ON EFFECTIVE DESCENT V -FUNCTORS AND FAMILIAL DESCENT MORPHISMS

. We study eﬀective descent V -functors for cartesian monoidal categories V with ﬁnite limits. This study is carried out via the properties enjoyed by the 2-functor V 7→ Fam ( V ) , results about eﬀective descent of bilimits of categories, and the fact that the enrichment 2-functor preserves certain bilimits. Since these results rely on an understanding of (eﬀective) descent morphisms in Fam ( V ) , we brieﬂy study those epimorphisms when V is a regular category.


Introduction
Let C be a category with pullbacks.For each morphism p : x → y, we have a change-of-base functor along p: p * : C/y → C/x Via these functors, we are able to provide a description of the basic bifibration of C. Thanks to the Bénabou-Roubaud theorem [4] (see also [21, p. 258] or [29,Theorems 7.4 and 8.5] for generalisations), the descent category for p with respect to the basic bifibration, denoted Desc(p), is equivalent to the Eilenberg-Moore category for the monad induced by the adjunction p ! ⊣ p * .This allows us to say that the morphism p is effective for descent if the comparison functor K p in the Eilenberg-Moore factorisation (1) (1) is an equivalence of categories; here, U p is the functor which forgets descent data.Janelidze-Galois theory [5] and Grothendieck descent theory [22,29] feature the use of effective descent morphisms, requiring the knowledge of some (or all) such morphisms in the category of interest, and are the main motivation to undertake the study of finding sufficient conditions or even characterising effective descent morphisms; see [20,21] for introductions to the subject.
As an example, if C is a locally cartesian closed category, or an exact category (in the sense of Barr [1]), the effective descent morphisms are precisely the regular epimorphisms.However, the characterisation of effective descent morphisms in a given category C is a notoriously difficult problem in general; for instance, see the characterisation in [33] and a subsequent reformulation [10] for the case C = Top.
Motivated by this reformulation, [8,7,9,11] study this characterisation problem for more general notions of spaces: these works provide various results about effective descent in (T, V)-categories (originally defined in [13]).Due to their concerns with topological results, the study was restricted to the case in which the enriching category V is a quantale.
From the perspective of internal structures, we have the work of Le Creurer [25], in which he studies the problem of effective descent morphisms for essentially algebraic structures internal to a category B with finite limits.In particular, the author provides sufficient conditions for effective descent morphisms in C = Cat(B), and confirms that these conditions provide a complete characterisation with the added requirement that B is extensive and has a (regular epi, mono)-factorization system.Generalisations of these results to internal multicategories were studied in [32].
Let − • 1 : Set → V be the "copower with the terminal object" functor, defined on objects by X → X • 1 = x∈X 1. Making use of Le Creurer's results, Lucatelli Nunes, via his study on effective descent morphisms for bilimits of categories, provides sufficient conditions for effective descent morphisms in C = V-Cat via the following pseudopullback (see [29,Lemma 9.10, Theorem 9.11]): (2) for suitable lextensive categories V, with the cartesian monoidal structure.
The central contribution of this paper is to extend [29,Theorem 9.11] to all cartesian monoidal categories V with finite limits, in Theorem 3.3.We highlight the use of the following three tools, used in the proof of Lemma 3.1, which are the skeleton of the argument: the properties of familial 2-functors, in particular, of the endo-2-functor Fam : CAT → CAT studied in [37]; results about effective descent morphisms in bilimits of categories (see [29,Theorem 9.2 and Corollary 9.5]); and preservation of pseudopullbacks via enrichment (Theorem 2.1).
Since Theorem 3.3 relies on understanding (effective) descent morphisms in Fam(V), it naturally raises the problem of studying these classes of epimorphisms in the free coproduct completion of V. Lemma 4.1 and Theorem 4.3 provide a couple of improvements, which we first illustrate in Theorem 4.7 for V a (co)complete Heyting lattice (a new proof for one implication of [8,Theorem 5.4]), and then we apply it to obtain the more general Theorem 4.9, providing a refinement of Theorem 3.3 for regular categories V.
In Section 1, we recall the notion of pseudopullback in the restricted context of the 2-categories Cat and MndCat, we fix some terminology and notation for (strong) monoidal functors, used in the proofs of the results in Section 2, and we recall a couple of results from [29] and [37], restated in a convenient form, which are part of our toolkit in Section 3.
Section 2 is devoted to establishing some technical results on preservation of pseudopullbacks (Theorem 2.1), full faithfulness (Lemma 2.2) by the 2-functor (−)-Cat : MndCat → CAT, and in the preservation of descent morphisms by suitable functors (Lemma 2.3), which complete our toolkit.
As mentioned earlier, we establish our main result in Section 3, in Theorem 3.3.We restate the main ideas here: if V is a cartesian monoidal category, the following composite of functors reflects effective descent morphisms.If F is a V-functor, we let F be the value of (3) at F .Then, if we denote by F n the underlying morphism in Fam(V) on the objects of n-tuples of composable morphisms, we have that if then F is an effective descent morphism in Cat(Fam(V)) by [25,Proposition 3.3], and by reflection, F is an effective descent morphism in V-Cat.
Indeed, these conditions on F are statements about (effective) (almost) descent morphisms in Fam(V), leading us to studying such morphisms in the coproduct completion of V. We devote Section 4 to provide tractable descriptions of these classes of epimorphisms, with an illustrative application to categories enriched in (co)complete Heyting lattices.We obtain Theorem 4.9, which refines Theorem 3.3 for regular categories, with further simplifications for infinitary coherent categories, exact categories or locally cartesian closed categories.
In Section 5, we provide some brief remarks regarding categories enriched in cartesian monoidal categories.We instantiate our results when the enriching category V is -the category CHaus of compact Hausdorff spaces, -the category Stn of Stone spaces, -a (Grothendieck) topos.Finally, we have a couple of concluding remarks in Section 6, where we sketch some possible lines of future research, with regard to extending the result to all symmetric monoidal categories, or to generalized multicategories.
Acknowledgments.The author is deeply grateful to Fernando Lucatelli Nunes and Maria Manuel Clementino for their helpful comments regarding this work.The valuable comments of the anonymous referee were also very appreciated.

Preliminaries
The purpose of this section is to give a concise summary of the terminology and notation used for our main results.We begin by reviewing the notion of pseudopullbacks, as our treatment of effective descent morphisms employs the techniques of [29] on commutativity of bilimits.Then, we recall the notion of free coproduct completion of a category, an important tool for our main insight.Finally, we fix some notation regarding monoidal and enriched categories.We remark that 2-pullbacks and pseudopullbacks are far from being equivalent in general.We consider the following cospan: where 1 is the terminal category, and (x ∼ = y) is a category with two isomorphic, but distinct, objects x, y, identified by the functors x, y : 1 → (x ∼ = y).The 2-pullback is the empty category, since x = y, while the pseudopullback is the discrete category whose objects are the isomorphisms from x to y in (x ∼ = y).
Regarding descent theory, we recall the following result of Lucatelli Nunes: Proposition 1.1 ([29, Corollary 9.6]).Suppose Diagram (5) below is a pseudopullback of categories with pullbacks and pullback-preserving functors 1.2.Free coproduct completion of a category: Let V be a category.The free coproduct completion of V, also known as the category of families of objects in V, is denoted by Fam(V).Its objects are set-indexed families (X j ) j∈J of objects X j in V, and a morphism (6) (X j ) j∈J → (Y k ) k∈K consists of a pair (f, φ) where f : J → K is a function on the underlying sets, and is a set-indexed family of morphisms φ j in V. We refer to [3,6,5] for more thorough introductions to this concept.Here, we simply recall that the "underlying set" functor Fam(V) → Set is a fibration, and that we have a canonical, fully faithful functor η V : V → Fam(V) identifying the one-element families.
More importantly, we recall the following observation of Weber: Proposition 1.2 ([37, Proposition 5.15]).The canonical, fully faithful functors η V : V → Fam(V) form a 2-natural cartesian transformation, that is, for all functors A final remark on notation: for a morphism (6) in Fam(V) with K ∼ = 1, the function on the underlying sets is uniquely determined.In this case, the underlying family (7) of morphisms in V is sufficient to determine the morphism in Fam(V).For this reason, we simply denote such morphisms as φ : (X j ) j∈J → Y .1.3.Monoidal categories: Let V = (V, ⊗, I) and W = (W, ⊗, I) be monoidal categories, whose coherence isomorphisms we omit, as they play no role in what follows.We recall that a monoidal functor F : V → W consists of a functor F between the underlying categories, preserving the unit object and tensor products only up-to-isomorphism.This means that we have -an isomorphism e F : I → F I, the unit comparison morphism, -and an isomorphism m F : F x ⊗ F y → F (x ⊗ y), for each pair x, y of objects, the tensor comparison morphism satisfying naturality and coherence conditions (see [2, p. 1889]).
We denote by MndCat the 2-category of monoidal categories, monoidal functors, and their natural transformations [14, p. 474].We further highlight that the forgetful 2-functor MndCat → Cat is pseudomonadic [26, Section 3.1], [28,Remark 4.3], and therefore it creates bilimits.In particular, the underlying category of the pseudopullback of a cospan of monoidal functors is the pseudopullback of the underlying ordinary functors, and fully faithful morphisms in MndCat are precisely those monoidal functors whose underlying functor is fully faithful in Cat.
We have a 2-functor (−)-Cat : MndCat → CAT, mapping each monoidal category V to the category V-Cat of small V-categories, and for each monoidal functor F : V → W, we have the direct image functor F ! : V-Cat → W-Cat.On a small V-category C, the W-category F ! C has the same underlying set of objects, and for each pair of objects x, y in C, Regarding notation, we will denote the unit and composition morphisms of a V-category C by u C : I → C(x, x) for each object x in C, and c C : C(y, z) ⊗ C(x, y) → C(x, z) for each triple x, y, z of objects in C. The unit and composition morphisms for F ! C are given by As it will prove to be convenient, we fix the following notation of composable pairs and triples of hom-objects of a V-category C: - For example, the composition morphism of C may be written as c C : C(x, y, z) → C(x, z).We may also write m F : (F !C)(x, y, z) → F (C(x, y, z)) for the tensor comparison isomorphism when hom-objects are concerned.Analogously, we may define for a V-functor Φ : C → D.

Preservation of bilimits and descent
The following result, present in a more general form in [15], is a helpful, labour-saving device in our work with pseudopullbacks: Theorem 2.1.The enrichment 2-functor (−)-Cat : MndCat → CAT preserves pseudopullbacks.
Proof.Let F : U → W and G : V → W be monoidal functors between monoidal categories.
We desire to confirm that -set of objects given by ob D Φ = ob B, -hom-object given by D Φ (x, y) = Φ x,y : F B(x, y) ∼ = GC(Φx, Φy) at x, y ∈ ob D Φ , -unit object and composition given by the pairs (u B , u C ), (c B , c C ) of the respective unit objects and compositions from B and C; these pairs are well-defined morphisms of PsPb(F, G), since F, G are monoidal functors and Φ is a W-functor.
To be more precise on this last point, we note that the following diagrams commute: (10) Since identity and associativity laws of D Φ are precisely those of B and C, it follows that D Φ is indeed well-defined.The underlying U -category of D Φ is B itself, while its underlying V-category is isomorphic to C: it is given by ob Φ on the sets of objects, and identity on the hom-objects.
Moreover, let X , Y be PsPb(F, G)-categories, and let H : X U → Y U be a U -functor and K : X V → Y V be a V-functor between the underlying U -categories and V-categories of X and Y respectively, such that ob H = ob K and We note that there exists a unique PsPb(F, G)-functor Φ : X → Y with underlying U -functor and V-functor given by H and K, respectively.Indeed, Φ : X → Y is given as follows: ob Φ = ob H, -Φ x,y is given by the pair H x,y , K x,y , which is a morphism in PsPb(F, G), due to (12).
The laws that make Φ into a PsPb(F, G)-functor are precisely given by the laws that make H into a U -functor and K into a V-functor.
If Ψ : X → Y is a PsPb(F, G)-functor with H as underlying U -functor and K as underlying V-functor, we necessarily get Φ = Ψ by comparing their hom-morphisms.
Proof.Let F : V → W be a fully faithful monoidal functor.To prove F ! : V-Cat → W-Cat is fully faithful, let C, D be V-categories, and let Ψ : F ! C → F ! D be a W-functor.It consists of the following data: -A function Ψ : ob C → ob D, -A morphism Ψ x,y : F C(x, y) → F D(Ψx, Ψy) for each pair x, y ∈ ob C. Since F is fully faithful, there exists a unique Φ x.y : C(x, y) → D(Ψx, Ψy) such that F Φ x,y = Ψ x,y .
With this, we define a V-functor Φ : C → D given -on objects by the function Φ = Ψ : ob C → ob D, -on morphisms by Φ x,y : C(x, y) → D(Φx, Φy) for each pair x, y ∈ ob C.This is a V-functor: we note that the following diagrams commute (13) so, by the full faithfulness of F , plus invertibility of e F and m F , we confirm that Φ is a V-functor.Moreover, by definition, it is the unique V-functor such that F ! Φ = Ψ, which concludes our proof.
Lemma 2.3.Given a string of adjoint functors L ⊣ F ⊣ R between categories with finite limits, if L (and therefore R) is fully faithful, then F preserves descent morphisms.
Before providing the proof, we recall that descent morphisms in categories with finite limits are precisely the pullback-stable regular epimorphisms (see, for instance, [20,25]).
Proof.Let p : x → y be a descent morphism.Since F is a left adjoint, we may conclude that F p is a regular epimorphism; we just need to prove that it is stable under pullback.
To do so, let f : z → F y be a morphism, and we consider the following pullback diagram: We wish to prove that f * (F p) is a regular epimorphism.Indeed, we note that F Lf * (F p) ∼ = f * (F p), and since F reflects pullbacks (via R), we have a pullback ( 16) ) is a regular epimorphism; which is preserved by F , hence f * (F p) must be a regular epimorphism, as desired.
Remark 2.4.We highlight one application of Lemma 2.3: for a category B with finite limits, the underlying object-of-objects functor (−) 0 : Cat(B) → B has fully faithful left and right adjoints: these assign to each object b of B its respective discrete and indiscrete internal categories with b as the underlying object of objects; see [19, 7.2.6].Thus, we conclude that (−) 0 preserves descent morphisms.Remark 2.4 can be used to verify that V-Cat → Cat(V) reflects effective descent morphisms for extensive categories V with finite limits with − • 1 : Set → V fully faithful, without requiring V to have a (regular epi, mono)-factorization system, using the same argument as in the proof of [29,Theorem 9.11].

Descent for cartesian enriched categories
Throughout this section, fix a cartesian monoidal category V with finite limits, and consider the canonical embedding η : V → Fam(V), as given in Subsection 1.2.
To conclude the proof, we note that since (−)-Cat is a 2-functor, it preserves adjoints, which, together with Lemma 2.2, guarantees that the functor Fam(V)-Cat → Set-Cat has fully faithful left and right adjoints, thus it preserves descent morphisms by Lemma 2.3.It is well-established that η !: Set → Set-Cat reflects descent morphisms, and descent morphisms in Set are effective for descent.This places us under the conditions of Proposition 1.1, so the result follows.
Lemma 3.2.The category Fam(V) is extensive with finite limits, and Proof.We have already confirmed that − • 1 : ; we consider the fibration Fam(V) → Set.The base category Set has all (finite) limits, and the fibers V J at a set have finite limits as well.The latter are preserved by the change-of-base functors f * : V K → V J for each function f : J → K. See also [18,Corollary 4.9], and [5, Sections 6.2, 6.3].Now, we are ready to prove our main result, Theorem 3.3.We begin by considering the following string of functors: By Lemma 3.1, the functor V-Cat → Fam(V)-Cat reflects effective descent morphisms.We observe that the same holds for the Fam(V)-Cat → Cat(Fam(V)): indeed, Lemma 3. Hence, the composite (19) reflects effective descent morphisms, that is, if F is the value of a Vfunctor F : C → D via (19), then F is an effective descent V-functor whenever F is an effective descent morphism in Cat(Fam(V)).
By Le Creurer's [25, Corollary 3.3.1]and [32,Lemma A.3], the problem of confirming F is an effective descent morphism can be reduced to showing that the actions of F on n-tuples of composable morphisms satisfy suitable "surjectivity" conditions for n = 1, 2, 3. To be precise, we denote by F n the morphism in Fam(V) given by the action of F on the objects of n-tuples of composable morphisms.For n = 1, 2, 3, we have and these are respectively given by the formal coproducts of the following morphisms indexed by the underlying functions If E is the class of effective descent morphisms, descent morphisms or almost descent morphisms in Fam(V), then E is closed under coproducts.
Therefore, if we are given a morphism (f, φ) : ) is an almost descent morphism in Fam(V) for all quadruples y 0 , y 1 , y 2 , y 3 of objects in D, then F is an effective descent morphism in V-Cat.
Proof.By taking the coproducts of the morphisms given in (I), (II) and (III), we conclude that (i), (ii) and (iii) are respectively an effective descent morphism, a descent morphism, and an almost descent morphism.Thus, it follows that F is an effective descent morphism in Cat(Fam(V)).
By reflecting along (19), we conclude that F is an effective descent morphism in V-Cat.

Familial descent morphisms
Theorem 3.3 raises the question of understanding (stable) regular epimorphisms and effective descent morphisms in Fam(V) for a category V with finite limits, with the goal of providing more tractable methods to verify conditions (I), (II), (III).
The key ideas for many of the applications are given in the next couple of lemmas.We begin by noting that the kernel pair of a morphism φ : for each i, j ∈ I.Then, the kernel pair of φ, denoted by ker φ, is given by where p n : I × I → I for n = 0, 1 is the projection which forgets the nth component.
Lemma 4.1.Let φ : (X i ) i∈I → Y be a morphism in Fam(V).We consider the diagram D φ : J I → V where ob J I = (I × I) + I, -for each pair i, j ∈ I, we have two arrows (i, j) → i and (i, j) → j, -the values of D φ at (i, j) → i and (i, j) → j are defined to be π 1 i,j and π 0 i,j , respectively.φ is a (stable) regular epimorphism if and only if D φ has a (stable) colimit and colim D φ ∼ = Y .
Proof.We begin by recalling that a morphism a category with finite limits is a regular epimorphism if and only if it is the coequalizer of its kernel pair.
The fibration Fam(V) → Set is a left adjoint functor, hence preserves colimits.In particular, if φ : I × I I 1 must be a coequalizer, and this is the case only when I is non-empty.
We note that we have a natural isomorphism which is fibered over K: an element from either set is completely determined by an element k ∈ K and a morphism ω : (X i ) i∈I → Z k in Fam(V) satisfying ω i • π 1 i,j = ω j • π 0 i,j for all i, j ∈ I. Given such an element, any morphism (q, ψ) : (Z k ) k∈K → (W l ) l∈L provides an element qk ∈ L and a morphism for all i, j.Thus, if ker φ has a colimit, its underlying set is necessarily a singleton by (21), so we denote it as an object Q of V. We have and since this isomorphism is fibered over K, we conclude Q is a colimit of D φ .
Conversely, if Q is a colimit of D φ , then we have which confirms Q is a colimit of ker φ.
To verify stability, we begin by assuming φ to be a regular epimorphism.Given a morphism ω : Z → Y , the colimits of ker ω * (φ) and D ω * (φ) are isomorphic whenever either exist, so the stability of one colimit is the equivalent to the other.Taking coproducts in Fam(V), we confirm the same holds for any morphism (Z k ) k∈K → Y .
Understanding effective descent morphisms in Fam(V) is a more difficult problem, as is to be expected.However, we can reduce the study of the category of descent data of a morphism φ : (X i ) i∈I → Y to the full subcategory of connected descent data, an idea made precise by the following result.Lemma 4.2.Let φ : (X i ) i∈I → Y be a morphism in Fam(V), with I non-empty.We have an equivalence Desc(φ) ≃ Fam(Desc conn (φ)), where Desc conn (φ) is the full subcategory of connected objects of Desc(φ).
Proof.Given descent data (f, γ), (h, ξ) as in the following diagram (22) we obtain descent data (f, h) for the unique morphism I → 1.Since I is non-empty, this morphism is effective for descent, so that K ∼ = J × I for a set J, and we may take f = p 0 : J × I → I and h = p 2 : J × I × I → J × I to be projections (recall p n forgets the nth component).Thus, taking the pullback of this descent data along ((j, −), id) : (W j,i ) i∈I → (W j,i ) j,i∈J×I , we obtain the following descent data for φ: for each j ∈ J. Now, we claim that the descent data of the form ( 24) More concretely, we wish to prove that any morphism of descent data (q, χ) : (V i ) i∈I → (W j,i ) j,i∈J×I factors through ((j, −), id) for some j ∈ J.This gives a morphism of descent data q : (id, p 1 ) → (p 0 , p 2 ) for the unique morphism I → 1.We note that q is uniquely determined by a function j : 1 → J, whose value provides the desired factorization.
Having verified that all descent data is a coproduct of connected descent data, the result follows.Proof.We note that the full subcategory of connected objects of Fam(V)/Y is precisely V/Y , and any object of Fam(V)/Y is a coproduct of such connected objects.Thus, Fam(V)/Y ≃ Fam(V/Y ), and we may conclude (b) =⇒ (a), since The converse relies on the fact that the comparsion Frames: Effective descent morphisms in V-Cat were studied in [8, Section 5], for Heyting lattices V.As an illustration of our tools, we provide a second proof that *-quotient morphisms in V-Cat (that is, surjective-on-objects V-functors that satisfy condition (26) below for all y 0 , y 1 , y 2 ) are effective for descent when V is a (co)complete Heyting lattice.
Let V be a thin category (ordered set).A morphism (X i ) i∈I → Y in Fam(V) is simply the assertion "for all i ∈ I, X i Y ".Thus, we simply write -It is an epimorphism if and only if I is non-empty.
-If it is an epimorphism, it is also stable.
-It is a regular epimorphism if and only if i∈I -If it is a regular epimorphism, it is stable if and only if the above join is distributive, that is, Proof.We note that (X i ) i∈I Y is an epimorphism if and only if the underlying function I → 1 is surjective, and this is the case exactly when I be non-empty.
So, if I is non-empty, we confirm (X i ) i∈I Y is a stable epimorphism: given Z Y we can produce an epimorphism (Z ∧ X i ) i∈I Z, since V has meets.By taking coproducts, the same holds for all (Z j ) j∈J Y .
We immediately deduce from Lemma 4.1, that (X i ) i∈I Y is a regular epimorphism if and only if i∈I X i ∼ = Y , and stability under pullbacks is exactly the condition (25), so there's nothing to verify.
We say a thin category V is a Heyting semilattice (also known as an implicative semilattice [31] or a Brouwerian semilattice [23]) if it has finite limits (has meets and bounded) and is cartesian closed (has implication).In particular, this means that a ∧ − is a left adjoint functor for each a ∈ V, which must preserve colimits (joins).As a corollary, we conclude that: Corollary 4.5.If V is a Heyting semi-lattice, regular epimorphisms in Fam(V) are stable.
Proof.Condition ( 25) is automatically satisfied, by the previous remark.
Corollary 4.6.If V is a (co)complete Heyting (semi-)lattice, then regular epimorphisms in Fam(V) are effective for descent.

Proof. Let (X i ) i∈I
Y be a regular epimorphism.Given connected descent data id : We note that it is enough to prove that X i ∧ Z W i for all i ∈ I. Indeed, by distributivity, we have Now, Theorem 4.3 and [32, Corollary 2.3] complete our proof.
With this, we obtain one direction of [8, Theorem 5.4]: Theorem 4.7.Let V be a (co)complete Heyting (semi-)lattice, and let F : C → D be a V-functor.If F is surjective on objects and we have an isomorphism for all y 0 , y 1 , y 2 , then F is effective for descent.
Proof.Due to Lemma 4.4, we may conclude that -condition (III) is satisfied, since F is surjective on objects, and -condition (II) is given by ( 26), plus the stability of regular epimorphisms provided by Corollary 4.5.
Condition (I) remains to be verified.Taking y 1 = y 2 above, so that D(y 1 , y 2 ) ∼ = 1, we have and since we have C(x 0 , x 1 ) D(y 0 , y 1 ) for all x i ∈ F * y i , we conclude that (C(x 0 , x 1 )) x i ∈F * y i D(y 0 , y 1 ) is a regular epimorphism in Fam(V), and therefore is effective for descent by Corollary 4.6.

Regular categories:
The ideas behind the previous results generalize to regular categories V, via their (regular epi, mono)-factorization system, which allow us to reduce statements about epimorphisms φ : (X i ) i∈I → Y in Fam(V) to families of monomorphisms.The following result makes this precise: Lemma 4.8.Suppose V is a regular category, and let φ : (X i ) i∈I → Y be a morphism in Fam(V).For each i ∈ I, consider the following factorization where π i is a regular epimorphism and ι i is a monomorphism for all i ∈ I.
φ is a (stable) epimorphism if and only if ι is a (stable) epimorphism.
φ is a (stable) regular epimorphism if and only if i∈I M i ∼ = Y (and the join is stable).
-If π i is an effective descent morphism in V for all i ∈ I, then φ is an effective descent morphism if and only if ι is an effective descent morphism.
Proof.The factorizations (27) for each i ∈ I give a factorization φ = ι • (id, π) in Fam(V).We note that (id, π) is a coproduct of stable regular epimorphisms, hence φ is a (stable) (regular) epimorphism if and only if ι is a (stable) (regular) epimorphism (see [20,Propositions 1.3,1.5]).Moreover, if π i is effective for descent for all i ∈ I, taking coproducts will guarantee (id, π) is effective for descent as well, meaning that φ is effective for descent if and only if ι is effective for descent (because the basic bifibration respects the BED, see [22,Section 4]).
Under this light, the results are immediate consequences of Lemma 4.1.
To prove our main result for this section, we let F : C → D be a V-functor, and we consider the following (regular epi, mono)-factorizations of the underlying morphisms of the families (I), (II), (III), as in ( 27): C(x 0 , x 1 , x 2 , x 3 ) M x 0 ,x 1 ,x 2 ,x 3 D(y 0 , y 1 , y 2 , y 3 ) respectively, for objects y i in D, and x i ∈ F * y i , for n = 0, 1, 2, 3.
Theorem 4.9.Let V be a regular category, and let F : C → D be a V-functor.If (i) P x 0 ,x 1 is an effective descent morphism in V for each pair of objects x 0 , x 1 , (ii) we have an equivalence V/D(y 0 , y 1 ) ≃ Desc conn (I) for all y 0 , y 1 , (iii) the join x i ∈F * y i M x 0 ,x 1 ,x 2 exists, is stable and is isomorphic to D(y 0 , y 1 , y 2 ) for all y 0 , y 1 , y 2 , (iv) I : (M x 0 ,x 1 ,x 2 ,x 3 ) x i ∈F * y i → D(y 0 , y 1 , y 2 , y 3 ) is an almost descent morphism in Fam(V) for all y 0 , y 1 , y 2 , y 3 , then F is effective for descent.
Proof.The goal is to verify that properties (I), (II) and (III) are satisfied, so we can apply Theorem 3.3.By Theorem 4.3, condition (ii) holds if and only if I : (M x 0 ,x 1 ) x i ∈F * y i → D(y 0 , y 1 ) is an effective descent morphism for all objects y 0 , y 1 in D. Now, by applying Lemma 4.8, we have that -(I) follows as a consequence of (i) and (ii), -(II) is a consequence of (iii), -(III) is a consequence of (iv), so the result follows.
In case V satisfies further properties, we can simplify the above list: -If V is infinitary coherent (has stable, arbitrary unions of subobjects), then the join in (iii) exists and is stable; one only needs to verify if the isomorphism exists.-If V is exact, or locally cartesian closed, then (i) is redundant, since regular epimorphisms are effective for descent.

Enrichment in cartesian monoidal categories
Theorem 3.3 generalizes Lucatelli's result [29] about effective descent V-functors, by not requiring V to be extensive, nor that the induced functor − • 1 : Set → V is fully faithful (if it even exists).Thus, we consider examples of enriching categories V not satisfying one of those aforementioned properties (excluding examples such as V = Set, Top, Cat), dedicating this section to the study of such categories V-Cat.
Thin categories: Thin categories V with cartesian monoidal structures are (essentially) bounded meet-semilattices, which we have previously discussed in Section 4, as an illustrative example.We only briefly repeat here that the result for (co)complete Heyting lattices V admits a particularly nice description (Theorem 4.7), which was already provided in [8] using other techniques.
However, 1//Cat is not an extensive category, since it doesn't have an initial object.It doesn't even have coproducts for any pair of objects: let (C 1 , c 1 ) and (C 2 , c 2 ) be pointed categories, and we assume this pair has a coproduct ( C, c) in 1//Cat, with coprojections Let F i : C i → D be functors, and we suppose we have morphisms f i : F c i → d for i = 1, 2. These define morphisms (F i , f i ) : (C i , c i ) → (D, d) for i = 1, 2, so the universal property guarantees there exists a unique morphism (G, g) : ( C, c) → (D, d) satisfying GI i = F i and f i = g • Gι i .
In fact, we can prove that Gc and by the universal property, (G, g) = (G, h), hence g = h.
But there is no reason for D to have such a coproduct: consider the category D given by the following graph and observe that the pair d 1 , d 2 does not have a coproduct.Thus, we obtain the desired contradiction by letting F i : C i → D be the constant functor to d i , for i = 1, 2.
A (1//Cat)-category is a 2-category B and -for each x, y, an object hom(x, y) ∈ B(x, y), -for each x, a morphism e x : 1 x → hom(x, x), -for each x, y, z, a morphism m x,y,z : hom(y, z) • hom(x, y) → hom(x, z), -the following diagrams commute for all w, x, y, z: Categories with zero object: Let V be a category with a zero object, which we denote by 1.Such categories are usually not extensive, for if the zero object were strict, we would have V ≃ 1.
For a V-category C, we write p x,y : 1 → C(x, y) for the uniquely determined morphism.In particular, this implies u x = p x,x for all x, and With this, we can confirm that all hom-objects must be isomorphic: the isomorphism is given by: Thus, we conclude V-Cat has objects the empty V-category plus pairs (non-empty set, V-monoid).
Eckmann-Hilton: We suppose V is the category of unital magmas.By the Eckmann-Hilton argument, a V-monoid is precisely a commutative monoid.Since V has a zero object, we conclude V-Cat essentially has objects the empty V-category plus pairs (non-empty set, commutative monoid), in which case the effective descent V-functors are given by the empty V-functor on the empty V-category, and pairs (surjective function, regular epimorphism of monoids).

Coextensive categories:
We say a category V with finite limits -has codisjoint products if V op has disjoint coproducts, -has a strict terminal object if V op has a strict initial object, -is finitely coextensive if V op is finitely extensive.As expected, finitely coextensive categories V have codisjoint products and a strict terminal object.This is the case for the categories of commutative R-algebras for a ring R, as a class of examples.
We verify that cartesian monoidal categories V with codisjoint products and strict terminal object do not provide an interesting enriching base with the cartesian monoidal structure: we shall confirm that V-Cat ≃ Set.Therefore, the effective descent V-functors are precisely those that are surjective on objects.
Let C be a V-category.For each x ∈ ob C, the unit morphism 1 → C(x, x) is an isomorphism, and for each pair x, y ∈ ob C, the composition morphism C(x, y) × C(y, x) → 1 is uniquely determined.Thus, the associativity condition for elements in C(x, y, x, y) translates to saying that the projections on the first and third component are equal.But since products are codisjoint, we must have C(x, y) ∼ = 1, for all x, y.
Categories of spaces: Since most varieties of algebras V seem to have an uninteresting V-Cat for the cartesian monoidal structure, we turn our attention to categories of spaces.We begin by instantiating our results when V = CHaus of compact Hausdorff spaces, which is a pretopos [30], and therefore is a Barr-exact category, but it is not infinitary extensive.Let F : C → D be a CHaus-functor between CHaus-categories, and consider the factorizations (28), ( 29) and ( 30) of the hom-morphisms of F .By Theorem 4.9, F is effective for descent if -CHaus/D(y 0 , y 1 ) ≃ Desc conn (I) for all objects y 0 , y 1 in D, -We have a stable join x 0 ,x 1 ,x 2 M x 0 ,x 1 ,x 2 ≃ D(y 0 , y 1 , y 2 ), for all y 0 , y 1 , y 2 , -I : (M x 0 ,x 1 ,x 2 ,x 3 ) x i → D(y 0 , y 1 , y 2 , y 3 ) is an almost descent morphism in Fam(CHaus) for all y 0 , y 1 , y 2 , y 3 .Similarly, since the category V = Stn of Stone spaces is a regular category [30], we can also say something about Stn-functors.Let F : C → D be a Stn-functor between Stn-categories, and we consider the factorizations (28), ( 29) and ( 30) of the hom-morphisms of F .Again by Theorem 4.9, F is effective for descent if -P x 0 ,x 1 is an effective descent morphism in Stn for each pair x 0 , x 1 , -Stn/D(y 0 , y 1 ) ≃ Desc conn (I) for all objects y 0 , y 1 in D, -We have a stable join x 0 ,x 1 ,x 2 M x 0 ,x 1 ,x 2 ≃ D(y 0 , y 1 , y 2 ), for all y 0 , y 1 , y 2 , -I : (M x 0 ,x 1 ,x 2 ,x 3 ) x i → D(y 0 , y 1 , y 2 , y 3 ) is an almost descent morphism in Fam(Stn) for all y 0 , y 1 , y 2 , y 3 .It would be worthwhile to explore whether analogous arguments are fruitful in the more general setting of (monad, frame)-enriched categories.More specifically, we may take V to be the category of ordered compact Hausdorff spaces [36], or the category of ultrametric compact Hausdorff spaces.
Finally, we may consider a topos V, so that the effective descent morphisms are exactly the epimorphisms.It is not always the case that there exists a functor − • 1 : Set → V, and when it does, it is not always fully faithful 1 , so we can use Theorem 3.3 expand our knowledge of effective descent V-functors to all toposes V. Let F : C → D be a V-functor between V-categories.If -(F x 0 ,x 1 ) x i ∈F * y i : (C(x 0 , x 1 )) x i ∈F * y i → D(y 0 , y 1 ) is an effective descent morphism in Fam(V) for all pairs y 0 , y 1 of objects in D, -(F x 0 ,x 1 ,x 2 ) x i ∈F * y i : (C(x 0 , x 1 , x 2 )) x i ∈F * y i → D(y 0 , y 1 , y 2 ) is a descent morphism in Fam(V) for all triples y 0 , y 1 , y 2 of objects in D, -(F x 0 ,x 1 ,x 2 ,x 3 ) x i ∈F * y i : (C(x 0 , x 1 , x 2 , x 3 )) x i ∈F * y i → D(y 0 , y 1 , y 2 , y 3 ) is an almost descent morphism in Fam(V) for all quadruples y 0 , y 1 , y 2 , y 3 of objects in D, then F is an effective descent V-functor.While these conditions can be refined in general with Theorem 4.9, if V is a Grothendieck topos, then we can take advantage of the fact that Fam(V) ≃ V ↓ Set is a Grothendieck topos as well, in which case F is an effective descent functor whenever (F x 0 ,x 1 ,x 2 ,x 3 ) x i ∈F * y i : (C(x 0 , x 1 , x 2 , x 3 )) x i ∈F * y i → D(y 0 , y 1 , y 2 , y 3 ) is an epimorphism in Fam(V) for all y 0 , y 1 , y 2 , y 3 .

Future work
Having established sufficient conditions for effective descent in V-Cat for cartesian monoidal categories V, an obvious continuation would be to extend this result to suitable monoidal categories V. We describe a strategy which would rely on the present work; we denote CartCat and SymMndCat for the 2-categories of cartesian (monoidal) categories and symmetrical monoidal categories.

CartCat SymMndCat
Cat where every 2-functor is forgetful.Both functors to Cat have left 2-adjoints which are easy to describe.So, if the existence of the left biadjoint F : SymMndCat → CartCat is guaranteed, we need to study the following questions: -What conditions on V guarantee existence of pullbacks in F V? -Is the unit η : V → F V fully faithful?After obtaining solutions to the above questions, we could then study the functor η !: V-Cat → F V-Cat, 1 Grothendieck toposes satisfying this property are said to be hyperconnected.
which raises the ultimate question: does it reflect effective descent morphisms?An affirmative answer would provide a string of functors that reflect effective descent.Then, since F V is hypothetically a cartesian monoidal category with finite limits, we obtain a more general result via Theorem 3.3.Combined with an adequate study of effective descent morphisms in F V, these results can be applied in the study of effective descent morphisms in V-Cat for any symmetrical monoidal category V.

1. 1 .
Pseudopullbacks and 2-pullbacks: Let F : C → E and G : D → E be functors.The pseudopullback of F, G, denoted by PsPb(F, G) may be succinctly defined as the full subcategory of the comma category (F ↓ G) whose objects are isomorphisms.To be explicit, PsPb(F, G) has -objects given by isomorphisms ξ : F c ∼ = Gd, where c ∈ C and d ∈ D, -morphisms (ζ : F a → Gb) → (ξ : F c → Gd) given by a pair of morphisms f : a → c and g : b → d such that ξ • F f = Gg • ζ. -identities and composition given componentwise from C and D. The 2-pullback of F, G is simply the ordinary pullback in the underlying category CAT.It may also be seen as the full subcategory of PsPb(F, G) whose objects are the identity morphisms; these are determined by pairs c ∈ C, d ∈ D such that F c = Gd.

Theorem 4 . 3 .
Let φ : (X i ) i∈I → Y be a morphism in Fam(V).Then the following are equivalent: (a) φ is effective for descent.(b) We have an equivalence V/Y ≃ Desc conn (φ).
fully faithful in Remark 2.4, because V has a terminal object.Moreover, extensivity of Fam(V) is well-established; see, for instance, [6, Proposition 2.4].Existence of finite limits is a direct corollary of [16, Proposition 4.1, Theorem 4.2] (p y,z • p x,y ) • p w,x p y,z • (p x,y • p w,x ) p x,z • p w,x p y,z • p w,y hom B ) → (C, hom C ) consists of a 2-functor F : B → C, and a Cfunctor Φ : F ! B → C, with the same underlying function on objects.