, one can check that given π 1 , π 2 : G → G , then [π 1 ] = [π 2 ] if and only if there is an automorphism τ : G → G with [τ] = [id] such that π 1 = τ • π 2 . We thank the referee for pointing out this clarification.
The equivalence class of a morphism π : (G, D) → (G , D ) in QDiv g,n will be denoted by [π] : where the vertical arrows are the essentially surjective and full (but non faithful) functors that are the identities on objects and send a morphism π into [π]. Finally, the above categories induce some natural partially ordered sets (posets) that we now introduce.
(i) Let QSG g,n (resp. SG g,n ) be the posets whose objects are isomorphism classes of quasi-stable (resp. stable) graphs of type (g, n) and G ≥ G ⇔ there exists a morphism π : G → G in QSG g,n (resp. in SG g,n ).
(ii) Let QDiv g,n be the poset whose objects are isomorphism classes of objects of QDiv g,n and Similarly, we can define the posets QDiv g,n,d , QDiv spl g,n,(d) and QDiv g,n (φ) .
In what follows we will often pass, implicitly, to skeleton categories of each of the categories defined above, picking one object arbitrarily for each isomorphism class. In particular, we will identify the objects in each category with the elements in the associated poset.
Remark 2.13. Observe that the binary relations ≥ in the above definition are clearly reflexive and transitive, while they are antisymmetric because any morphism π : G → G of (n-marked vertex-weighted) graphs is either an isomorphism or is such that π * E : E(G ) → E(G) is a proper inclusion. This also implies that the posets | SG g,n |, | QSG g,n |, | QDiv g,n,(d) | and | QDiv spl g,n,(d) | are graded with respect to the rank function given by the cardinality of the edge set. By following the lines of the proof of Proposition 4.11 and the first part of Theorem 4.15 in [AP20], one gets that | QDiv g,n (φ)| is also graded if φ is general (we do not include a proof here since it is not important for what follows). The lengths of the above posets with respect to this rank function are equal to l | SG g,n | = 3g − 3 + n, l | QSG g,n | = l | QDiv g,n,(d) | = 2(3g − 3 + n), l | QDiv

The universal tropical Jacobian as a cone stack
The aim of this subsection is to construct the universal tropical Jacobian as a cone stack, in the sense of [CCUW20, Section 2.1], endowed with a forgetful morphism to the cone stack M trop g,n of tropical curves, constructed in [CCUW20, Section 3]. By slight abuse of notation, by a rational polyhedral cone we will mean the data of a pair (σ , N ), consisting of a lattice N and a full-dimensional, strictly convex rational polyhedral cone in N R : an intersection σ of finitely many half spaces in N R which contain no line and span all of N R . The effect of this definition is that a rational polyhedral cone always comes with an integral structure, namely the intersection N ∩ σ , and a dual monoid Conversely, the rational polyhedral cone is recovered from S σ as Equivalently, it corresponds to a homomorphism of monoids S σ → S σ . Denote the category of rational polyhedral cones by RPC. We usually suppress the reference to N and simply write σ for a rational polyhedral cone.
One may think of a rational polyhedral cone σ as a combinatorial analogue of an affine scheme and of S σ as its ring of functions. Quite like a scheme is glued from affine schemes, we may think of a rational polyhedral cone complex as a combinatorial object that is glued from rational polyhedral cones along their faces. Rational polyhedral cone complexes form a category that we denote by RPCC. We refer the interested reader to [CCUW20, Definition 2.1] for a precise definition of this notion. The category RPCC carries a natural Grothendieck topology that is generated by embeddings of rational polyhedral cones as faces, the face topology.
Let us begin with the definition of the universal tropical Jacobian as a stack over the category RPCC of rational polyhedral cone complexes endowed with the face topology, or equivalently as a category fibered in groupoids over the category RPC of rational polyhedral cones (see [CCUW20, Proposition 2.3]). We go through the effort of using this formalism, rather than, say, the formalism of generalized cone complexes, as this is the formalism that will allow us to compare the universal tropical Jacobian with its logarithmic analogue.
(i) Let J trop g,n (RPC) be the category fibered in groupoids over RPC such that: n-markings and of genus g (or simply of type (g, n)), i.e. a pair consisting of a quasi-stable graph G(Γ ) of type (g, n) (called the underlying graph) and a (generalized) metric d : is a morphism in QDiv g,n such that π is compatible with f as in [CCUW20, Definition 3.2], i.e. such that for any e ∈ E(G(Γ )) we have that π contracts e if and only if f • the fibration J trop g,n (RPC) → RPC sends an object (Γ /σ , D) into σ and sends a morphism (ii) Given σ ∈ RPC, we will denote by J trop g,n (σ ) the essential fiber of J trop g,n (RPC) → RPC over σ , i.e. the groupoid whose objects are (Γ /σ , D) ∈ J trop g,n (RPC) and whose morphisms are the ones of the , the underlying morphism of graphs, is an isomorphism. (iii) The universal tropical Jacobian (over RPCC), denoted by J trop g,n , is the unique stack over RPCC (by [CCUW20, Proposition 2.3]) whose restriction to RPC coincides with J  g,n by taking only the objects (Γ /σ , D) such that, respectively, (G(Γ ), D) ∈ QDiv g,n,d , QDiv spl g,n,(d) and QDiv g,n (φ). The functor σ → S σ gives an equivalence of categories between rational polyhedral cones and the category of sharp, fine and saturated monoids. We can thus also think of J trop g,n (and similarly of J trop g,n,d , J trop, spl g,n,(d) and J trop g,n (φ)) as a category fibered in groupoids over the category of sharp fine and saturated monoids. We may expand J trop g,n to a category fibered in groupoids over the category ShpMon op of sharp integral and saturated monoids, whose fiber over a sharp integral and saturated monoid P is the groupoid of pairs (Γ , D) consisting of a quasistable tropical curve Γ of genus g with n marked legs, whose edges are metrized by the monoid P , and an admissible divisor D on G(Γ ). We refer the interested reader to [CCUW20, Section 5.2] for more background on this procedure.
The universal tropical Jacobian comes equipped with a morphism towards the stack M → S σ is defined (using the notation below (2.2)) by We will now prove that the universal tropical Jacobian J trop g,n is a cone stack, i.e. a geometric stack over RPCC (see [CCUW20, Definition 2.7]). We will achieve this by describing it as a combinatorial cone stack in the sense of [CCUW20, Definition 2.15], and then use that cone stacks are equivalent to combinatorial cone stacks by [CCUW20, Proposition 2.19]. The analogous description for M trop g,n is proved in [CCUW20, Section 3.4] and it realizes M trop g,n as the cone stack associated to the combinatorial cone stack (2.10) . Here RPC f denotes the category of rational polyhedral cones with face inclusions as morphisms.
(i) J trop g,n is the cone stack associated to the combinatorial cone stack g,n coincides with the morphism of combinatorial cone stacks from (Γ QDiv g,n : QDiv opp g,n → RPC f ) to (Γ SG g,n : SG opp g,n → RPC f ) induced by the functor F : QDiv g,n → SG g,n of (2.6) together with the following collection of contravariant and surjective morphisms of cones (for any (G, D) ∈ QDiv g,n ) .
The above Theorem remains true for the cone substacks J g,n by replacing QDiv g,n , respectively, with its full subcategories QDiv g,n,d , QDiv spl g,n,(d) or QDiv g,n (φ).
Proof. Let us prove part (i). Denote temporarily by J comb g,n the cone stack associated to the combinatorial cone stack (2.11) and let J comb g,n (RPC) be its restriction to RPC. According to [CCUW20, Definition 2.15, Proposition 2.19], an object of J comb g,n (RPC) is a triple (G, D, l : ≥0 is a morphism in RPC whose image is not contained in any proper face of R E(G) ≥0 . By composing with the projections along the coordinates, the morphism l is equivalent to the datum of a collection of surjective morphisms {l e : σ → R ≥0 } e∈E(G) in RPC. By passing to the toric monoids, this is equivalent to giving a collection of injective morphisms of monoids {l * e : N → S σ } e∈E(G) , which is indeed determined by the collection of non-zero elements {l * e (1) ∈ S σ } e∈E(G) . Therefore, the triple (G, D, l : σ → R E(G) ≥0 ) gives rise to (and it is completely determined by) an object (Γ /σ , D) of J trop g,n (RPC) such that Γ /σ is the n-marked genus g quasi-stable tropical curve whose underlying graph is G(Γ ) := G and whose metric is given by is the datum of a morphism of cones f : σ → σ and a morphism π : (G , D ) → (G, D) in QDiv g,n such that the following diagram commutes where the left arrow is induced by the inclusion π * E : E(G) → E(G ) that identifies the edges of G with the edges of G that are not contracted by π. In terms of the objects (Γ /σ , D) and (Γ /σ , D ) of J trop g,n (RPC) associated to, respectively, (G, D, l : ≥0 ) as explained above, the commutativity of the above diagram means exactly that π is compatible with f in the sense of Definition 2.14, and therefore we get a (uniquely defined) morphism (f , π) : The above discussion shows that we have an isomorphism J comb g,n (RPC) J trop g,n (RPC) of categories fibered in groupoids over RPC, which then gives rise to an isomorphism of stacks J comb Let us now prove part (ii). We will temporarily denote by M comb g,n the cone stack associated to the combinatorial cone stack (2.10), by M comb g,n (RPC) its restriction to RPC and by Φ comb : J comb g,n → M comb g,n the morphism of cone stacks induced by the morphism of combinatorial cone stacks described in part (ii). By definition, the morphism Φ comb is given on objects by .
In terms of the isomorphisms J comb g,n J trop g,n and M comb g,n M trop g,n , this corresponds to sending an object The above discussion shows that, under the isomorphisms J comb

Compactified universal Jacobian
The aim of this section is to introduce and study the compactified universal Jacobian over the moduli stack of stable curves. In particular, we give a combinatorial description of its toroidal stratification. We refer the reader to [CC19] for an alternative description of the toroidal stratification in degrees g and g − 1 (without marked points).
(i) The compactified universal Jacobian (of type (g, n)) is the algebraic stack J g,n parametrizing pairs (C → S, L) consisting of a family C → S of quasi-stable curves of type (g, n), i.e. n-pointed nodal projective and connected curves of arithmetic genus g whose dual graph is a quasi-stable graph, and an admissible line bundle L on C, i.e. a line bundle that has degree 1 on any exceptional component of every geometric fiber of C → S. (ii) The universal Jacobian (of type (g, n)) is the open and dense substack J g,n ⊂ J g,n parametrizing objects (C → S, L) ∈ J g,n (S) such that C → S is a family of smooth curves.
The decomposition of J g,n into connected components is where J g,n,d is the algebraic stack parametrizing pairs (C → S, L) ∈ J g,n (S) such that L has relative degree d on C → S.
We will be considering the following open substacks (for any universal stability condition φ ∈ V g,n ): is the open substack parametrizing pairs (C → S, L) ∈ J g,n (S) such that for any geometric point s of S the line bundle L s is φ-semistable on C s (i.e. its multidegree deg(L s ) is a φ-semistable divisor on the dual graph G(C s ) of the curve C s ), while J spl g,n is the open substack parametrizing pairs (C → S, L) ∈ J g,n (S) such that for any geometric point s of S the quasi-stable curve C s is simple, i.e. it remains connected when we remove its exceptional components, or equivalently its dual graph G(C s ) is simple. The stack J g,n (φ) is connected and it is contained in J g,n,|φ| , while J spl g,n admits the decomposition into connected components The stack J g,n comes equipped with a forgetful-stabilization morphism to M g,n that sends an object (C → S, L) ∈ J g,n (S) into the stabilization (C st → S) ∈ M g,n (S) of the family C → S of quasi-stable curves. If we denote by M qs g,n the stack of quasi-stable curves of type (g, n), then the forgetful morphism Φ factors as where the forgetful morphism Φ qs sends The algebraic group G m injects functorially into the automorphism group scheme Aut S (C/S, L) of every object (C → S, L) ∈ J g,n (S) as scalar multiplication on the line bundle L. Hence, we may form the Note that the forgetful morphism Φ qs (and hence also the forgetful-stabilization morphism Φ), factors through the G m -rigidification. By a slight abuse of notation, we will denote by Φ : J g,n G m → M g,n the induced morphism, as well as its restrictions to the open substacks J spl g,n G m and J g,n (φ) G m .
Remark 3.2. It follows from [EP16] that J g,n is isomorphic to the algebraic stack parametrizing pairs (X → S, I ) consisting of a family X → S of n-marked genus g stable curves and a coherent sheaf I on X , flat over S, whose geometric fibers are rank-1 torsion-free sheaves. In this alternative description: • the open substack J spl g,n parametrizes objects (X → S, I ) as above such that the geometric fibers of I are simple sheaves; • for any φ ∈ V g,n , the open substack J g,n (φ) corresponds to the stack defined in [KP19, Definition 4.2] (using [MV12, Theorem 6.1]).
• if φ ∈ V g,n is general, then J g,n (φ) is isomorphic to the Esteves' universal compactification J E, ss g,n (see [Est01]) for a suitable choice of a universal vector bundle E on the universal family over M g,n (see [Mel15,Proposition 4.17]). • the morphism Φ : J g,n → M g,n is the morphism that sends objects (X → S, I ) as above into (X → S) ∈ M g,n .
In the following Proposition, we collect all the properties of the stack J g,n and of the forgetful morphism Φ, that will be used in the sequel. (1) The G m -gerbe J g,n,d → J g,n,d G m is trivial if and only if either n > 0 or n = 0 and gcd(d − g + 1, 2g − 2) = 1. In particular, under the above numerical conditions, the universal line bundle over the universal family of J g,n,d descends to the universal family on the rigidification J g,n,d G m .
(2) The algebraic stack J spl g,n G m is the DM-locus of J g,n G m .
(3) The morphism Φ : J g,n → M g,n of (3.3), as well as its restriction to the open subsets J spl g,n and J g,n (φ) (for any φ ∈ V g,n ), satisfies the existence part of the valuative criterion for properness.
In particular, the stacks J g,n and J spl g,n satisfy the existence part of the valuative criterion, and J g,n (φ) (for any φ ∈ V g,n ) is universally closed.
(4) For a universal stability condition φ ∈ V g,n , the following conditions are equivalent (i) The stack J g,n is smooth and the boundary ∂J g,n : Part (5) follows since the obstruction to deform a line bundle on a quasi-stable curve C lies on H 2 (C, O C ), which is zero because C has dimension one.
Part (6) follows from part (5) and the well-known fact that M qs g,n is smooth and that ∂M qs g,n defined as M qs g,n \ M g,n is a normal crossing divisor. Alternatively, one could use the explicit description of the completed local rings of J g,n given in [BFV12, Section 2.3] or [CMKV15, Section 3] (ii) The poset of strata {J G,D } (G,D)∈QDiv g,n is anti-isomorphic to the poset | QDiv g,n | of Definition 2.12, i.e.
(iii) The forgetful-stabilization morphism Φ : J g,n → M g,n of (3.3) is toroidal and, for any G ∈ SG g,n , we have that where M G is the locally closed (reduced) substack of M g,n whose k-points consist of curves C ∈ M g,n (k) such that G(C) G.
Parts (i) and (ii) have been proved for J g,n (φ), in the special case n = 1 and for specific choices of φ ∈ V g,n , by Abreu-Pacini in [AP20, Proposition 6.4].
Proof. Let us first prove part ((i)). Since the toroidal embedding (J g,n ⊂ J g,n ) is the pull-back of the toroidal embedding (M g,n ⊂ M qs g,n ) via the smooth morphism Φ qs : J g,n → M qs g,n (see Proposition 3.3), the toroidal stratification of (J g,n ⊂ J g,n ) is formed by the connected components of the pull-backs of the strata that form the toroidal stratification of (M g,n ⊂ M qs g,n ). It is well-known that the toroidal stratification of (M g,n ⊂ M qs g,n ) is given by where M G is the locally closed (reduced) substack of M qs g,n whose k-points are C ∈ M qs g,n (k) such that G(C) G. It remains to observe that we have a disjoint union

The category of strata
We now want to show that the (anti-)isomorphism of posets in Proposition 3.4 is induced by an (anti-)equivalence of categories. In order to do that, we have to show that the poset of strata {J (G,D) } is induced by a category, namely the category of strata of (J g,n ⊂ J g,n ), that we are now going to define in a more general setting.
Assume that we have a toroidal embedding of Artin stacks (U ⊂ X ), locally of finite type over k. Consider the lisse-étale sheaves D X and E X over X such that, for every smooth morphism V → X with V a scheme, D X (V ) (resp. E X (V )) is the group of Cartier divisors on V (resp. the submonoid of effective Cartier divisors on V ) that are supported on V \ U where U := V × X U . Consider the strata {W i } of the toroidal embedding U ⊂ X (in particular each W i is irreducible) and pick a geometric generic point w i in each stratum W i . The stalk D X ,w i is a finitely generated free abelian group and the stalk E X ,w i is a sharp, saturated, and finitely generated submonoid that generates D X ,w i as a group.
Definition 3.5. The category of strata of the toroidal embedding (U ⊂ X ), denoted by Str(U ⊂ X ) or simply by Str(X ), is the category whose objects are the toroidal strata {W i } and whose morphisms W i → W j are generated by the following two classes of morphisms: • the étale specializations w i w j (see [CCUW20, Appendix A]) whenever W i W j ; • the image H W i (called the monodromy group of the stratum W i ) of the natural monodromy representation πé t 1 (W i , w i ) → Aut mon (E X ,w i ) whenever W i = W j . The category Str(X ) comes also equipped with the structure of a category fibered in groupoids over the category RPC f of rational polyhedral cones with face inclusions as morphisms. Namely, to any stratum W i ∈ Str(X ), we associate the rational polyhedral cone Moreover, any morphism W i → W j in Str(X ) induces a surjective monoid homomorphism E X ,w j E X ,w i (and it is completely determined by it), and hence it induces a face morphism σ (W i ) → σ (W j ) of rational polyhedral cones. In this way we get a functor (3.8) which is easily checked to be a category fibered in groupoids.
In the following Proposition 3.6, we are going to relate Str(J g,n ) with the category QDiv E g,n introduced in Definition 2.11. Observe also that the category fibered in groupoids Γ QDiv g,n : QDiv opp g,n → RPC f of (2.11) factors through a category fibered in groupoids (3.9) where the face inclusion R ≥0 is induced by the injection π * E : E(G ) → E(G). The following Proposition 3.6 is crucial for our understanding of both the analytic and the logarithmic tropicalization map.
Proposition 3.6. Let Str(J g,n ) be the category of strata of the toroidal embedding (J g,n ⊂ J g,n ). There is an equivalence of categories such that Γ QDiv E g,n = Γ Str(J g,n ) • E. Before giving a proof, we will need the following Lemma where we describe a rigidification of each stratum J (G,D) of J g,n . For a similar description of the strata of M g,n (and also of their closure), see [ACG11, Chapter XII, Proposition (10.11)] (or [EF12, Proposition 4.6] for a more algebraic proof).
Let (G, D) ∈ QDiv g,n . Consider the category fibered in groupoids J (G,D) over the category of schemes, whose fiber over a scheme S consists of the groupoid of objects (C → S, L) ∈ J g,n (S) together with isomorphisms φ s : (G, D) − → (G(C s ), degL s ) in QDiv g,n for every geometric point s → S, that are compatible with étale specializations in the following sense: for any étale specialization f : t s of geometric points, with induced morphism π f : (G(C s ), degL s ) → (G(C t ), degL t ) in QDiv g,n , we require that φ t = π f • φ s .
The group Aut(G, D) (considered as a constant group scheme over k) acts on J (G,D) by precomposition with the isomorphisms φ s . A slightly different presentation of the strata of J g,n (φ), in the special case n = 1 and for specific choices of φ, is given in [AP20, Proposition 6.1].
Proof. Let us first prove part (i). Consider the category fibered in groupoids M G over the category of schemes, whose fiber over a scheme S consists of families of quasi-stable curves (C → S) ∈ M qs g,n (S) together with an isomorphism φ s : G − → G(C s ) in QSG g,n for every geometric point s → S, that are compatible with étale specializations in the following sense: for any étale specialization f : t s of geometric points, with induced morphism of graphs π f : We claim that we have an isomorphism of categories fibered in groupoids which implies that M G is a smooth and irreducible algebraic stack of finite type over k.  In order to prove this, consider the commutative diagram where M G is the locally closed (reduced) substack of M qs g,n whose k-points are C ∈ M qs g,n (k) such that G(C) G, the morphism Φ G forgets the isomorphisms {φ s }, the morphism Ψ (G,D) forgets the line bundle, the square is cartesian and the morphism F (G,D) is induced by the universal property of the fibered product. Now, property (b) will follow from the following two properties: (b1) Φ G is representable, finite and unramified.
Indeed, it is enough to show that Φ G is representable, finite and unramified. It follows from (3.11) (and its proof) that Φ G is a composition of clutching morphisms which are known to be representable, finite and unramified by [Knu83, Cor. 3.9]. Proof of Proposition 3.6. First of all, we need a local description of the boundary ∂J g,n ⊂ J g,n around a geometric point (C, L) ∈ J g,n (K). The morphism Φ qs : J g,n → M qs g,n is smooth and the boundary ∂J g,n of J g,n is the pull-back of the boundary ∂M qs g,n of M g,n . From the well-known deformation theory of nodal curves it follows that (C, L) admits a smooth chart V (C,L) → J g,n such that the pull-back ∂V (C,L) of the boundary ∂J g,n is a normal crossing divisor whose irreducible components are {D e } e∈E(G(C)) , where D e parametrizes all the partial smoothings of (C, L) for which the node n e corresponding to the edge e ∈ E(G(C)) persists.
From this local description of ∂J g,n , it follows that the stalk of the sheaf E J g,n at the geometric point (C, L) ∈ J g,n (K) is canonically isomorphic to (3.13) E J g,n ,(C,L) = N E(G(C)) .
In particular, the pull-back of the sheaf E J g,n to the finite and étale cover J (G,D) → J (G,D) of Lemma 3.7 is equal to the constant sheaf Recall that our goal is to define an equivalence of categories E : (QDiv E g,n ) opp −→ Str(J g,n ) such that we have the equality of functors (3.17) Γ QDiv E g,n = Γ Str(J g,n ) • E. We define E on objects by sending (G, D) ∈ QDiv E g,n into J (G,D) ∈ Str(J g,n ), so that E will be essentially surjective by Proposition 3.4(i). Moreover, (3.16) says that equality (3.17) holds true at the level of objects.
It remains to define E on morphisms in such a way that it is fully faithful and (3.17) is satisfied at the level of morphisms. Since any morphism in QDiv g,n (and hence also on QDiv E g,n ) is a composition of edge contractions and automorphisms and any morphism in Str(J g,n ) is a composition of étale specializations (between the geometric generic points of the strata) and automorphisms, it will be enough to treat these two classes of morphisms separately.
Edge contractions and étale specializations. Fix an element (G, D) ∈ QDiv E g,n . Given a subset S ⊆ E(G), denote by [π S ] : (G, D) → (G , D ) the unique morphism in QDiv E g,n induced by the morphism of graphs π S : G → G/S := G which is the contraction of the edges belonging to S (so that D := (π S ) * (D)). Note that . On the other hand, to a subset S ⊆ E(G) we can associate the étale specialization sp S : η (G ,D ) η (G,D) such that, on a smooth chart V (G,D) of the geometric point η (G,D) (as explained at the beginning of the proof), the pull-back of η (G ,D ) is the geometric generic point of the intersection e∈S D e (that is irreducible if the chart is small enough). Using the identification (3.15), the associated surjective monoid homomorphism In this way, we get f S : J (G ,D ) → J (G,D) , a morphism in Str(J g,n ) corresponding to the étale specialization sp S , with the property that, using the identification (3.16), the face inclusion of cones

is induced by the inclusion E(G ) = E(G) \ S → E(G).
We now define E([π S ]) = f S for any S ⊆ E(G) and we are done since any edge contraction in QDiv E g,n with domain (G, D) is equal to [π S ] for a unique S ⊆ E(G) and any morphism in Str(J g,n ) with codomain J (G,D) and which is induced by étale specializations (between the geometric generic points of the strata) is equal to f S for a unique S ⊆ E(G). Automorphisms. By the definition of the morphisms in QDiv E g,n (see Definition 2.11), the automorphism group in QDiv E g,n of an object (G, D) ∈ QDiv E g,n is equal to where Aut(G, D) is the automorphism group of (G, D) in QDiv g,n and S E(G) is the permutation group on the set E(G). Moreover, by the definition (3.9) of the functor Γ QDiv E g,n , the homomorphism

is given by the restriction of the natural action of S E(G) on R E(G)
≥0 by permutation of the extremal rays (note that indeed we have that Aut  such that Γ SG E g,n := Γ Str(J g,n ) • F : (SG E g,n ) opp −→ RPC f is the factorization of the category fibered in groupoids Γ SG g,n of (2.10) through the quotient category SG opp g,n → (SG E g,n ) opp .
Moreover, the two equivalences of categories (3.10) and (3.20) are compatible with the functor F E : QDiv E g,n → SG E g,n of (2.7) and the functor induced by the toroidal morphism Φ : J g,n → M g,n (see Proposition 3.4(iii)).

Logarithmic tropicalization
The aim of this section is to define and study a logarithmic tropicalization map from a logarithmic universal Jacobian to a universal tropical Jacobian defined over the category of logarithmic schemes.

The logarithmic universal Jacobian
Throughout the section, all logarithmic schemes (or stacks) are fine, saturated and locally of finite type over the base field k. A logarithmic scheme (or stack) will be denoted by X = (X, α X : M X → O X ), or simply by X = (X, M X ), where X is the underlying scheme, M X is a sheaf of monoids and α X : M X → (O X , ·) is a morphism of sheaves of monoids which is an isomorphism over the subsheaf (O * X , ·) ⊂ (O X , ·) of invertible functions. We will denote by M X := M X /α −1 X (O * X ) the characteristic monoid sheaf of X. The category of logarithmic schemes (resp. logarithmic algebraic stacks) will be denoted by LSch (resp. LSta).
The forgetful functor from logarithmic algebraic stacks to algebraic stacks is also called the trivial logarithmic structure on X. The unit of the above adjoint pair of functors gives rise to a morphism of logarithmic algebraic stacks (4.1) Υ X : X = (X, M X ) −→ X = (X, O * X ) for any logarithmic algebraic stack X.
As usual, the 2-category LSta embeds into the 2-category of categories fibered in groupoids over LSch by sending X ∈ LSta into HOM LSta (−, X). Observe that the category fibered in groupoids over LSch associated to X is given by X(S) = X(S) for any S ∈ LSch.
Definition 4.1. The logarithmic universal Jacobian, denoted by J log g,n , is the category fibered in groupoids over LSch whose fiber over S is the groupoid whose objects are pairs consisting of • a quasi-stable logarithmic curve X → S of type (g, n), i.e. an integral and saturated logarithmically smooth morphism X → S such that the underlying morphism X → S is a family of quasi-stable curves of type (g, n); • an admissible line bundle L on X.
Remark 4.2. Logarithmic curves (quasi-stable or not) have been characterized in [Kat00, Theorem 1.3]. In particular, given a logarithmic curve π : X → S, the stalk of the characteristic monoid M X at a geometric point x of X has one of the following forms: • M X,x M S,π(x) if x is a smooth point of X → S which is not a marked point; which is called the smoothing parameter of the node x.
Denote by M qs g,n the algebraic stack of quasi-stable curves of type (g, n) and by M log, qs g,n the logarithmic algebraic stack of quasi-stable logarithmic curves of type (g, n). If we consider J g,n and M qs g,n as logarithmic algebraic stacks (by endowing them with the trivial logarithmic structure, as explained above), then it is an immediate consequence of Definition 4.1 that In the next proposition, we will denote by M ∂J g,n the divisorial logarithmic structure on J g,n associated to the normal crossing divisor ∂J g,n (see Proposition 3.36). Moreover, the natural morphism Υ J log g,n of (4.1) is given by for any logarithmic scheme S.
Proof. The arguments in [Kat00] imply that M  Explicitly, the stack J Explicitly, the stack J log g,n (φ) parametrizes those objects (X → S, L) in J log g,n such that for every geometric point s of S the line bundle L s on X s is φ-semistable, while the stack J log, spl g,n,(d) parametrizes those objects (X → S, L) in J log g,n such that for every geometric point s of S the curve X s remains connected when we remove its exceptional components (resp. and the line bundle L s has total degree d on X s ). where M log g,n is the logarithmic algebraic stack parametrizing stable logarithmic curves of type (g, n), which is representable by (M g,n , M ∂M g,n ) by [Kat00], and the logarithmic stabilization morphism st log is defined in [CCUW20, Section 8.4].

Lifting the universal tropical Jacobian
We next define a tropical universal Jacobian as a category fibered in groupoids over the category of logarithmic schemes.
Definition 4.5. The tropical universal Jacobian over LSch, denoted by J trop g,n , is the category fibered in groupoids over the category of logarithmic schemes such that the fiber over S is the groupoid whose objects consists of is a morphism in QDiv g,n ) and such that π f is compatible with f * : M S,s → M S,t , i.e. such that for any e ∈ E(G(Γ s )) we have that π f contracts e if and only if f * (d Γ s (e)) = 0; -if π f (e) = e ∈ E(G(Γ t )) then f * (d Γ s (e)) = d Γ t (e ).
We can define the subcategories J is defined (using the notation below (2.2)) by There is a close relationship between the universal tropical Jacobian J trop g,n over RPCC (see Definition 2.14) and the above defined universal tropical Jacobian J trop g,n over LSch, that we now want to explain. As explained in [CCUW20, Section 6], there is an equivalence of 2-categories (4.5) a * : Cone stacks − −→ Artin fans over k , where an Artin fan over k is a logarithmic algebraic stack that is étale locally isomorphic to the stack quotient of a toric k-variety by its big torus (endowed with its natural toric logarithmic structure). Explicitly, the above equivalence a * sends a rational polyhedral cone σ to . More generally, the equivalence a * sends the cone stack associated to a combinatorial cone stack Ψ : C → RPC f (in the sense of [CCUW20, Section 2]) to the Artin fan which is equal to the following colimit in the category of logarithmic algebraic stacks where all the morphisms appearing in the above colimit are open embeddings. We refer the interested reader to [ACMW17, ACM + 16, Uli19, CCUW20, Uli21] for more details on the theory of Artin fans. Proof. Our proof is a generalization of the proof of [CCUW20, Lemma 7.9]. Let S be a logarithmic scheme. Following the discussion after Definition 2.14 we may consider J The second assertion, namely that Φ trop a * Φ trop , follows from the compatibility of this isomorphism with stabilization. Proof. Using Proposition 4.6, it is enough to exhibit the desired stratification for a * J trop g,n . Using Theorem 2.16(i) and (4.6), we get that a * J trop g,n is equal to the following colimit in LSta where the morphisms in LSta appearing in the above colimits are given as follows: to a morphism π : (G, D) → (G , D ) of QDiv g,n we associate the morphism in LSta induced by the map π * E : E(G ) → E(G) that identifies E(G ) with the subset of E(G) consisting of edges of G that are not contracted by π : G → G .
Therefore, we get a stratification of a * J trop g,n into locally closed subsets by considering, for any (G, D) ∈ QDiv g,n , the image in , and J trop g,n (φ) = a * J trop g,n (φ) . Moreover, the colimit description (4.8) and the stratification (4.7) hold true for the above mentioned subcategories provided that we substitute the category QDiv g,n by, respectively, QDiv g,n,d , QDiv spl g,n,(d) or QDiv g,n (φ).

A modular tropicalization morphism
We can now define a modular logarithmic tropicalization map. where δ n e is the smoothing parameter (in the sense of Remark 4.2) of the node n e of X s corresponding to e. The pair (G(X s ), d π ) defines a quasi-stable tropical curve over M S,s of type (g, n), that we denote by Γ (X s )/M S,s . We denote by deg(L s ) the multidegree of the line bundle L s := L |X s on X s , which is naturally an admissible divisor on G(X s ). Moreover, given an étale specialization f : t s of geometric points of S, the family π : X → S induces a natural morphism of graphs π f : G(X s ) → G(X t ) that is compatible with f * : M S,s → M S,t and such that (π f ) * (deg(L s )) = deg(L t ). Then we define (4.10) From the above definition of trop J log g,n , the explicit descriptions of the maps Φ log of (4.3) and Φ trop of (4.4), we get the commutativity of the following diagram Before proving the main properties of the map trop J log g,n in the following Theorem, recall [ACMW17, Proposition 3.2.1] that for any logarithmic stack X (fine, saturated and locally of finite type over a base field k) there exists an Artin fan A X with faithful monodromy together with a natural strict morphism of logarithmic algebraic stacks (that we like to call the functorial logarithmic tropicalization morphism of X ) (4.12) trop X : X −→ A X that is initial among all strict morphisms to an Artin fan with faithful monodromy. Clearly, the map trop X is functorial with respect to strict morphisms of logarithmic algebraic stacks, i.e. for any strict morphism φ : X → Y of logarithmic stacks there exists a strict morphism A(φ) : A X → A Y of Artin fans that fits into a commutative diagram for every (G, D) ∈ QDiv g,n . In particular, we have that • ( trop J log ) −1 (J g,n (φ)) for any universal stability condition φ ∈ V g,n .
Note that the morphisms of logarithmic algebraic stacks Φ log and Φ trop are not strict, so that the existence of the morphisms of Artin fans A( Φ trop ) and A(Φ log ) in part (iii) does not come automatically from the functoriality of the logarithmic tropicalization morphism trop X .
Proof. Part (i) follows along the lines of the proof of [CCUW20, Theorem 7.12]. Given a quasi-stable tropical curve Γ over a monoid P together with an admissible divisor D on G(Γ ), we can always find a quasi-stable logarithmic curve X together with an admissible line bundle L such that the dual tropical curve of X is equal to Γ and the multidegree of L is equal to D. This shows that trop J log g,n is surjective.
In order to show that trop J log g,n is strict, we need to check that for every scheme S together with a morphism From the proof of [CCUW20, Theorem 7.12] we obtain such an (X, M X ) by modifying the logarithmic structure on (X, M X ) and we set L = L in order to find an (X, M X , L ) with the desired properties.
Since J trop g,n is logarithmically étale over the base, we have that trop J log g,n is logarithmically smooth, since J log g,n is logarithmically smooth over k. Since trop J log g,n is also strict, the map trop J log g,n is smooth (in the classical non-logarithmic sense), since strict logarithmically smooth morphisms are automatically smooth.
Let us now prove part (ii). It follows from [Uli19, Proposition 4.5] that A J log g,n is equal to the Artin fan associated, via the equivalence (4.5), to the (combinatorial) cone stack Γ Str(J g,n ) : Str(J g,n ) → RPC f of (3.8). On the other hand, since J trop g,n is the Artin fan associated to the (combinatorial) cone stack Γ QDiv g,n : QDiv opp g,n → RPC f of (2.11) (by Proposition 4.6), it follows from [Uli19, Cor. 4.6] that A J trop g,n is the Artin fan associated to the (combinatorial) cone stack Γ QDiv E g,n : (QDiv E g,n ) opp → RPC f of (3.9). Proposition 3.6, together with the modular description of trop J log Let us prove part (iii). The commutativity of the left square of (4.14) has been already observed in (4.11). We now define a morphism A( Φ trop ) : A J trop g,n → A M trop g,n of Artin fans making commutative the central square of (4.14). Consider the commutative diagram of combinatorial cone stacks (4.16) Γ QDiv g,n : QDiv where the two horizontal arrows are the natural factorizations of Γ QDiv g,n (see (2.11)) and Γ SG g,n (see (2.10)) through the quotient categories in Definition 2.11, the left vertical arrow is given in Theorem 2.16(ii) and the right vertical arrow is the induced morphism. Passing to the Artin fans associated to the above combinatorial cone stacks, we get the required commutative diagram

Analytic tropicalization
The aim of this section is to describe the relation between the universal tropical Jacobian and the analytification of the universal compactified Jacobian. In this section, we work over a fixed algebraically closed field k on which we put the trivial valuation.

Analytification and functorial analytic tropicalization
In this subsection, we are going to review the definition of the (Berkovich) analytification of an Artin stack and of the functorial analytic tropicalization map of a toroidal embedding of Artin stacks. We will mostly follow the presentation and notation of [Uli19] (that generalizes the previous works of [Thu07] and [ACP15]) although we will describe the tropicalization map in the special case of toroidal embeddings of stacks and not for arbitrary logarithmic stacks.
Let X be an Artin stack locally of finite type over k. It is shown in [Uli19, Section 5.1] (generalizing the construction of Thuillier [Thu07] for schemes) that one can associate to X (in a functorial way) a strict k-analytic stack X , i.e. a stack in the category of strict analytic k-spaces endowed with the G-étale topology having representable diagonal and admitting a G-smooth atlas. We will call X the beth-analytification of X .
We refrain from giving the definition of X (since we will not need it in this paper), but we recall that the topological space associated to X admits the following explicit description where R varies among all the rank-1 valuation rings containing k and the equivalence relation ∼ is defined as follows: we say that Spec R → X is equivalent to Spec R → X if there exists another rank-1 valuation ring R containing both R and R , and a 2-isomorphism between the two natural morphisms Spec R → Spec R → X and Spec R → Spec R → X . In particular, note that every point of X can be represented by a morphism Spec R → X where R is a complete rank-1 valuation ring with an algebraically closed residue field. The topological space |X | admits an anticontinuous surjective map, called the reduction map, to the topological space |X | underlying the stack X which is defined by sending a point [φ : Spec R → X ] ∈ X to the image via φ of the special point o ∈ R. More precisely, we have where m R is the maximal ideal of R.
Remark 5.1. Note that if X is proper then, by the valuative criterion of properness, we may describe the topological space underlying X as where K varies among all the rank-1 valuation fields extending k with trivial valuation, and the equivalence relation is defined as above. This is the underlying topological space |X an | of the non-Archimedean analytification X an associated to X in the sense of [Uli17, Section 2.3].
Assume now that we have a toroidal embedding of Artin stacks (U ⊂ X ), locally of finite type over k. Associated to (U ⊂ X ) there is a canonical topological space Σ(X ), together with a canonical compactification Σ(X ), and a functorial analytic tropicalization map trop an X : |X | → Σ(X ), that we are now going to review following [Uli19] (which deals with the more general situation of logarithmic Artin stacks) and [ACP15] (which deals with the special case of toroidal embeddings of DM-stacks).
As in §3.2, consider the lisse-étale sheaves D X and E X over X such that, for every smooth morphism V → X with V a scheme, D X (V ) (resp. E X (V )) is the group of Cartier divisors on V (resp. the submonoid of effective Cartier divisors on V ) that are supported on V \ U where U := V × X U . Consider the category of strata Str(X ) = {W i } of the toroidal embedding U ⊂ X (as defined in §3.1) and pick a geometric generic point w i in each stratum W i . By composing the functor (3.8) with the natural inclusion RPC f ⊂ Top, we get a functor (5.4) We can also upgrade the above functor Γ X to a functor taking values on compact topological spaces as follows. To any stratum W i ∈ Str(X ), we associate the extended cone which is a canonical compactification of σ (W i ). Moreover, any morphism W i → W j in Str(X ) induces a surjective monoid homomorphism E X ,w j E X ,w i , and hence it induces an extended face morphism σ (W i ) → σ (W j ) of extended cones. In this way, we get a functor (5.5) The generalized cone complex Σ(X ) and the generalized extended cone complex Σ(X ) (in the terminology of [ACP15, Section 2]) of the toroidal embedding U ⊂ X are given by where the colimit is with respect to the two functors (5.4) and (5.5). We have a stratification into locally closed subsets (see [ACP15, Proposition 2.6.2]) and H W i is the monodromy group of the stratum W i . The functorial analytic tropicalization map trop an X : X → Σ(X ) is defined as follows. Consider a point [ψ : Spec R → X ] of |X |, where R is an integral domain which is complete with respect to a rank-1 valuation val R : R → R ≥0 ∪ {∞}. Let x be the image via ψ of the closed point of Spec R and let W i be the toroidal stratum containing x. Since E X is étale locally constant along each stratum W i (see [ACP15, Proposition 6.2.1]), any étale specialization s : w i x induces by pull-back an isomorphism s * : E X ,x − → E X ,w i , which is well-defined up to the action of the monodromy group H W i on E X ,w i . Fix an étale specialization s : w i x and consider the chain of monoid homomorphisms where the morphism ψ sends an effective Cartier divisor D with local equation f ∈ O X ,x at x to the element ψ (f ) ∈ R (well-defined up to units) which is not a unit of R since x ∈ W i , and val R is induced by the valuation val R : R → R ≥0 ∪ {∞} using that the units R * are the only elements of R having valuation zero. Then trop an X [ψ : Spec R → X ] is the image in Σ(X ) of the monoid homomorphism (5.8) which is a well-defined element of σ (W i ) o /H W i . The map trop an X is continuous, surjective and proper, and it is functorial with respect to toroidal morphisms.

Remark 5.2. The functorial analytic tropicalization map trop an
X has a section J X : Σ(X ) → X such that the composition p X = J X • trop an X : X −→ X is a strong deformation retraction onto the non-archimedean skeleton of |X |, see [Thu07], [ACP15], and [Uli19, Proposition 6.3], as well as [Ran17, Section 2.6] for a generalization to toroidal Artin stacks.

Analytic tropicalization of the universal Jacobian
By applying the construction of the previous section §5.1 to the forgetful morphism Φ : J g,n → M g,n , which is toroidal by Proposition 3.4(iii), we get the following commutative diagram of continuous maps of topological spaces (5.9) J g,n trop an Remark 5.3. The decomposition (3.1) of J g,n into connected components induces the following decomposition into connected components trop an Moreover, we have that can be identified with the closed subspace red −1 J g,n |J spl g,n | ⊂ J g,n ; • Σ(J Analogous statements hold for J g,n (φ) for any universal stability condition φ ∈ V g,n .
The aim of this subsection is to describe the above diagram in terms of tropical geometry. First of all, let us introduce the relevant tropical objects.

Definition 5.4 ((Quasi-)stable (extended) tropical curves).
(1) A (quasi-)stable (resp. extended) tropical curve Γ of type (g, n) is a (quasi-)stable graph GΓ ) of type (g, n) together with a metric d Γ : (2) The stabilization of a quasi-stable (resp. extended) tropical curve Γ of type (g, n) is the stable (resp. extended) tropical curve Γ st of type (g, n) such that • the underlying graph is G( is defined (using the notation below (2.2)) by We now introduce the generalized (extended) cone complex associated to J trop g,n .
(1) The generalized (resp. extended) cone complex associated to J trop g,n is defined as the colimit which are obtained as colimits in the category of topological spaces Top of, respectively, the diagram (5.11) Γ top QDiv g,n : QDiv where Γ QDiv g,n is the functor (2.11), and its natural extension (5.12) Γ top QDiv g,n : QDiv opp g,n −→ Top, using extended cones.
(2) We denote by Φ trop the forgetful-stabilization morphism of generalized (resp. extended) cone complexes induced by, respectively, the morphism of diagrams Γ QDiv g,n : QDiv opp g,n → RPC f → Γ SG g,n : SG opp g,n → RPC f described in Theorem 2.16(ii) composed with the inclusion RPC f ⊂ Top, and its natural extension using extended cones.
Remark 5.6. From the definition (2.11) of the functor Γ QDiv g,n : QDiv opp g,n → RPC f , it follows that the morphism Γ QDiv g,n (π) : R ≥0 in RPC f induced by a morphism π : (G, D) → (G , D ) in QDiv g,n depends only from π * E : E(G ) → E(G). This implies that the functor Γ QDiv g,n factors through the functor Γ QDiv E g,n of (3.9) and hence that J trop g,n can also be described as the colimit of the diagram (5.14) Γ top QDiv E g,n : QDiv Similarly, the functor Γ top QDiv g,n of (5.12) factors through a functor Note that J trop g,n (resp. J trop g,n ) naturally parametrizes pairs (Γ , D), where Γ is a quasi-stable (resp. extended) tropical curve of type (g, n) and D is an admissible divisor on the underlying graph G(Γ ). Therefore, using also the presentations (5.10), J  D). Moreover, since the maps appearing in the colimits (5.10) are face inclusions (and hence closed embeddings), we deduce that, given (G, D), (G , D ) ∈ QDiv g,n , we have that Similarly, M trop g,n (resp. M trop g,n ) naturally parametrizes stable (resp. extended) tropical curves of type (g, n) (see [BMV11] and [ACP15]) and, moreover, the map Φ trop sends (Γ , D) ∈ J and J trop g,n (φ) ⊂ J trop g,n (φ) for any universal stability condition φ ∈ V g,n , by replacing QDiv g,n with, respectively, QDiv g,n,d , QDiv spl g,n,(d) and QDiv g,n (φ). The stratification (5.16) and its modular description extends easily to these new generalized (resp. extended) cone complexes.
From Remark 2.8 and Definition 5.5, it follows that: • we have a decomposition into connected components • J trop, spl g,n,(d) (resp. J trop, spl g,n,(d) ) is a generalized (resp. extended) cone subcomplex, and hence a closed subspace, of J trop g,n,(d) (resp. J trop g,n,(d) ).
We now define a modular analytic tropicalization map.
Definition 5.8. The modular analytic tropicalization map (for J g,n ) is the map (5.18) trop an J g,n : J g,n −→ J trop g,n defined as follows. Consider a point Spec R → J g,n ∈ J g,n , where R is an integral domain which is complete with respect to a rank-1 valuation val R : R → R ≥0 ∪ {∞} and having an algebraically closed residue field, and let (C → Spec R, L) be the induced family of quasi-stable curves endowed with an admissible line bundle. On the dual graph G(C s ) of the special fiber C s of C → Spec R we consider the metric where f e is an element of R, well-defined up to units and which is not a unit, such that an étale local equation for C at the node n e of C s is given by xy = f e , and val R is induced by the valuation val R using that the units R * of R are the only elements of valuation zero. The pair (G(C s ), d C ) defines a quasi-stable extended tropical curve of type (g, n), that we denote by Γ (C s ), and we set • ( trop an J g,n ) −1 J trop g,n (φ) = J g,n (φ) = (red J g,n ) −1 |J g,n (φ)| for any universal stability condition φ ∈ V g,n .
Parts (i), (ii) and (iii) for J g,n (φ), in the special case n = 1 and for specific choices of φ ∈ V g,n , have been proved by Abreu-Pacini in [AP20, Theorem 6.9]. Both their proof and ours follow the blueprint provided by [ACP15] and go by giving an explicit description of the toroidal stratification (see Proposition 3.6 above).
Proof. Let us first prove part (iv): consider a point Spec R → J g,n ∈ |J g,n |, where R is an integral domain which is complete with respect to a rank-1 valuation val R : R → R ≥0 ∪ {∞} and having an algebraically closed residue field k, and let (C → Spec R, L) be the induced family of quasi-stable curves endowed with an admissible line bundle. By (5.2), we have that where s is the special point of Spec R. On the other hand, by Definition 5.8, we have that where Γ (C s ) is the quasi-stable tropical curve of type (g, n) whose underlying graph is G(C s ) and whose metric is the metric d C in (5.19). In particular, we have that By combining (5.24), (5.25) and (5.26), and recalling the definition of J (G,D) from Proposition 3.4(i) and of J trop (G,D) from (5.16), we conclude that (for any given (G, D) ∈ QDiv g,n ) Spec R → J g,n ∈ ( trop an J g,n ) −1 (J trop (G,D) ) ⇔ Spec R → J g,n ∈ (red J g,n ) −1 |J (G,D) | .
The last assertion follows from what we already proved together with Remarks 5.3 and 5.7. Part (i) follows immediately by combining Remark 5.6, the definitions (5.6) and Proposition 3.6. Let us now prove part (ii), using the notation already introduced in the proof of parts (iv) and (i). Fix (G, D) ∈ QDiv g,n such that (C s , L s ) ∈ J (G,D) (k). From the the proof of Proposition 3.6 (see (3.13) and the deformation-theoretic arguments that precedes it), it follows that we have a canonical identification E J (G,D) ,(C s ,L s ) = N E(G(C s )) and that the monoid homomorphism appearing in (5.8) is induced by the morphism E(G(C s )) → (R \ R * )/R * that sends e ∈ E(G(C s )) into [f e ] ∈ (R \ R * )/R * where f e ∈ R \ R * is such that an étale local equation for C at the node n e is given by xy = f e . We deduce that the monoid homomorphism is induced by the metric d C : E(G(C s )) → R >0 ∪ {∞} of (5.19). From this, it follows that the isomorphism Ψ J g,n constructed in (i) sends the element trop an

Fibers of the universal tropical Jacobian
In this subsection, we will study the "fibers" of the forgetful-stabilization morphism More precisely, we will describe the fiber product of Φ trop : J Definition 6.1. Let Γ /σ be a stable tropical curve over σ ∈ RPC and let G := G(Γ ) ∈ SG g,n be its underlying stable graph. The Jacobian cone space of Γ /σ is the (combinatorial) cone space Jac Γ /σ associated to the category fibered in groupoids QDiv opp G −→ RPC f given by the following: • To any object (G, D, ρ) ∈ QDiv G , we associate the following fibered product over σ of rational polyhedral cones over σ : where the fibered product (σ × R ≥0 R 2 ≥0 ) e is with respect to the following morphism of cones: the morphism R 2 ≥0 → R ≥0 is the addition map that sends (a, b) into a + b, while the morphism σ → R ≥0 is dual to the following morphism of toric monoids where ρ * E (e) ∈ E(G) is the inverse image of e ∈ E exc (G st ) via ρ : G − → G st and d Γ : E(G) → S σ is the generalized metric corresponding to the tropical curve Γ /σ . • Let π : (G, D, ρ) → (G , D , ρ ) be a morphism in QDiv G . For e ∈ E(G st ) and e ∈ E(G st ) with e = (π st ) * E (e ) we have non-zero maps C(π) e ,e : C(G , D , ρ ) e → C(G, D, ρ) e such that the following holds: -If e and e are either both non-exceptional or both exceptional then C(π) e ,e = id; -Otherwise, we must have that e ∈ E nex (G st ) and e ∈ E exc (G st ) and we define C(π) e ,e : σ −→ (σ × R ≥0 R 2 ≥0 ) e to be induced by the i-th face inclusion R ≥0 −→ R 2 ≥0 where i = 1, 2 is the index such that e i is not contracted by π (while e 3−i is necessarily contracted by π). We associate to π the morphism C(π) = e∈E(G st ) e ∈E(G st ) C(π) e ,e : C(G , D , ρ ) −→ C (G, D, ρ) induced by the C(π) e,e .
In a similar way, we can define Jac Γ /σ ,d , Jac spl Γ /σ , (d) or Jac Γ /σ (φ) using, respectively, the full subcategories QDiv G,d , QDiv spl G, (d) or QDiv G (φ) of QDiv G . Proof. Observe that by Definition 6.1, there is a natural morphism Jac Γ /σ → σ and moreover, for any morphism u : τ → σ in RPC, we have that τ × σ Jac Γ /σ Jac u * (Γ /σ ) . Hence, in order to show the statement, it is enough to show that there is an isomorphism of groupoids between the objects associated to the two objects in (6.1).
Since the above construction can be reversed, it follows that the morphism (6.2) is an isomorphism and we are done.

The universal tropical Jacobian as a topological stack
The aim of this subsection is to construct a topological realization of Φ trop : J trop g,n → M trop g,n and to study its fibers.
As explained in [CCUW20, Section 5.3], we can extend (2.9) to a morphism of real cone stacks (i.e. geometric stacks over the category PC of (non necessarily rational) polyhedral cones): As in [CCUW20, Proposition 5.9], the fiber of the above morphism over a given σ ∈ PC can be described as follows: • J trop, R g,n (σ ) is the groupoid of pairs (Γ , D) consisting of a quasi-stable tropical curve Γ of type (g, n) with edge lengths over the dual cone σ ∨ , i.e. a quasi-stable graph G(Γ ) of type (g, n) endowed with a generalized metric d Γ : E(G(Γ )) → σ ∨ \{0}, and D is a divisor on G(Γ ) such that (G(Γ ), D) ∈ QDiv g,n ; • M trop, R g,n (σ ) is the groupoid of stable tropical curves Γ of type (g, n) with edge lengths over the dual cone σ ∨ , i.e. a stable graph G(Γ ) of type (g, n) endowed with a generalized metric  D) consisting of a quasi-stable tropical curve Γ of type (g, n) with real edge lengths, i.e. a quasi-stable graph G(Γ ) of type (g, n) endowed with a metric d Γ : E(G(Γ )) → R >0 , and a divisor D on G(Γ ) such that (G(Γ ), D) ∈ QDiv g,n ; • M We now want to describe the fiber of Φ trop : J trop g,n → M trop g,n over a tropical curve Γ ∈ M trop, R g,n ( ). Definition 6.3. Let Γ ∈ M trop, R g,n ( ) be a stable tropical curve with real edge lengths and let G := G(Γ ) ∈ SG g,n be its underlying stable graph. The Jacobian topological space of Γ is the topological space Jac Γ obtained as the colimit associated to the functor where the possibly non-zero maps P (π) e ,e : P (G , D , ρ ) e → P (G, D, ρ) e are those for which e = (π st ) * E (e ). In this case we have that -if e and e are either both non-exceptional or both exceptional then P (π) e ,e = id; -otherwise, we must have that e ∈ E nex (G st ) and e = e v ∈ E exc (G st ) for v ∈ V exc (G) and we define if e 1 v is contracted by π, d(ρ * E (e)) if e 2 v is contracted by π.
In a similar way, we can define Jac Γ ,d , Jac spl Γ , (d) or Jac Γ (φ) using, respectively, the full subcategories QDiv G,d , QDiv spl G, (d) or QDiv G (φ) of QDiv G . We can choose a rational polyhedral cone σ , a map of cones u : R ≥0 → σ and a stable tropical curve Γ /σ over σ such that G( Γ ) = G(Γ ) and the metric d Γ of Γ is equal to the following composition This is equivalent to saying that the morphism of real cone stacks R ≥0 strata from Section 3 above to identify the non-Archimedean skeleton associated to this polystable model (in the sense of [Ber99]) with the tropical Jacobian J trop g (Γ X ) of the tropicalization Γ X of X. This constitutes another path towards the main result of [BR15] saying that the skeleton of the Jacobian is the Jacobian of the skeleton.