ONTHETOPOLOGIESOFTHE EXPONENTIAL annacepek damienlejay

Factorization algebras haveAbstract been defined using three different topo- logies on the Ran space. We study these three different topologies on the exponential, which is the union of the Ran space and the empty configur- ation, and show that an exponential property is satisfied in each case. As a consequence we describe the weak homotopy type of the exponential Exp(X) for each topology, in the case where X is not connected. We also study these exponentials as stratified spaces and show that the metric exponential is conically stratified for a general class of spaces. As a corollary, we obtain that locally constant factorization algebras defined by Beilinson-Drinfeld are equivalent to locally constant factoriz- ation algebras defined by Lurie.

1 Exponentials3 contents1.1 Exponential functors3 1.2 The exponential functor6 2 Topologies on the exponential9 2.1 The topological exponential9 2.1.1 First steps towards exponentiability9 2.1.2 The topological exponential is not an exponential 10 2.1.3 The topological exponential is almost an exponential 11 2.1.4 The topological exponential is a restricted exponential 11 2.2 The metric exponential 12 2.3 The minimal exponential 14 2.4 Weak homotopy type of the exponentials 15 3 Interlude: minimal enclosing balls 17 arXiv:2107.11243v1 [math.AT] 23 Jul 2021 4 Stratification of the exponentials 19 4.1 The exponentials as stratified spaces 19 4.2 Cones and joins 20 4.3 Conical stratification 21

This work was supported by IBS-R003-D1.

1 Roughly speaking, a factorization algebra on a space X with values in a A symmetric monoidal is a gadget associating to each finite subset C⊗ of points S X an object , such that ⊂ AS ∈ C

i I Si  i I Si for every finite family I of disjoint finite A⊔ ∈ ⊗ ∈ A subsets S X; i ⊂ factorizationthe assignment S be continuous. 7→ AS Yet, to be able to say that varies continuously with S, one first needs to AS answercontinuity the question:

What is the topology on the set of all finite subsets S X? ⊂ The set of all finite subsets of X is called the exponential Exp(X) of X. The literature on factorization algebras provides three different candidates to topologize Exp(X): In Chiral Algebras [1, 3.4.1], Beilinson and Drinfeld endow Exp(X) with a colimit topology; 2004 In Derived Algebraic Geometry VI [2, 3.3.2], Lurie endows Exp(X) with a topology reminiscent of the metric topology introduced by Hausdorff 2009on the space of compact subsets of a ; In Factorization Algebras in Quantum Field Theory [3, 1.4.1], Costello and Gwilliam use yet another topology to define factorization algebras, 2016this time using coverings inspired from Weiss. For a given separated X, the three topologies above are given from finest to coarsest and one then obtains three different levels of strength for the continuity requirement of a factorization algebra. It has been conjectured that the three different definitions agree in the special case of locally constant factorization algebras which are roughly speaking, those factorization algebras for which is “homotopic” to when x,y X both belong to the same A Ax Ay ∈ contractible open subset. The set of all finite subsets of X is called the exponential of X because its algebraic properties resemble that of exponential functions. This is the subject of the first section, where we define exponential functors in general and give some general properties. In the second section, we introduce the three different topologies giving rise to the topological exponential (B&D version), the metric exponential (Lurie version) and the minimal exponential (C&G version). We show how each exponential listed above is an exponential functor in the sense of the definition given in the first section. From this we deduce the weak homotopy type of each exponential in the case where X is not connected, extending contractibility results of Handel [4, 4.3] and Curtis & Nhu [5]. Finally, the last two sections are dedicated to the study of the stratification of the metric exponential. The goal is to show that it is conically stratified (in the sense of Lurie) for a general class of spaces. For this we need to solve an optimization problem: finding the smallest enclosing ball of a finite number of points in a general normed space; this is the content of the third section. Using a companion article from the second author [6], one can then deduce from the conical stratification of the metric exponential that the notions of locally constant factorization algebras from Beilinson & Drinfeld and Lurie coincide.

2 1 exponentials Any1.1 continuousExponential functors f : R R satisfying f (x + y) = f (x)f (y) must be an → exponential x ax with base a = f (1). The exponential is traditionally the 7→ one with base e defined X xn ex B n! n N ∈ using power series. An analogous theory can be described in the realm of categories. The set R can be replaced by any category , functions can be replaced by functors, sums C can be replaced by coproducts and products can be replaced by categorical products. However, there shall be two main differences between exponential functors and exponential functions. First, what was a property of a function in the realm of set theory shall become a structure on a functor in category theory. An exponential functor shall be a symmetric monoidal functor

E C⊔ C× between endowed with the coproduct symmetric monoidal structure and C C endowed with the product symmetric monoidal structure. Let us see some of the first obvious consequences. First, since each object X admits a map X with source the initial object of , one gets a map ∅C → C  E( ) E(X) ∗C ∅C −→ with source the terminal object of , so each E(X) is a pointed object of . Since C C every object X is a commutative monoid with product map X X X the ∈ C ⨿ → fold map, it follows that E(X) is also a commutative monoid with composition

E(X) E(X)  E(X X) E(X) × ⨿ −→ and with unit the pointing already described. Second, one needs to replace continuity with an equivalent notion. There is already a notion of continuity for functors in category theory: a functor is (co)continuous if it preserves small (co)limits. This is unreasonable to ask. Instead, let us rewrite an equivalent definition for the continuity of an exponential function: a function f : R R for which f (x + y) = f (x)f (y) for → P every x,y R is continuous if and only if for every converging series x = x, ∈ Q n the sequence with general term i⩽n f (xi) converges to f (x). In category theory convergence is replaced by existence and “limit of a sequence” can be replace with the “colimit of filtered diagram”. To make this precise, we first recall the notion of the finite product of an infinite family. Definition 1.1 (Finite product). Let be a category with finite products and C filtered colimits. Given a small family of pointed objects Xi i I in , let { } ∈ C Yf Y Xi B lim Xj i I −−→J I j J ∈ J finite⊂ ∈ denote the finite product (or weak product, or restricted product) of the family obtained by taking the colimit over all finite subsets J I. ⊂

3 Example 1.2. In the category of vector spaces (seen as pointed via their zero vector), the finite product of a small family Vi i I , { } ∈ Yf M Vi = Vi i I i I ∈ ∈ coincides with their direct sum. If Xi i I is a small family of pointed topological spaces, then one has { } ∈ Yf Y box Xi Xi i I i I ∈ ∈ a continuous injection from the finite product to the product, endowed with the box topology. When the points X are all open, this map becomes an ∗ → i open embedding. In this case a basis of opens of the finite product is given by Q the images of the products j J Uj with J I finite and Uj Xj open. ∈ ⊂ ⊂ Construction 1.3. Let E : be a symmetric monoidal functor with C⊔ → C× C having enough limits and colimits. Let Xi i I be a small family of elements { } ∈ of , then for each finite subset J I, one gets a map C ⊂ Q `  `  i J E(Xi)  E i J Xi E i I Xi ∈ ∈ −→ ∈ using the functoriality of E and its monoidal structure. Moreover for any K J, the following diagram ⊂ Q  `  i J E(Xi) E i J Xi ∈ ∈

`  E i I Xi ∈

Q  `  i K E(Xi) E i K Xi ∈ ∈ commutes by functoriality of E and its monoidal structure. One then obtains a canonically defined map   Yf a  E(X ) E  X  i −→  i i I i I ∈ ∈ from the finite product of E(Xi) i I . { } ∈ Definition 1.4 (Exponential functor). Let be a category with finite products, C filtered colimits and small coproducts. An exponential functor of the category is a symmetric monoidal functor C E C⊔ C× between endowed with the coproduct symmetric monoidal structure and C endowed with the product symmetric monoidal structure, for which in C addition, the canonical map Qf `  i I E(Xi) E i I Xi ∈ −→ ∈ is an , for every small family Xi i I of objects of . A { } ∈ C between two exponential functors is the data of a monoidal natural transform- ation.

4 Remark 1.5. One could define exponential functors in the following more abstract way. The coproduct is an infinitary symmetric monoidal structure: the operations Xi i I i I Xi are associative, symmetric and unital in an { } ∈ 7→ ⨿ ∈ obvious way. Similarly, the finite product endows the category of pointed objects with another infinitary symmetric monoidal structure. C• An exponential functor can then be defined as an infinitary symmetric f monoidal functor ⊔ × . Here it happens that any (finitary) lax monoidal C → C• f functor gives rise to an infinitary lax monoidal functor C⊔ → C× C⊔ → C× which allows us• to define exponential functors without first having to• develop the theory of infinitary monoidal categories.

The classification result for exponential functions on R has an equivalent form in the case of exponential functors on the category of sets: these are classified by their base.

Definition 1.6 (Exponential of base A). Let A be a commutative monoid. The exponential of base A is the endofunctor of the category of sets defined by

1 Exp (X) B φ: X A φ− (0) is cofinite A { → | } for any set X. If f : X Y is a function and φ: X A is almost null, its image → → by Exp (f ) is the function ψ : Y A, where A → X ψ(y) B φ(x) x f 1(y) ∈ − for any y Y . The function ψ is well defined and almost null since A is ∈ commutative and φ is almost null. The exponential structure of ExpA is straightforward.

Theorem 1.7. The assignment A Exp induces an equivalence 7→ A Commutative monoids = Set exponentials between the category of commutative monoids and the category of exponential functors of the category of sets.

Proof. The assignment A Exp is functorial, its inverse takes an exponential 7→ A E and extracts the commutative monoid E( ). By construction one has a ∗ canonical isomorphism Exp ( ) = A and the maps Exp ( ) Exp ( ) and A ∗ A ∅ → A ∗ Exp ( ) Exp ( ) = Exp ( ) Exp ( ) recover the commutative monoid A ∗ × A ∗ A ∗ ⨿ ∗ → A ∗ structure on A. Conversely if E is an exponential, let A denote the commutative monoid Qf E( ). Then one has E(X)  A = Exp (X) for every set X. Lastly, let us ∗ X A show that E(f )  Exp (f ) for every function f : X Y . The case where Y A → is a singleton and X is finite is true by construction and corresponds tothe Q monoid structure A A of A. Taking the colimit over finite subsets gives Q X → us the case f A A where X is infinite. Finally, the general case is obtained X → by writting a function f : X Y as a disjoint union fy : Xy y with y Y : Qf → → { } ∈ E(f )  y Y E(fy)  ExpA(fy)  ExpA(f ). The natural isomorphism ExpA  E we have just∈ described, is monoidal by construction. It is straightforward to check that Exp ( ) = A is natural in A and Exp  A ∗ E( ) E is natural in E. ∗

5 Example 1.8. The exponential of base N

Exp (X) = X0 X X2 X3 N ⨿ ⨿ S2 ⨿ S3 ⨿ ··· is the exponential functor corresponding to the analyst exponential of base e. Let I2 be the idempotent commutative monoid on two elements. Then the exponentials of base I2 and Z2 have identical sets

Exp (X) = Exp (X) = S X S is finite I2 Z2 { ⊂ | } but their monoid structures are different: for the exponential of base I2, the pair ( x , x ) is sent to x whereas for that of base Z , it is sent to . { } { } { } 2 ∅ Remark 1.9. One can extend the definition of an exponential functor to accom- modate any monoidal structure on the target. For example, the classification theorem above also holds for exponential functors Vect⊕ Vect⊗: they are R → R equivalent to unital commutative R-algebras. The exponential of base R[X] is the symmetric algebra functor. The expo- nential of base R[Z2] is the antisymmetric algebra functor.

As1.2 isThe apparentexponential from the functor definition of the exponential functors with bases, there is a preferred exponential, the exponential of base I2, which we shall refer to as the exponential functor, and denote it simply by Exp. For any set X, Exp(X) can be identified with the set of all finite subsets of X. For each function f : X Y , the associated function Exp(f ): Exp(X) → → Exp(Y ) sends a finite subset S X to f (S) Y . ⊂ ⊂ The exponential can also be described as a particular colimit. This is the definition one can use to define the exponential in a general category.

Definition 1.10. Let be a category admitting finite products and small C colimits. Let Fin↠ denote the category of finite sets and surjections. Given an object X and a surjection φ: I ↠ J between two finite sets, one gets a split ∈ C monomorphism Xφ : XJ , XI → op which means that X defines a functor Fin↠ . → C The exponential functor on is the colimit C I Exp(X) B lim op X I Fin↠ −−→ ∈ op of the functor X : Fin↠ , with the convention that X is the terminal object → C ∅ of X, for every X . ∈ C We shall first make a remark about the structure of this colimit andthen show its universal property.

Definition 1.11. Let ω denote the poset ∗ ω B 0 1 < 2 < < n < ∗ { } ⨿ { ··· ···} and let denote by ⩽ the associated partial order. ∗

6 The opposite category of the category of finite sets and surjections admits a canonical functor to ω sending a finite set I to its cardinal. Hence the colimit defining Exp can ∗be computed in two steps.

Notation 1.12. Let Exp⩽ n(X) lim XI ∗ B I ⩽ n −−→| | ∗ for every natural n. For example, if X Set, Exp⩽ n(X) is the set of all non- ∈ ∗ empty finite subsets S X having at most n elements. ⊂ Since ω has an isolated point, we shall let ∗ Exp (X) lim XI ∗ B 0< I −−→ | | be the subobject called the “Ran space” of X in some parts of the literature.

Remark 1.13. By construction

Exp(X) = lim Exp⩽ n(X) and Exp (X) = lim Exp⩽ n(X) n ω ∗ ∗ n>0 ∗ −−→ ∈ ∗ −−→ Theorem 1.14 (The exponential is an exponential). Let be a category with C finite products and small colimits. Assume moreover that Y X Y commute 7→ × with all small colimits for every X . Then the exponential Exp: has the ∈ C C → C canonical structure of an exponential functor.

Proof. For each finite set I, the functor X XI commutes with filtered colim- 7→ its. Hence we have `  `  lim J K Exp j J Xj = Exp k K Xk . ⊂ −−→J finite ∈ ∈ So it shall be enough to show that Exp is a symmetric monoidal functor. Let Xk k K be a finite family of elements of . One has a sequence of { } ∈ C canonical

`  ` I Exp k K Xk = lim k K Xk ∈ I ∈ −−→ ` Q Ik = lim (I K) k K Xk (by distributivity) I → ∈ −−→ Q I = lim X k (by cofinality, see 3) (I K) k K k −−→ → ∈ Q Ik = lim k K Xk (by isomorphy, see 2) Ik k K ∈ −−→{ } ∈ Q Ik = k K lim Xk (by distributivity) ∈ Ik Q −−→ = k K Exp(Xk) ∈ where:

1 1. given a map of finite sets φ: I K, we let I B φ− (k) for each k K; → k ∈ 2. the coproduct induces an isomorphism of categories

K Fin = Fin/K ` sending Ik k K to k K Ik K; { } ∈ ∈ →

7 3. given a small category and an object K , let p : denote the C ∈ C CK/ → C forgetful functor. Then for every x , the canonical map ∈ C 1 p− (x) p −→ /x 1 is cofinal. Moreover the fiber p− (x) is discrete. Thus for any functor F with source the coslice , its colimit can be computed as CK/ ` lim F = lim lim F = lim lim 1 F = lim (K x) F. (K x) x p/x x p− (x) x −−→ → −−→ −−→ −−→ −−→ −−→ → It is straightforward to check that these canonical isomorphisms endow Exp with the structure of an exponential functor.

Remark 1.15. If Y X Y only commutes with coproducts, then one can 7→ × Qf show that we still get structural maps Exp( i I Xi) i I Exp(Xi) turning the ⨿ ∈ → exponential into what one would call an oplax infinitary∈ symmetric monoidal functor. Corollary 1.16. Under the same assumptions, for every X , Exp(X) is the free ∈ C idempotent commutative monoid on X. Proof. Since Exp is an exponential functor, Exp(X) is a commutative monoid as explained earlier. It is idempotent for the following reason: for every finite set I, the diagram

diagonal XI XI XI (X X)I I fold XI I × ⨿ ⊔ ⊔

Exp(X) Exp(X) Exp(X)  Exp(X X) Exp(X) × ⨿ where all vertical maps are canonical maps, commutes. Moreover the top composite map equals the map XI XI I induced by the fold map I → ⊔ ⊔ I I. Hence the full top right composite map is again the canonical map → XI lim XJ . Since this is true for every finite I, this shows that the bottom → J composition−−→ is the identity of Exp(X). Let (M,µ) be a commutative and idempotent monoid in . Assume a given C map ψ : X M. Then for every finite I, one has a well defined map → ψI µ XI MI M because µ is associative and commutative. Moreover, because µ is idempotent, the diagram ψI XI MI µ Xφ Mφ ψJ µ XJ MJ M commutes for every φ: I ↠ J. One thus gets a map ψ˜ : Exp(X) M extending → ψ. The map ψ˜ is obviously unital. To show that ψ˜ is compatible with µ is to show that

ψ˜ ψ˜ Exp(X) Exp(X) × M M × × µ ψ˜ Exp(X) M

8 commutes, which can be done by precomposing with XI XJ for all I,J finite × sets. We then only need to show that

ψI ψJ µ µ XI XJ × MI MJ × M M × × ×

I J (ψ ψ) ⊔ (X X)I J ⨿ (M M)I J µ ⨿ ⊔ ⨿ ⊔ fold fold

I J I J ψ ⊔ I J µ X ⊔ M ⊔ M commutes. The left two squares commute by functorality. The right square commutes because µ is unital, commutative and associative.

In this section we shall review three different topologies on the set Exp(X) of finite2 topologies subsets S X of on a topological the exponential space X, with the goal of transforming ⊂ Exp into an exponential endofunctor of the category of topological spaces.

As2.1 explainedThe topological in the previous exponential section, the exponential can be computed with the help of a colimit ranging over the opposite category of the category of finite sets and surjections. Computing the colimit in the category of topological spaces, one obtains the topological exponential, of which we give a simpler definition.

Definition 2.1 (Topological exponential). The topological exponential of a topological space X is the topological space denoted ExpT(X) with set of points Exp(X), the set of finite subsets S X, endowed with the finest topology such ⊂ that the canonical maps Xn Exp(X) −→ given by sending each tuple (x ,...,x ) to the subset [x ,...,x ] X it repres- 1 n 1 n ⊂ ents, be continuous for every n ⩾ 0.

The topological exponential is used by Beilinson and Drinfeld [1, 3.4.1] to define factorization algebras on a topological space. As we shall soonsee, the topological exponential suffers one drawback: it is not an exponential functor because the functor Y X Y commutes with colimits only when X 7→ × is core-compact.

Proposition 2.2. Let Xi i I be a small family of topological spaces. The 2.1.1 First steps towards{ } ∈ exponentiability `  Qf ExpT i I Xi i I ExpT(Xi) ∈ −→ ∈ is continuous.

9 Proof. To show that this map is continuous it is enough to check that its K composition with the projections ( i I Xi) ExpT( i I Xi) is continuous ⨿ ∈ → ⨿K∈ for every finite set K. For such a K, the space ( i I Xi) is a disjoint union Q Kj ⨿ ∈ of spaces of the form j J Xj with J I finite, and each projection map ∈ ⊂ Q Kj Q Qf j J Xj j J ExpT(Xj ) i I ExpT(Xi) is continuous. ∈ → ∈ → ∈ Lemma 2.3 [4, 2.4 & 2.5]. Given a separated space X, the projection map Xn Exp (X) −→ T factors as a composite

n ⩽ n X Exp ∗ (X) Exp (X) −→ T ⊂ T of a closed quotient map followed by a closed embedding, for every n ⩾ 0.

Lemma 2.4. For any small family of separated topological spaces Xi i I and every { } ∈ natural n, the canonical map

Qf ⩽ n `  i I ExpT∗ (Xi) ExpT i I Xi ∈ −→ ∈ is continuous. Proof. By definition of the finite product, it is enough to show itfor I finite. n ↠ ⩽ n Since each Xi is separated, the quotient map Xi ExpT∗ (Xi) is perfect [2.3], Q n Q ⩽ n and hence a perfect map. Thus the product i I Xi i I ExpT∗ (Xi) is again ∈ → ∈ Q n a perfect map, and, in particular, a quotient map. Then, the map i I Xi I I ∈ → ( i I Xi) ⊔···⊔ ExpT( i I Xi) is continuous and by the previous observation, ⨿ ∈ → ⨿ ∈ Q ⩽ n factors as a continuous map through the quotient i I ExpT∗ (Xi). ∈

1 Lemma 2.5. The topological exponential ExpT(S ) contains a copy of the infinite bouquet2.1.2 The of circles topologicalωS1. exponential is not an exponential ∨ Proof. We shall build a sequence of closed embeddings

S1 ... n S1 ... ∨i=1

⩽ 1 1 ⩽ n 1 ExpT∗ (S ) ... ExpT∗ (S ) ... which shall lead to a closed embedding ωS1 , Exp (S1). For this, we ∨ → T embed two circles into the torus T2 via the vectors (0,1) and (1,1), three circles in T3 via the vectors (0,0,1), (0,1,1) and (1,1,1) etc. This defines a n 1 n compatible family of closed embeddings i=1S , T . Since moreover the n ⩽ n 1 ∨ → projection map T ExpT∗ (S ) is closed, we get continuous closed maps n 1 ⩽ n 1→ S Exp ∗ (S ). By construction, they are injective and fit as expected ∨i=1 → T in the diagram of closed embeddings above.

Theorem 2.6 (The topological exponential is not an exponential). The canon- ical continuous bijection     Exp Q S1 Exp (Q) Exp S1 T ⨿ −→ T × T is not a homeomorphism.

10 Proof. Using the same tori embeddings as in the previous lemma, one can fit a n 1 ⩽ n+1 1 copy of Q S in Exp ∗ (Q S ) for every n ⩾ 1. One can then check × ∨i=1 T ⨿ that the continuous bijection Exp (Q S1) Exp (Q) Exp (S1) restricts T ⨿ → T × T to the continuous bijection lim Q n S1 Q ωS1 which is not a 0

As we have just seen, the canonical continuous bijection Exp (X Y ) T ⨿ → 2.1.3Exp (XThe) Exp topological(Y ) is not exponential always a homeomorphism. is almost an exponential However when X and Y T × T are separated, its inverse is still sequentially continuous. Remark 2.7 (Converging sequences in a colimit topology). Let Z , , 0 → ··· → Z , be a sequence of closed embeddings between T topological spaces p → ··· 1 and let Z denote its colimit. Then every morphism K Z with K compact → factors through one Z Z [8, 2.4.2]. More generally this is true if Z is the p ⊂ colimit of an ordinal sequence of closed embeddings. As a consequence, if X is separated, a sequence (Sn)n N in ExpT(X), con- ∈ verges only if the sequence of cardinals S ω is bounded. | n|n< Proposition 2.8. Let Xi i I be a small family of separated spaces, the canonical { } ∈ bijection Qf `  i I ExpT(Xi) ExpT i I Xi ∈ −→ ∈ is sequentially continuous.

Proof. Let S be a sequence in ExpT( i I Xi). Since Sn is a finite subset of i I Xi ⨿ ∈ ⨿ ∈ for each natural number n, the union ωS intersects only a countable ∪n< n number of Xi, and we can thus reduce to the case where I is countable. Given a sequence X0,X1,... of separated spaces, the sequence Exp (X ) , Exp (X ) Exp (X ) , Exp (X ) Exp (X ) Exp (X ) , ... T 0 → T 0 × T 1 → T 0 × T 1 × T 2 → is made of closed embeddings between separated spaces. Thus ωS inter- ∪n< n sects only a finite number of Xi [2.7]. Then, we only need to consider the case of two separated spaces X and Y . In that case, the sequence S is then comprised of a pair of two sequences S(X) and S(Y ). Because ExpT(X) is a union of closed embeddings between separated spaces, S(X) is bounded in cardinality [2.7]. The same is true ⩽ n ⩽ n for S(Y ). We conclude using that Exp ∗ (X) Exp ∗ (Y ) Exp (X Y ) is T × T −→ T ⨿ continuous for every n [2.4].

As we have explained earlier, the topological exponential is not an exponential functor2.1.4 The because topological the functor exponentialY X is aY restricteddoes not exponentialcommute with colimits in 7→ × general, for a given X. One might ask whether the exponential property could still hold if restricted to core-compact spaces, i.e., the spaces X for which Y X Y commutes with colimits. The answer to this question is 7→ × non-obvious as ExpT(X) is usually not going to be core-compact, even when X is.

Proposition 2.9. Let Xi i I be a small family of separated and core-compact { } ∈ topological spaces. The canonical bijection

Qf `  i I ExpT(Xi) ExpT i I Xi ∈ −→ ∈

11 is a homeomorphism. Proof. We only need to show that the above map is continuous [2.2]. By definition of the finite product, we can reduce to the case ofafinite I. Since n ⩽ n each Xi is separated, the projection map Xi ExpT∗ (Xi) is a perfect map and ⩽ n → thus ExpT∗ (Xi) is core-compact. Because sequential unions of core-compact spaces commute with finite products [7], one has canonical homeomorphisms Q Q ⩽ n Q ⩽ n i I ExpT(Xi) = i I lim ExpT∗ (Xi) = lim i I ExpT∗ (Xi) ∈ ∈ n ω n ω ∈ −−→ ∈ ∗ −−→ ∈ ∗ and the map Q Q ⩽ n   i I ExpT(Xi) = lim i I ExpT∗ (Xi) ExpT i I Xi ∈ n ω ∈ −→ ⨿ ∈ −−→ ∈ ∗ is continuous, as a colimit of continuous maps [2.4].

Corollary 2.10. Let X be a separated and core-compact topological space. Then ExpT(X) is the free idempotent commutative topological monoid on X.

2.2In additionThe metric to not exponential being an exponential functor, the topological exponential also does not preserve metrizability. In fact ExpT(X) is almost never metriz- able. Proposition 2.11. Let X be a metrizable topological space. If X has an accumula- tion point, then ExpT(X) is not metrizable.

Proof. Pick a metric D inducing the topology on ExpT(X). Let x X be an n ∈ ⩽ n accumulation point. For every n ⩾ 0, using the quotient map X Exp ∗ (X), → T one can find a subset S X made of exactly n elements such that D(S , x ) ⩽ n ⊂ n { } 1/n. In other words, Sn n + x and Sn n + + which is forbid- → → ∞ { } | | → → ∞ ∞ den [2.7].

One way to remedy this is to compute the colimit defining the exponential not in the category of topological spaces but rather in the category Met of generalized metric spaces. ∞ A generalized metric space is a metric space whose distance function is allowed to have the value + . in Met are the metric maps: ∞ ∞ the maps f :(M,d ) (N,d ) such that d (f (x),f (y)) ⩽ d (x,y) for every M → N N M x,y M. ∈ The main advantage of the category of generalized metric spaces is that it admits all small limits and colimits [9, 4.5(3)]. Computing the colimit defining the exponential in Met , one obtains the metric exponential, of which we give a concrete definition. ∞ Definition 2.12 (Metric exponential of a metric space). Given a (generalized) metric space (X,d), its metric exponential is the generalized metric space (Exp(X),D), where   maxs S mint T d(s,t) D(S,T ) B max ∈ ∈ maxt T mins S d(s,t). ∈ ∈

We shall denote the metric exponential by ExpM(X). In particular, one has D([ ],T ) = D(T,[ ]) = + when T is not empty. ∅ ∅ ∞

12 Remark 2.13. The metric subspace Exp∗ (X) Exp (X) is used by Lurie as M ⊂ M an intermediate tool to deal with locally constant non-unital factorization algebras which are locally constant cosheaves on ExpT∗ (X) [2, 3.3.2]. In a remark in Higher Algebra, he suggests using a variant of the exponential Exp (X) where D([ ],T ) = D(T,[ ]) = 0 for every T X, to deal with unital M ∅ ∅ ⊂ factorization algebras. The metric exponential has also been used by Knudsen [10] in his work extending the constructions of Francis and Gaitsgory [11] to the topological setup. The topology of the metric exponential admits a basis given by opens of the form [Ui]i I where ∈ S [Ui]i I i I,S Ui , . ∈ ∈ ⇔ ∀ ∈ ∩ ∅ This allows us to define the metric exponential ExpM(X) when X is only a topological space. Definition 2.14 (Metric exponential of a topological space). For a topological space X, its metric exponential ExpM(X) consists of the set Exp(X) endowed with the coarsest topology including all [Ui]i I for every finite set I of open subsets U X. ∈ i ⊂ This definition is functorial: if f : X Y is a continuous map, the preim- 1 → age of [Ui]i I by Exp(f ) equals [f − (Ui)]i I . ∈ ∈ Before looking at the exponential property of ExpM, we shall discuss how some limits and colimits are computed in Met . Given a small family ∞ (Xi,di) i I of (pointed) metric spaces, their { } ∈ is the disjoint union of sets i I Xi endowed with the distance ⨿ ∈ d for which   coproduct di(xi,yi), if i = j d(xi,yj ) =  + , if i , j ∞ Q is the product set i I Xi endowed with the sup metric ∈ d( xi , yi ) B supi I di(xi,yi) product { } { } ∈ is the finite product set endowed with the sup metric. In other words, in that case, the natural map finite product Qf Q i I Xi i I Xi ∈ −→ ∈ is an isometric embedding.

Proposition 2.15 (Exponential property). The metric exponential ExpM is an exponential functor for both Met and Top. ∞ Proof. Starting with the metric case: let (Xi,di) i I be a small family of metric { } ∈ spaces. We only need to show that the in the exponential structure of Exp are isometric. Let S and T be two finite subsets of the union in (X,d) B i I (Xi,di) and write Si B S Xi and Ti B T Xi for every i I. By ⨿ ∈ ∩ ∩ ∈ construction of the disjoint union, if s S and t T do not belong to the same ∈ ∈ component Xi, their distance d(s,t) in X is infinite. As a consequence the distance D(S,T ) in ExpM(X) becomes   maxs S mint T d(s,t) = supi I maxs S mint T di(s,t) D(S,T ) = max ∈ ∈ ∈ i ∈ i  ∈ maxt T mins S d(s,t) = supi I maxt Ti mins Si di(s,t) ∈ ∈ ∈ ∈ ∈

13 and thus D(S,T ) = supi I Di(Si,Ti). ∈ Let Xi i I be a small family of topological spaces. Because finite sets { } ∈ can only intersect a finite number of connected components, the open sets of the form [Uj,k]j J,k K where J I and each Kj are finite, and where each ∈ ∈ j ⊂ Uj,k Xj is open, form a basis of the topology of ExpM( i I Xi). It corresponds ⊂ Qf ⨿ ∈ Q bijectively to the base open set inside i I ExpM(Xi) given by j J [Uj,k]k Kj . Qf ∈ ∈ ∈ Thus the bijection ExpM( i I Xi) i I ExpM(Xi) is a homeomorphism. ⨿ ∈ → ∈ Proposition 2.16. Let X be a (generalized) metric space. Then ExpM(X) is the free idempotent commutative metric monoid on X.

Proof. The canonical map X Exp (X) is an by construction. → M So the only thing to show is that for (A,d) an idempotent and commutative Q metric monoid, the map Exp (A) A sending S A to s A — which M → ⊂ s S ∈ is well defined because A is commutative and is a monoid map∈ because A is idempotent — is a metric map. Q Q Given two finite subsets S,T A, we need to show that d( s, t) ⩽ ⊂ s S t T D(S,T ). If S or T is empty, it is immediate. Because A is an∈ idempotent∈ metric monoid d(a,bc) = d(aa,bc) ⩽ max(d(a,b),d(a,c)) for every a,b,c A. By ∈ straightforward induction, one gets the case where either S or T has a unique element. Let n be an integer and assume that the inequality has been shown for every S,T with S + T ⩽ n. Let S,T A with S + T = n + 1. Without loss | | | | ⊂ | | | | of generality, we can assume that there exists x S such that D(S,T ) = d(x,T ). ∈ Let S0 denote the complement of x in S. Then Q Q Q Q d( s, t) = d(x s, t) s S t T s S0 t T ∈ ∈ ×  ∈ ∈   ⩽ Q Q Q max d x, t T t ,d s S s, t T t (A is metric) ∈ ∈ 0 ∈ ⩽ max(d(x,T ),D(S0,T )) (by hypothesis) = D(S,T ) (by definition of x) ending showing that Exp (A) A is a metric map. M → Proposition 2.17. For every topological space X, the identity

Exp (X) Exp (X) T −→ M is a continuous map, which restricts to homeomorphisms

⩽ n ⩽ n ExpT∗ (X) = ExpM∗ (X) for every n ω , whenever X is separated. ∈ ∗ Proof. Let U X be an open subset. Let n ⩾ 1 be an integer and let σ denote ⊂ the permutation (1 n). Then the preimage along Xn Exp(X) of [U] is the i n 1··· → set ⩽ σ (U X ) which is open. It follows that [U] is open in Exp (X). For ∪i n × − T a finite I,[Ui]i I = i I [Ui] is then also open in ExpT(X). ∈ ∩ ∈

Definition2.3 The minimal 2.18. Given exponential a topological space, the minimal exponential Exp (X) is the set Exp(X) endowed with the coarsest topology containing the subsets∨ Exp(U) Exp(X) for all open subsets U X. ⊂ ⊂

14 Remark 2.19. One distinctive feature of the minimal exponential is that the point presenting the empty configuration [ ] is dense. ∅ Families of open subsets Ui V i I for which Exp(Ui) Exp(V ) i I is a { ⊂ } ∈ { ⊂ } ∈ cover in the minimal exponential, were introduced by Weiss in his work on the embedding calculus [12]. This notion of covering is used by Costello and Gwilliam to define factorization algebras in general [3, 1.4.1]. It is also used by Ayala and Francis in their study of factorization homology [13, 2.6].

Proposition 2.20 (Exponential property). The minimal exponential is an expo- nential functor on the category of topological spaces.

Proof. Let Xi i I be a small family of spaces. Since [ ] is open in the minimal { } ∈ ∅ topology, the finite product of the Exp (Xi) is a subspace of the product endowed with the box topology. ∨ Given a family of open subsets Ui Xi i I , one has bijections { ⊂ } ∈ `  Qf Q  Qf  Exp i I Ui = i I Exp(Ui) = i I Exp(Ui) i I Exp(Xi) ∈ ∈ ∈ ∩ ∈ showing the correspondence between the two bases of open sets between Qf Exp ( i I Xi) and i I Exp (Xi). ∨ ⨿ ∈ ∈ ∨ Remark 2.21 (Minimality). The functors ExpT, ExpM and Exp preserve open embeddings between topological spaces. In the category of exponential∨ func- tors of Top having this preservation property, Exp is a final object. ∨

⩽ n The2.4 functorsWeak homotopyX Exp type∗ (X of) have the exponentials interesting homotopy properties as shown 7→ T by Handel. In particular, he showed that for X a separated and path connected space, ExpT∗ (X) is weakly contractible [4, 4.3]. Curtis & Nhu showed that ExpM∗ (X) is homeomorphic to a linear space, whenever X is a connected, locally path connected metric space, which is a countable union of finite dimensional compact spaces [5]; it is in particular contractible in the strong sense. In a simpler proof, Lurie showed that ExpM∗ (M) is weakly contractible when M is a connected manifold [2, 3.3.6]. Here we shall enhance these results by describing the weak homotopy type of each exponential for any separated and locally path connected space. Since [ ] is dense in Exp (X), it follows that Exp (X) is contractible for any space X. ∅ ∨ ∨ Hence, we shall focus on the metric and the topological exponentials. We start with a lemma due to Beilinson and Drinfeld.

Lemma 2.22. Let G be a group endowed with an extra operation : G G G ∧ × → such that is associative, idempotent and such that ab cd = (a c)(b d) for any ∧ ∧ ∧ ∧ a,b,c,d G. Then G is a trivial group. ∈ Proof. For every g G one has ∈ g g = g = (1g) (1g) = g = (1 g)2 = g. ∧ ⇒ ∧ ⇒ ∧ So for every h G, letting g = 1 h, ∈ ∧ (1 h)2 = (1 1 h)2 = 1 h ∧ ∧ ∧ ∧ since G is group, we get 1 h = 1 and h = (1 h)2 = 1 for every h G. ∧ ∧ ∈

15 Lemma 2.23. If X is path connected, then ExpT∗ (X) is path connected. As a consequence, ExpM∗ (X) is also path connected. Proof. Given two proper finite subsets S,T X, there exists a large enough ⊂ positive n N and two tuples (s ,...,s ) and (t ,...,t ) representing respect- ∈ 1 n 1 n ively S and T . Since X is path connected, there exists a path between those two tuples in Xn and since the map Xn Exp (X) is continuous by construction → T and factors through ExpT∗ (X), this gives us a continuous path between S and T in ExpT∗ (X).

In what follows, let us denote by I2 the commutative idempotent monoid with two elements ( 0,1 , ) and endow it with the discrete topology. { } ∨ Theorem 2.24. Let X be a locally path connected topological space. The monoid map M ExpM(X) ∃ I2 π0(X) sending a finite subset S X to the family i i π (X) with i = 0 if and only if no ⊂ {∃ } ∈ 0 ∃ element of S belongs to the connected component X X, is continuous and a weak i ⊂ homotopy equivalence. Moreover, if X is also separated, the induced continuous map M ExpT(X) ∃ I2 π0(X) is also a weak homotopy equivalence.

Proof. Using that ExpM is an exponential and the fact that for each connec- ted component X X, Exp (X ) is the disjoint union of [ ] and Exp∗ (X ), i ⊂ M i ∅ M i ExpM(X) splits as Qf ` Q Exp (X) = Exp (X ) = π Exp (X ) M i π0(X) M i J 0(X) j J M∗ j ∈ J⊂finite ∈ immediately showing that the map is continuous and that π ( ) is a bijection. ∃ 0 ∃ Let S X be a finite subset. Then, since S S = S, the monoid structure of ⊂ ∪ ExpM(X) induces an associative and idempotent map π (Exp (X),S) π (Exp (X),S) π (Exp (X),S) n M × n M −→ n M which satisfies the exchange property, for every n > 0. As a consequence each of these groups is trivial [2.22]. When X is separated, the canonical bijection Qf ExpT(X) i π (X) ExpT(Xi) −→ ∈ 0 is sequentially continuous with continuous inverse [2.8] and thus one has Qf  π0(ExpT(X)) = π0 i π (X) ExpT(Xi) ∈ 0 because the segment [0,1] is a sequential space. Since spheres and balls are also sequential spaces, the sequentially continuous map Exp (X) Exp (X) T × T → ExpT(X) still induces maps π (Exp (X),S) π (Exp (X),S) π (Exp (X),S) n M × n M −→ n M so using the same proof as for the metric case, we see that : Exp (X) ∃ T → i π (X)I2 is a weak equivalence. ⊕ ∈ 0

16 Given a normed vector space V and a proper finite subset S V , a minimal ⊂ enclosing3 interlude: ball for S is minimal a closed ball enclosingB V which contains ballsS and such that any ⊂ other ball containing S have a bigger radius.

The enclosing circle of a finite set of points in the plane.

UsingFigure classic 1: convex optimization results, one can show the existence and uniqueness of a minimal enclosing ball in the case of rotund reflexive normed vector spaces [14]. One may wonder whether the center cS and the radius rS of the minimal enclosing ball of a proper finite subset S vary continuously with S. This question is naturally posed using the metric exponential ExpM(V ). For such a general space as a reflexive vector space, one can only show that the center cS varies continuously with S for the weak topology of V . The continuity of the center becomes strong if one instead considers a restricted version of the minimal enclosing ball problem. This is what we shall see here. Definition 3.1 (Restricted minimal enclosing ball). Let V be a normed vector space and let S V be a proper finite subset of V . A restricted minimal ⊂ enclosing ball is a closed ball B V containing S and whose center belongs to ⊂ the convex hull Conv(S) of S, such that, any other ball with center in Conv(S) and containing S have a bigger radius. Remark 3.2. In a Hilbert space H, the restricted minimal enclosing ball of S H coincides with its minimal enclosing ball. ⊂ Restricted minimal enclosing balls might not be unique for a given norm. We shall then restrict our attention to spaces that can be endowed with a norm with strictly convex unit ball. Definition 3.3 (Rotund vector space). We shall say that a topological vector space is rotund if its topology can be induced by a norm for which the closed unit ball is strictly convex: the equation x + y x = y = ∥ ∥ 2 holds only when x = y. By extension, we shall say that such a norm is rotund. Example 3.4. Finite dimensional vector spaces are rotund. More generally sep- arable complete normable spaces are rotund [15, Thm. 9]. The space ℓ∞ is not rotund [16, Thm. 8]. Every reflexive normed vector space is rotund [17, Cor. 1 (i)]. Lemma 3.5. Let V be a normed vector space. The correspondence sending S ∈ Exp∗ (V ) to its convex hull Conv(S) V is continous. M ⊂

17 Proof. It is upper hemicontinuous: let v V and let ϵ > 0. One has ∈ Convu(B(v,ϵ)) B S Conv(S) B(v,ϵ) = B( v ,ϵ) { | ⊂ } { } showing that the upper inverse image preserves opens. It is lower hemicontinuous: let

S Convl(B(v,ϵ)) B S Conv(S) B(v,ϵ) , ∈ { | ∩ ∅} then there exists s S such that s v < ϵ. Let δ = ϵ s v , then B(S,δ) ∈ − − − ⊂ Convl(B(v,ϵ)) showing that the lower∥ inverse∥ image preserves∥ ∥ opens.

Lemma 3.6. The function

V Exp∗ (V ) R × M +

(v,S) maxs S v s ∈ ∥ − ∥ is continuous.

Proof. Consider a converging sequence (vn,Sn) n (v,S). For ϵ > 0 small → →∞ enough, if Sn is at distance less than ϵ from S, then Sn must have more points that S and for each x S , there is a unique s S such that x s ⩽ ϵ. ∈ n x ∈ ∥ − x∥ This gives us a partition of Sn as Sn = s S Sn(s). Then for vn v ⩽ ϵ and ∪ ∈ ∥ − ∥ D(Sn,S) ⩽ ϵ one has

⩽ max v s max vn t max v s max vn t s S ∥ − ∥ − t Sn ∥ − ∥ s S ∥ − ∥ − t Sn(s) ∥ − ∥ ∈ ∈ ∈ ∈ ! ⩽ max v vn + max s t s S ∥ − ∥ t S (s) ∥ − ∥ ∈ ∈ n ⩽ 2ϵ showing that (v,S) maxs S v s is continuous. 7→ ∈ ∥ − ∥ Theorem 3.7 (Solution to the restricted minimal enclosing ball problem). Let V be a vector space endowed with a rotund norm. Then every proper finite subset S V admits a restricted minimal enclosing ball of radius ⊂ rS B inf max s v v Conv(S) s S ∥ − ∥ ∈ ∈ and this ball is unique. Moreover, the function

Exp∗ (V ) V R M −→ × + mapping a proper finite subset S V to the pair (c ,r ) where c denotes the center ⊂ S S S of the restricted minimal enclosing ball of S, is continuous. Proof. Because S is finite, its convex hull Conv(S) is compact in V . As a consequence rS is finite and S admits a restricted minimal enclosing ball. It is unique: because the norm is rotund, the function v maxs S s v is strictly 7→ ∈ − convex, so its infimum on the convex hull Conv(S) is attained∥ at∥ a unique point. The continuity of cS and rS can be obtained using the Maximum The- orem [18, 17.31]: the correspondence S Conv(S) is continuous with com- 7→ pact values and the function (v,S) maxs S s v is continuous by the pre- 7→ ∈ − vious lemmas. ∥ ∥

18 The exponential of a set X admits a natural counting function Exp(X) N → sending4 stratification each finite subset ofS X theto its exponentials cardinal S . In this section we study ⊂ | | the exponentials endowed with the stratification given by this counting func- tion. We shall show that, under some conditions on X, the metric exponential Exp (X) is conically stratified. As a consequence the -categories of con- M ∞ structible hypersheaves on ExpM(X) and ExpT(X) are equivalent, which leads to the following statement: locally constant factorization algebras on X in the sense of Beilinson-Drinfeld are equivalent to locally constant factorization algebras on X in the sense of Lurie.

There4.1 The are exponentialsseveral inequivalent as stratified definitions spaces of stratified spaces. The following one is a mild one, introduced by Lurie [2, A.5.1].

Definition 4.1 (Stratified space). A stratified space is the data of a poset P , endowed with the topology whose open sets are the upward-closed subsets, and a continuous map f : X P . For p P , we shall write Xp for the fiber 1 → ∈ f − (p). A morphism of stratified spaces is a commutative square

X Y

P Q where the top map is continuous and the bottom map is a poset map.

In our case, we select the poset ω . Since the empty configuration is dense in Exp (X), the minimal exponential∗ shall never be stratified over ω or even ω as soon∨ as X is not empty. We shall thus only have a look at the two∗ other exponentials.

Proposition 4.2. When X is separated, the canonical maps

ExpT(X) ExpM(X) ω −→ −→ ∗ are continuous. Moreover, one has homeomorphisms

⩽ n ⩽ n ExpT∗ (X) = ExpM∗ (X) for every n ω . ∈ ∗ ⩽ n Proof. We need to show that ExpM∗ (X) is a closed subset of ExpM(X) for every n ω . For n = 0 this is obvious. Let S be a non-trivial finite subset of X. ∈ ∗ Since X is separated, one can find a disjoint family of open neighborhoods s Us s S . Then [Us]s S becomes an open neighborhood of S in ExpM(X) { ∈ } ∈ ∈ ⩽ n which lies in the complement of ExpM∗ (X). When X is separated, Handel has shown that the opens of the form [Ui]i I ⩽ n ⩽ n ∈ ∩ ExpT∗ (X) form a basis of the topology of ExpT∗ (X) [4, 2.11], giving us the ⩽ n ⩽ n homeomorphism ExpT∗ (X) = ExpM∗ (X) for every n ω . ∈ ∗

19 Definition4.2 Cones and 4.3 (Open joins cone). For a topological space X, the open cone of X is the set C(X) B 0 (R∗ X) { } ⨿ + × with topology defined as follows: a subset U C(X) is open if and only if ⊂ U (R X) is open, and if 0 U, then (0,ε) X U for some positive real ∩ +∗ × ∈ × ⊂ number ε. If X is stratified over a poset P , then C(X) is naturally stratified over the poset P ◁ obtained from P by adding a new element smaller than every other element of P . Warning 4.4. One should not confuse the cone on X with the collapsed rectangle defined as the quotient R X/ 0 X. When X is compact and separated, + × { } × the cone on X and the collapsed rectangle on X are homeomorphic. This is no longer true in the general case: the cone on the open interval (0,1) can be embedded in R2, whereas the collapsed rectangle on (0,1) is not metrizable. If (X,d) is a metric space, the topology of the cone C(X) is metrizable by letting d((λ,x),(µ,y) = max( λ µ ,d(x,y)) and by adding d(0,(λ,x)) = λ. | − | Definition 4.5 (Join). Given two posets P and Q, their join is the poset

P Z Q B P (P Q) Q ⨿ × ⨿ where one adds to the disjoint sum the additional relations p < (p,q) and q < (p,q) for every (p,q) P Q. ∈ × Let X P and Y Q be two stratified topological spaces. Their join → → X Z Y is the set X Z Y B X (X (0,1) Y ) Y ⨿ × × ⨿ where a basis of opens is given by the opens U X (0,1) Y together with ⊂ × × opens X X (0,ε) V with V Y open, and opens W (δ,1) Y Y with ⨿ × × ⊂ × × ⨿ W X open. ⊂ It is naturally stratified over P Z Q. Warning 4.6. Similarly to what we just said about the cone, when X and Y are both separated and compact, the join of X and Y is homeomorphic to the collapsed brick X [0,1] Y/R where R is the relation identifying X 0 Y X × × ×{ }× ∼ and X 1 Y Y . In general, this is no longer the case. × { } × ∼ Proposition 4.7. Let X P and Y Q be two stratified spaces. Then there isa → → homeomorphism C(X) C(Y )  C(X Z Y ) × over the canonical isomorphism P ◁ Q◁ = (P Z Q)◁. × Proof. The map sends bijectively tuples (λ,(x,t,y)) C(X Z Y ) to tuples ∈ ((λt,x),(λ(1 t),y)) C(X) C(Y ) and obviously respects the isomorphism − ∈ × P ◁ Q◁ = (P Z Q)◁. Let us see why it is open: there are four different cases to × look at. Case 1: open neighborhoods of the tip of C(X Z Y ). Let ε > 0, then the open 0 (0,ε) (X Z Y ) is mapped to the open ( 0 (0,ε) X) ( 0 (0,ε) Y ). { } ⨿ × { } ⨿ × × { } ⨿ × Case 2: open neighborhoods of C(X Z Y ) not containing the tip but in- cluding X. An open of the form (α,β) (X X (0,ε) V ) with 0 < α < β is × ⨿ × × mapped to the open ( 0 (0,εα) X) ((1 ε)α,(1 ε)β) V . { } ⨿ × × − − ×

20 Case 3: open neighborhoods of C(X Z Y ) not containing the tip but includ- ing Y . Confere supra. Case 4: a general open of C(X Z Y ). Let U X, V Y opens, 0 < α < β, ⊂ ⊂ 0 ⩽ t < s ⩽ 1. Then (α,β) U (t,s) V is mapped to the open (tα,sβ) U × × × × × ((1 s)α,(1 t)β) V . − − × Since the image of basis neighborhoods form a basis of neighborhoods of C(X) C(Y ), one can see that it is a homeomorphism. × Remark 4.8. A very similar proposition has been given by Ayala, Francis and Tanaka using the collapsed rectangle instead of the cone and the collapsed brick instead of the join [19, 3.8]. Of course, both propositions agree in the case where both X and Y are compact and separated.

There4.3 Conical are many stratification inequivalent notions of “goodness” for stratified space. The definition below is a mild one introduced by Lurie [2, A.5.5].

Definition 4.9 (Conically stratified space). Let f : X A be a stratified to- → pological space. One says that X is conically stratified whenever for each p A and each x X , there exists an open neighborhood U X of x and a ∈ ∈ p p ⊂ p stratified space L over Pp< such that Up Xp can be extended to a stratified ◁ ⊂ space over the poset map P = P ⩽ P . p< p ⊂ We have already seen that the minimal exponential Exp (X) is never strati- fied. Even though the topological exponential is always∨ a stratified spaceover ω when X is separated, it is usually impossible for the topological exponential ∗ ExpT(X) to be conically stratified; conical opens would allow sequences with unbounded cardinality to converge [2.7][6, 2.14]. We shall then restrict our attention to the metric exponential ExpM(X) and show that it is conically stratified for a large class of spaces X.

Lemma 4.10. Let V be a normed vector space, then the function R V + × × Exp (V ) Exp (V ) sending a triple (λ,v,S) to the configuration M → M λS + v B λs + v s S { | ∈ } is continuous.

Proof. One has

D(λS + v,µT + w) ⩽ v w + λD(S,T ) + λ µ D(0,T ) ∥ − ∥ | − | which shows that the function is continuous.

Proposition 4.11. Let V be a rotund vector space and let us denote by S (V ) M ⊂ ExpM∗ (V ) the subspace of configurations whose minimal enclosing ball has center 0 and radius 1. Since such a configuration must have at least two points, SM(V ) is naturally stratified over the poset ω2⩽. Then, one has a canonical homeomorphism

Exp∗ (V ) = V C(S (V )) M × M ◁ over the isomorphism ω1⩽ = ω2⩽.

21 Proof. In both cases the map sends one point configurations v V to the ∈ tuple (v,0) where 0 represents the tip of the cone, and sends multiple point 1 configurations S V to the tuple (c ,(r ,r− (S c ))). The inverse map simply ⊂ S S S − S sends tuples (v,(λ,S)) to λS + v. By the previous lemma and since S c and S r are continuous, it 7→ S 7→ S is clear that the bijection restricts to a homeomorphism between the open subspace of non-punctual configurations on one side and the product of V with the interior of the cone on the other side. Finally, if S v is a converging sequence with limit a punctual configur- n → ation, then by continuity c v and r 0, which means that the image of Sn → Sn → Sn converges to (v,0) by definition of the topology of the cone. Conversely, if (vn,(λn,Sn)) is a sequence converging to (v,0), this means by definition of the topology of the cone that λ 0 and since S is bounded, then λ S 0 so n → n n n → that v + λ S v in Exp∗ (V ). n n n → M

Theorem 4.12. The exponential ExpM(X) is conically stratified, whenever X is a separated topological space locally homeomorphic to a rotund vector space.

Proof. Since the emptyset is a disjoint point from the rest of the space, it emits a conical neighborhood trivially. Let S Exp∗ (M) so that S > 0. By ∈ M | | assumption, for each s S, one can find an open embedding V , X carrying ∈ s → the origin of a rotund vector space V to s X. Moreover these can be chosen s ∈ to be disjoint in X. Since ExpM is an exponential which also preserves open embeddings, one can build a stratified open embedding

Q Q  ` s S ExpM∗ (Vs) s S ExpM(Vs) ExpM ( s S Vs) ExpM(M) ∈ ∈ ∈

Q = Q + = s S ω1⩽ s S ω1⩽ ω ω ∈ ∈ ∗ ∗ whose image contains S. Since a finite product of cones is again homeomorphic to a coneasa stratified space [4.7] and since Exp∗ (V ) is homeomorphic to V C(S (V )) M s s × M s for every s S [4.11], one gets stratified homeomorphisms ∈ Q  Q  Q s S ExpM∗ (Vs) s S Vs C(SM(Vs)) s S Vs C(Zs S SM(Vs)) ∈ ∈ × ∈ × ∈

Q  Q ◁  ◁ s S ω1⩽ s S (ω2⩽) (Zs S ω2⩽) ∈ ∈ ∈ which concludes the proof.

Example 4.13. Since Fréchet manifolds are locally homeomorphic to Hilbert spaces [20, 6.1] and Hilbert spaces are rotund, ExpM(V ) is conically stratified when V is a Fréchet manifold.

Remark 4.14. Since ExpM(X) is conically stratified, it follows that each trun- ⩽ n cated version ExpM∗ (X) is also conically stratified. This truncated result was previously obtained by Ayala, Francis and Tanaka for X a smooth mani- fold [21, 3.7.5].

22 Corollary 4.15. Let X be a metrizable space, locally homeomorphic to a rotund topological vector space. Then, the -categories of ω -constructible hypersheaves ∞ ∗ of spaces on ExpT(X) and ExpM(X) are canonically equivalent. Moreover, both can be represented as the -category of functors from the exit ∞ path -category Exitω (ExpT(X)) = Exitω (ExpM(X)) to the -category of spaces. ∞ ∗ ∗ ∞ Proof. Since we know that ExpM(X) is conically stratified, we only need to check the other axioms of the main theorem of Constructible hypersheaves via exit paths [6, 3.13]. Since X is metrizable, ExpM(X) is also metrizable and thus paracompact. We now prove that each stratum is locally of singular shape. Being a local property, we can reduce to the case where X is homeomorphic to a separated locally convex topological vector space V [2, A.4.16]. The stratum 0 ω ∈ ∗ amounts to a single point so there is nothing to prove. Assume n ⩾ 1. By assumption the convex open sets form a basis of the topology of V which is stable under finite intersections. As a consequence, the n opens of the form [Cs]s S ExpM(V ) where Cs S is a family of S = n disjoint ∈ ∩ { } | |n convex open subsets of V , form a basis of the topology of ExpM(V ) which n is stable under intersection. It is then enough to see that [Cs]s S ExpM(V ) ∈ ∩ has singular shape [2, A.4.14], which immediately follows from the fact that n Q [Cs]s S ExpM(V ) is homeomorphic to s S Cs and is thus contractible. ∈ ∩ ∈ The metrizability axiom above cannot be easily removed as shown by the following proposition. Proposition 4.16. There exists a paracompact topological space X for which neither ExpT(X) nor ExpM(X) is paracompact. Proof. Let X be the set of real numbers R endowed with the lower limit topology: the topology whose basis of opens is made of the half open intervals [a,b). This space is paracompact but the product X2 is not even normal [22]. 2 2 2 Since the quotient map X X is closed (S2 being finite), X is also not → S2 S2 normal. As it is a closed subset of both ExpT(X) and ExpM(X) by the previous lemma, neither can be paracompact.

TheAcknowledgments authors would like to thank Mathieu Anel, David Ayala, Damien Calaque, Lucas Geyer, Grégory Ginot Jarek Kwapisz, and Yat-Hin Suen.

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