What should a generic object be?

Abstract Jacobs has proposed definitions for (weak, strong, split) generic objects for a fibered category; building on his definition of (split) generic objects, Jacobs develops a menagerie of important fibrational structures with applications to categorical logic and computer science, including higher order fibrations, polymorphic fibrations, 
$\lambda2$
 -fibrations, triposes, and others. We observe that a split generic object need not in particular be a generic object under the given definitions, and that the definitions of polymorphic fibrations, triposes, etc. are strict enough to rule out some fundamental examples: for instance, the fibered preorder induced by a partial combinatory algebra in realizability is not a tripos in this sense. We propose a new alignment of terminology that emphasizes the forms of generic object appearing most commonly in nature, i.e. in the study of internal categories, triposes, and the denotational semantics of polymorphism. In addition, we propose a new class of acyclic generic objects inspired by recent developments in higher category theory and the semantics of homotopy type theory, generalizing the realignment property of universes to the setting of an arbitrary fibration.


Introduction
Since the latter half of the 20th century, fibered category theory or the theory of fibrations has played an important background role in both the applications and foundations of category theory [17,8].Fibered categories, also known as fibrations, are a formalism for manipulating categories that are defined relative to another category, generalizing the way that ordinary categories can be thought of as being defined relative to the category of sets.
The sense in which ordinary category theory is pinned to the category of sets can be illustrated by considering the definition of when a category "has products": Definition 1.A category has products when for any indexed family { ∈ } ∈ of objects, there exists an object ∈ ∈ together with a family of morphisms : ∈ such that for any family of morphisms ℎ : there exists a unique morphism ℎ : ∈ factoring each ℎ through .
In the above, the dependency on the category Set is clear: the indexing object is a set.If we had required to be drawn from a proper subcategory of Set (e.g.finite sets) or a proper supercategory (e.g.classes), the notion of product defined thereby would have been different.The purpose of the formalism of fibered categories is to explicitly control the ambient category that parameterizes all indexed notions, such as products, sums, limits, colimits, etc.
Remark 2 (Relevance to computer science).The ability to explicitly control the parameterization of products and sums is very important in theoretical computer science, especially for the denotational semantics of polymorphic types of the form ∀ .[ ].Such a polymorphic type should be understood as the product of all [ ] indexed in the "set" of all types, but a famous result of Freyd [14] shows that if a category has products of this form parameterized in Set, then must be a preorder.Far from bringing to an early end the study of polymorphic types in computer science, awareness of Freyd's result sparked and guided the search for ambient categories other than Set in which to parameterize these products [37,21,23].Fibered category theory provides the optimal language to understand all such indexing scenarios, and the textbook of Jacobs [24], discussed at length in the present paper, provides a detailed introduction to the applications of fibered category theory to theoretical computer science.

Introduction to fibered categories
Before giving a general definition, we will see the way that fibered categorical language indeed makes parameterization explicit by considering the prototype of all fibered categories, the category Fam( ) of Set-indexed families of objects of a category .Construction 3 (The category of families).We define Fam( ) to be the category of Set-indexed families in , such that 1. an object of Fam( ) is a family { ∈ } ∈ where is a set, 2. a morphism { } ∈ { } ∈ in Fam( ) is given by a function : together with for each ∈ a morphism ¯ : .
More abstractly, the fiber category Fam( ) is the following pullback:

Set Construction 5 (Reindexing functors). For any function :
, there is a corresponding reindexing functor * : Fam( ) Fam( ) that restricts an -indexed family into a -indexed family by precomposition.
With the reindexing functors in hand, can now rephrase the condition that has products (Definition 1) in terms of Fam( ).

Proposition 6. A category has products if and only if for each product projection function , : ×
, the reindexing functor , : Fam( ) Fam( ) × has a right adjoint ( , ) : Fam( ) × Fam( ) such that the following Beck-Chevalley condition holds: for any function : , the canonical natural transformation The characterization of products in terms of the category of families may seem more complicated, but it has a remarkable advantage: we can replace Fam( ) Set with a different functor satisfying similar properties in order to speak more generally of when one category has "products" that are parameterized in another category.The properties that this functor has to satisfy for the notion to make sense are embodied in the definition of a fibration or fibered category; a functor ℰ ℬ will be called a fibration when it behaves similarly to the functor projecting the parameterizing object from a category of families of objects.We begin with an auxiliary definition of cartesian morphism: Definition 7. Let : ℰ ℬ and let be a morphism in ℰ, which we depict as follows: In the diagram above, we say that lies over .We say that is cartesian in when for any morphism in ℰ and such that the former lies over the composite → → in ℬ, there exists a unique morphism lying over such that lies over the composite as depicted below: Remark 8 (Explication of cartesian maps).Returning to our example of the category of families Fam( ) over Set, we can make sense of the notion of a cartesian map.Given a function : of indexing sets, the reindexing functor * takes -indexed families to -indexed families.Given an -indexed family { } ∈ , we may define a morphism * { } ∈ { } ∈ in whose first component is : and whose second component is the identity function at each ∈ .The morphism * { } ∈ { } ∈ is then cartesian in : Fam( ) Set.
In fact, we can strengthen Exercise 9 to give an extrinsic characterization of cartesian morphisms in Fam( ): cartesian morphisms are exactly the "fiberwise isomorphisms".Exercise 10.Let = { } ∈ and = { } ∈ be objects of Fam( ), and let : be a morphism between them.Show that the following are equivalent: 1. the morphism : is cartesian; 2. for each ∈ , the component : is an isomorphism.
The name "cartesian morphism" is inspired by pullbacks, as we see in Exercise 11 below.
Exercise 11.Let ℬ → be the arrow category of ℬ, whose objects are morphisms of ℬ and whose morphisms are commuting squares between them; let cod : ℬ → ℬ be the codomain functor that projects the codomain of a map .Show that a morphism ∈ ℬ → is cartesian if and only if the corresponding square in ℬ is a pullback square (also called a cartesian square).
The relationship between cartesian morphisms and pullback squares expressed by Exercise 11 suggests a generalization of the conventional "pullback corners" notation to an arbitrary fibration, which we shall use liberally.
Convention 12 (Generalized "pullback corners").Let : ℰ ℬ be a functor; when we wish to indicate that a morphism in ℰ is cartesian over , we will display it using a "pullback corner" notation as follows: Finally we may give the definition of a fibration.

Definition 13. A functor : ℰ
ℬ is called a fibration when for any object ∈ ℰ and morphism ∈ ℬ there exists a cartesian morphism lying over . The cartesian morphism is often called the cartesian lift of at .
Convention 14.We will depict a fibration : ℰ ℬ using triangular arrows.When we wish to leave the functor implicit, we refer to ℰ as a fibered category over ℬ.In the same way that one writes × for the apex of a product diagram, we will often write ¯ : * for the cartesian lift of : at as depicted below: * ¯ In the case of Fam(ℰ), the existence of cartesian lifts for each : corresponds to the reindexing functors * : Fam( ) Fam( ) .
Exercise 15.Verify that the functor Fam( ) Set is a fibration.
Exercise 16.Conclude from Exercise 11 that the codomain functor ℬ → ℬ is a fibration if and only if ℬ has all pullbacks.

When the codomain functor ℬ →
ℬ is a fibration, we will refer to it as the fundamental fibration, written P ℬ following Streicher [52].

Small categories, internal categories
An ordinary category need not have a set of objects -for instance, the category Set of all sets has a proper class of objects.Likewise, it is possible to find categories such that between two objects there may be a proper class of morphisms (e.g. the In this paper, we are somewhat agnostic about set theoretic foundations.Our discussion is compatible with the viewpoint of ZFC, in which classes are taken to be formulas at the meta-level; our discussion is, however, also compatible with other accounts of the "set-class" distinction, such as NBG set theory, MK set theory, or the universe-based approaches of Grothendieck [2] and Mac Lane [29].category of sets and isomorphism classes of spans between them).A category that has hom sets is called locally small, and we will refer to a category that has, up to equivalence, a set of objects as globally small.A category that has both these properties is equivalent to a small category in the ordinary sense.Small categories are very useful: for instance, if C is a small category then the category of functors [C, Set] is a Grothendieck topos.Functor categories of this kind play an important role in theoretical computer science [e.g.32,38,11].
The idea of a (globally, locally) small category can be relativized from Set to another category in two a priori different ways that ultimately coincide up to equivalence.The simplest and move naïve way to think of a small category C in a category ℬ is as an algebra for the sorts and operations of the theory of a category internal to ℬ, which we develop below; the more sophisticated way is to view C as a fibration over ℬ satisfying a generalization of the global and local smallness conditions.Definition 17.Let ℬ be a category that has pullbacks.An internal category or category object in ℬ is given by: 1. an object C 0 ∈ ℬ of objects, 2. and an object C 1 ∈ ℬ of morphisms, 3. and source and target maps , : C 1 C 0 , 4. and a morphism : C 0 C 1 choosing the identity maps, such that . and a morphism : 6. (plus associativity and unit laws for composition and identity) Observation 18.When ℬ = Set we obtain exactly the ordinary notion of a small category, i.e. a small category is the same thing as an internal category or category object in Set.
In the previous section, we have argued that fibrations are a fruitful way to think about categories defined relative to another category.Indeed, we may view an internal category C as a fibration via a process called externalization.This proceeds in two steps; first we construct a presheaf of categories on ℬ, and then we use the Grothendieck construction to turn it into a fibration.

Construction 19 (The presheaf of categories associated to an internal category).
Let C be an internal category in ℬ.We may define a presheaf of categories C • : ℬ op Cat like so: 1. for ∈ ℬ, an object of C is given by a morphism : C 0 , Observe that the presheaf of categories associated to an internal category is, in each fiber , the category object in Set obtained by restricting along the functor : 1 ℬ.
Definition 22 (Externalization of an internal category).Let C be an internal category in ℬ; its externalization is defined to be the Grothendieck construction As promised we may now isolate the properties of the fibered category [C] that correspond (up to equivalence) to arising by externalization from an internal category.

Definition 24. A fibered category : ℰ
ℬ is called globally small if there is an object ∈ ℰ such that for any ∈ ℰ there exists a (not necessarily unique) cartesian map .
The property of global smallness described above is often phrased as : ℰ ℬ having a "generic object" ∈ ℰ, but the reason for this paper's existence is that the precise definition of "generic" means in this context is somewhat controversial, and the bulk of the present paper is devoted to justifying the precise meaning for genericity that we have chosen (in agreement with Definition 24).Note that our Definition 24 does not agree with that of Jacobs [24], who assumes additional (somewhat rare) properties of the object , namely that for ∈ ℰ there is exactly one morphism lying underneath a cartesian morphism .We will show Section 3.1 that ours is the correct definition.
Definition 25 (Bénabou [7], Streicher [52]).A fibered category : ℰ ℬ is called locally small when for any ∈ ℬ and , ∈ ℰ there exists a span Set is both globally small and locally small.One of the fundamental results of fibered category theory is that, up to equivalence, global and local smallness in the sense of Definitions 24 and 25 suffice to detect internal categories, which we recall from Bénabou [7]: the externalization of an internal category E if and only if it is both globally and locally small.
Although we do not include the (standard) proof of Proposition 27, it is instructive to understand the object ∈ [E] in the externalization of an internal category E that renders [E] globally small.Recalling the definition of the externalization via the Grothendieck construction, we define the weak generic object to be the pair E 0 , id E 0 : E 0 E 0 given by the object of objects and its identity map.
Example 28.When E is a small category, then the weak generic object of [E] = Fam(E) can be written as the family { } ∈E 0 .

Cleavages and splittings
We briefly recall the definitions of cleavages and splittings for a fibration, as they are relevant to the rest of this paper.

Definition 29.
A cleavage for a fibration : ℰ ℬ is a choice of cartesian liftings, assigning to each morphism : ∈ ℬ in the base an object ∈ ℰ over and a cartesian morphism : over .
The data of a cleavage extends for each : ∈ ℬ to a reindexing functor : ℰ ℰ .

Definition 30.
A cleavage for a fibration : ℰ ℬ is called split when the assignment of reindexing functors ↦ → strictly preserves identities and compositions.
Convention 31.When the cleavage is understood, we will often write * for .
A fibration equipped with a cleavage is called a cloven fibration; we may use the axiom of choice to equip any fibration with a (non-canonical) cleavage.When the cleavage associated to a cloven fibration is split, we speak of split fibrations.Observe that a splitting allows one to view a fibration ℰ ℬ as a presheaf of categories ℬ op Cat sending each ∈ ℬ to the fiber ℰ .Recalling Construction 19, we see that the externalization of any internal category is split.

Goals and structure of this paper
Although the property stated in Definition 24, that there exist cartesian morphisms for any , is the most that can be required for an arbitrary internal category, more restrictive notions of generic object have proved important in practice for different applications.Unfortunately, over the years a number of competing definitions have proliferated throughout the literature -and some of the more established of these definitions lead to false conclusions when taken too literally, as we point out in Section 3.1 in our discussion of Jacobs' mistaken Corollary 9.5.6.
The goal of this paper, therefore, is to argue for a new alignment of terminology for the different forms of generic object that is both internally consistent and reflects the use of generic objects in practice.Because generic objects play an important role in several areas of application (categorical logic, algebraic set theory, homotopy type theory, denotational semantics of polymorphism, etc.), we believe that we have sufficient evidence today to correctly draw the map.
• In Section 2, we recall the definitions of several variants of generic object by Jacobs [24]; our main observation is that a split generic object in the sense of op.cit.need not be a generic object in the same sense.
• In Section 3, we analyze the consequences of the definitions discussed in Section 2 for the use of generic objects in internal category theory, tripos theory, denotational semantics of polymorphism, algebraic set theory, and homotopy type theory.
• In Section 4, we propose new unified terminology and definitions for all extant forms of generic object (as well as one new one).Our proposal is summarized and compared with the literature in Table 1.

Four kinds of generic object
We begin by recalling Definition 5.2.8 of Jacobs [24], from which we omit some additional characterizations that will not play a role in our analysis.
Jacobs [24] then defines split generic objects for split fibrations in Definition 5.2.1, paraphrased below: A split fibration : ℰ ℬ has a split generic object if there is an object Ω ∈ ℬ together with natural isomorphism : where the presheaf ob ℰ • is defined using the splitting.
A useful characterization of split generic objects is given in Lemma 5.2.2 of op.cit.: A split fibration : ℰ ℬ has a split generic object if and only if there is an object ∈ ℰ with the property that ∀ ∈ ℰ.∃! : . * = .[24] Comment 32.The weak and strong generic objects of Jacobs [24] are referred to by Phoa [34] as generic objects and skeletal generic objects.Phoa [34] does not consider the intermediate notion.On the other hand, Phoa [34] defines strict generic objects relative to an arbitrary (non-split) cleavage; a split generic object is indeed a strict generic object in the sense of Phoa, but even for a split cleavage, a strict generic object need not be a split generic object.We will discuss Phoa's terminology more in Comment 58.

Separating generic objects from strong generic objects
Jacobs [24] notes that generic and strong generic objects coincide in fibered preorders, but they may differ otherwise -the difference emanating from the presence of non-trivial vertical automorphisms.
Example 33.Let be a group containing two distinct elements ≠ , and let B be the groupoid with a single object whose hom set is itself.The family fibration Fam(B ) Set is a fibered category whose objects are just sets, but such that a morphism : ∈ Fam(B ) is a pair ( , ) of a function : ∈ Set together with a generalized element : .Moreover, every morphism in Fam(B ) is cartesian as B is a groupoid.The unique object ∈ Fam(B ) 1 Set is clearly generic, but not strong generic.Indeed, we have two distinct cartesian morphisms given by the pairs (id 1 Set , ) ≠ (id 1 Set , ).
Example 34.Another class of examples comes from considering skeleta of full subcategories of Set.For instance, one may take the skeleton of a Grothendieck universe and then externalize to obtain a fibration that has a generic object that is not strong.The weak generic object of the externalization of an internal category defined in Construction 36 obviously need not be a strong generic object, but it may be more surprising to learn that it also need not be a generic object at all.This can happen, for instance, when the internal category C has two distinct isomorphic objects; the following concrete example illustrates the problem:

A split generic object need not be a generic object
Example 37 (A split generic object that is not a generic object).Let be a set of sets containing two distinct elements , with the same cardinality, and Set ⊆ Set be the full subcategory of Set spanned by .Then the family fibration Fam(Set ) Set has a split generic object given by the pair = ( , id ), but is nonetheless not a generic object.Indeed, we have two cartesian morphisms (1 Set , ) lying over distinct elements ≠ : 1 Set respectively.

Corollary 38.
A split generic object is not necessarily a generic object.

Skeletal and gaunt small categories
We recall from Example 28 that for a small category C, the family fibration Because C is gaunt and thus skeletal, we know that ℎ 0 = ℎ 1 ; thus ℎ 0 assigns to each ∈ an isomorphism ℎ 0, : ℎ 1 which is (by assumption) necessarily the identity.Thus both ℎ 0 and ℎ 1 must send every ∈ to the identity map on and are thus equal.(2b) Conversely we assume that is strong generic to check that any isomorphism in C is an identity map; since is necessarily also generic, it follows by the first case of the present lemma that we may consider just the automorphisms in C, considering Lemma 48.To show that any automorphism : in C is the identity morphism, we proceed exactly as in the proof of Lemma 49 by observing that the two cartesian maps corresponding to the identity map and the automorphism respectively are necessarily equal by our assumption that is strong generic.

Generic objects from weak generic objects
Although the externalization of an internal category necessarily has a (split) weak generic object, in some cases it may also have a generic object that embodies the skeleton of C as in Example 34, but is usually different from the .
Construction 43 (Computing the skeleton of a small category).Suppose that ℬ = Set and thus C is an ordinary small category.Then we may consider the quotient C 0 / of the objects of C under isomorphism; in other words, this is the set of isomorphism classes of C-objects.Using the axiom of choice, we may arbitrarily choose a section : C 0 / C 0 to the quotient map; moreover, we may choose a function associating to each ∈ C 0 an isomorphism Lemma 44.The pair = (C 0 / , ) is a generic object for Fam(C).
1. First of all, it is clear that there exists a cartesian map lying over in the correct configuration.

Fixing :
such that there exists a cartesian map ( , ) lying over , it remains to show that = .This follows because such a cartesian map ensures that and are the same family of isomorphism classes of objects.
Lemma 45.If the defined above is a split generic object, then C is skeletal.
Proof.We have already seen that is a split generic object, hence if is also split generic we have isomorphisms C 0 and so we have C 0 C 0 / .

Weak generic objects are the correct generalization of split generic objects
It is clear that any split generic object is in particular a weak generic object; but the converse also holds in a certain sense that we make precise below.
Construction 50 (Presheaf of categories).Let : ℰ ℬ be a fibered category equipped with a cleavage , and let ∈ ℰ be a weak generic object for .We may construct a presheaf of categories ℰ • : ℬ op Cat like so: 1. an object of ℰ is a morphism : , 2. a morphism in ℰ is a vertical map ℎ : over .

Construction 51. Let E : ℬ op
Cat be a presheaf of categories; then the Grothendieck construction :
Lemma 52.There is a fibered equivalence from Proof.Fixing ∈ ∫ ℰ • , we must check that there exists a unique : such that * = .Unfolding definitions, we fix ∈ ℬ and : to check that there is a unique : such that ( , ) = ( , ).Of course, this is true with = .
Thus a weak generic object for a fibration ℰ ℬ generates in a canonical way a new equivalent split fibration ℰ ℬ that has a split generic object.This is the correspondence between weak generic and split generic objects.
We recall several definitions from Jacobs [24] below in order to illustrate a pattern.
1.A higher order fibration is a first order fibration with a generic object and a cartesian closed base category.Such a higher order fibration will be called split if the fibration is split and all of its fibered structure (including the generic object) is split.[24,Definition 5.3.1] 2. A tripos is a higher order fibration ℰ Set over Set for which the induced products and coproducts along an arbitrary function satisfy the Beck-Chevalley condition.[24,Definition 5.3.3] 3. A polymorphic fibration is a fibration with a generic object, with fibered finite products and with finite products in its base category.It will be called split whenever all this structure is split.[24,Definition 8.4.1].

A polymorphic fibration with Ω in the base as a generic object will be called (a) a →-fibration if it has fibered exponents;
(b) a 2-fibration if it has fibered exponents and also simple Ω-products and coproducts; (c) a -fibration if it has fibered exponents, simple products and coproducts, and exponents in its base category.[24,Definition 8.4.3] 5. Let be →, 2, .A -fibration will be called split if all of its structure is split.In particular, its underlying polymorphic fibration is split.[24,Definition 8.4.4]A split generic object need not be a generic object as we have seen in Section 2.2, and indeed, is only quite rarely one.Consequently, a split polymorphic fibration is not the same thing as a split fibration with a split generic object and split fibered finite products.This disorder suggests that a change of definitions is in order, which we propose in Section 4.

Consequences for internal category theory
The somewhat chaotic status of generic objects vis-à-vis split generic objects has led to an erroneous claim by Jacobs [24] that the externalization of an internal category has a generic object.What is actually proved by op.cit. in Lemma 7.3.2 is that the externalization of an internal category has a split generic object, but this is later claimed (erroneously) to give rise to a generic object in the proof of Corollary 9.5.6.Thus the claimed result of Corollary 9.5.6, that a fibration is small (equivalent to the externalization of an internal category) if and only if it is globally small (has a generic object) and locally small, is mistaken.To give a correct definition in the terminology of op.cit., one would define globally small categories to assume a weak generic object, as we have done in Definition 24.
We do not recall the definition of first order fibrations here, as the specifics are not important for the present paper.

Consequences for tripos theory
Jacobs [24] sketches in Example 5.3.4 the standard construction of a tripos from a partial combinatory algebra, referring to Hyland et al. [22] for several parts of the construction (including the generic object); the standard definition of a tripos involves a weak generic object only [22,36,35], a discrepancy that has already been noted by Birkedal [9,p. 110].
This difference is quite destructive, as the weak generic object of a realizability tripos will not generally be generic, as pointed out by Streicher [51,Theorem 4.6].Thus we conclude that definition of tripos proposed by Jacobs [24] rules out the main examples of naturally occurring triposes.Remark 54.A final subtlety: over the category of assemblies, the fibration of regular subobjects does indeed have a strong generic object, but this is not part of the structure of the tripos.

Consequences for polymorphism
The reception in the community studying polymorphism has been to either avoid or tacitly correct the definition of generic object.For instance, Hermida [18] speaks of weak generic objects and strong generic objects, and gives the correct definition of →, 2, fibration in terms of weak generic objects.On the other hand Birkedal et al. [10], Johann and Sojakova [25], Sojakova and Johann [44], Ghani et al. [15] deal mainly with the split generic objects, and thus do not seem to run into problems.It can be seen, however, that the examples of (split) -fibrations in the cited works are not in fact (split) -fibrations in the sense of Jacobs [24], because they do not have generic objects.Of course, the problem lies with the definitions rather than the examples.

Consequences for type theory and algebraic set theory
The idea of a universe in a category has been abstracted from Grothendieck's universes [2] by way of the contributions of a number of authors including Bénabou [6], Martin-Löf [30], Street [49], Joyal and Moerdĳk [27], Hofmann and Streicher [19], Streicher [50].In fibered categorical language, a universe usually is a full subfibration that has a weak generic object -potentially equipped with additional structure.
In certain cases, the weak generic object of a universe is also a generic object that is nonetheless not strong; one example is Voevodsky's construction of a universe of well-ordered simplicial sets in the context of the simplicial model of homotopy type theory, reported by Kapulkin and Lumsdaine [28]: it is strong generic for the fibered category of well-ordered families of simplicial sets and order-preserving morphisms, but only generic when considering morphisms that need not preserve the well-orderings.On the other hand, universes of propositions, such as the subobject classifier of a topos or the regular subobject classifier of a quasitopos, are the main source of strong generic objects in nature.
The theory of universes as applied to type theory on the one hand and algebraic set theory on the other hand motivates two additional variants of generic object: 1.In applications to type theory, it has been increasingly important for universes to have a weak generic object that satisfies an additional re- alignment property relative to a class of monomorphisms ℳ; in particular, given a span of cartesian maps → where ( ) ∈ ℳ, we need an extension → factoring → through .This realignment property has proved essential for the semantics of univalent universes in homotopy type theory [28,40,41,16] as well as Sterling's synthetic Tait computability [47], an abstraction of Artin gluing and logical relations.In Section 4.3 we will discuss the generalization of the realignment property to an arbitrary fibered category as the acyclic generic object, which lies strictly between weak generic and generic objects.[27], universes are considered that enjoy a form of generic object that is even weaker than weak generic object.For each ∈ ℰ one only "locally" has a morphism into ; the "very weak" generic objects of algebraic set theory seem to relate to weak generic objects in the same way that weak completeness relates to strong completeness in the context of stack completions [21,23], with important applications to the theory of polymorphism.As these very weak generic objects seem to play a fundamental role, we discuss them in more detail in Section 4.2 as part of our proposal for a new alignment of terminology.

A proposal for a new alignment of terminology
Based on the data and experience of the applications of fibered categories in the theory of triposes, polymorphism, and universes in type theory and algebraic set theory, we may now proceed with some confidence to propose a new alignment of terminology for generic objects that better reflects fibered categorical practice.In this section, we distinguish our proposed usage from that of other authors by underlining.
We will define several notions of generic object in conceptual order rather than in order of strength; in Table 1 we summarize our terminology, and compare it to several representatives from the literature.In Fig. 1 we compare the strength of the different kinds of generic object.

Generic objects: ordinary, skeletal, and gaunt
Let : ℰ ℬ be a fibered category.Our most basic definition below corresponds to the weak generic objects of Jacobs [24].Definition 55.A generic object for : ℰ ℬ is defined to be an object ∈ ℰ such that for any ∈ ℰ there exists a cartesian morphism .Comment 58.Our skeletal and gaunt generic objects are the same as Jacobs' generic objects and strong generic objects respectively.Our terminology is inspired by a comparison between the properties of an internal category C and its externalization: in particular, the generic object over C 0 is skeletal when C is a skeletal category and it is gaunt when C is a gaunt category.Unfortunately, Phoa [34] has used the word "skeletal" to describe what we call gaunt generic objects; but this usage accords with op.cit.'s unconventional definition of a skeletal category: usually a skeletal category is one in which any two isomorphic objects are equal, but Phoa defines it to be one in which the only isomorphisms are identity maps.Thus Phoa's skeletal categories would ordinarily be referred to as gaunt categories.
To illustrate the comparison between (skeletal, gaunt) categories and (skeletal, gaunt) generic objects respectively, we recall Lemma 42, which is restated in the new terminology as follows: Corollary 59.Let C be a small category; then C is (respectively skeletal, gaunt) if and only if the generic object = { } ∈C 0 for Fam(C) is (respectively skeletal, gaunt).

Weak generic objects and stack completions
In this section, we are concerned with stacks -a 2-dimensional variation on the concept of a sheaf.Definition 60.Suppose that a pretopos ℬ is equipped with the structure of a site.Then a fibration ℰ ℬ is a stack when for any object ∈ ℬ and covering sieve ⊆ ℬ ↓ , the canonical restriction functor ℰ hom Fib ℬ ( , ℰ) is an equivalence.
For more on stacks, see The Stacks Project [45] or the tutorial of Vistoli [55].
Our examples will mainly concern stacks on a pretopos ℬ equipped with its regular topology.
Definition 61 (Frey [13,  Remark 64.The notion of weak generic object defined above should be thought of as an internal version of the property of being a generic object -internal to a fibration that is a stack for the regular topology.Indeed, our explicit definition ought to be the translation of ordinary genericity through a generalization of Shulman's stack semantics [43] that is stated for stacks other than the fundamental fibration P ℬ = ℬ → ℬ.The same (informal) translation is used by Hyland et al. [23] to correctly define weak equivalences and weak completeness for categories fibered over a regular category.
Weak generic objects in the sense of Definition 62 arise very naturally.
Example 65.Let : be a morphism in a topos ℬ; then the class of maps arising as pullbacks of : determines a full subfibration [ ] ⊆ P ℬ of the fundamental fibration P ℬ = ℬ → ℬ for which : is a generic object.The stack completion of [ ] is a weakly (but not necessarily strongly) equivalent full subfibration { } ⊆ P ℬ , and it can be computed like so: an object of { } is a morphism such that there exists a regular epimorphism ˜ such that the pullback × ˜ ˜ lies in [ ] ˜ .In other words, we have the pullback squares in the following configuration: × ˜ ˜ Unless [ ] was already a stack, it is not necessarily the case that is a generic object for the stack completion { }.But Streicher [50] points out that is nevertheless a weak generic object for { }, essentially by definition.
Scenarios of the form described in Example 65 are easy to come by; a canonical example is furnished by the modest objects of a realizability topos as described by Hyland et al. [23], with critical implications for the denotational semantics of polymorphism.
Example 66 (Hyland et al. [23]).Let Eff be the effective topos [20], and let N be its object of realizers, i.e. the partitioned assembly given by N such that ∈ N ⇔ = .Write P ⊆ Ω N×N for the the object of ¬¬-closed partial equivalence relations on N and P ∈ Eff ↓ P for the generic ¬¬-closed subquotient of N; internally, this is the subquotient : P { : N | }/ .The family : P P then induces a full subfibration [ ] ⊆ P Eff spanned by morphisms that arise by pullback from ; an element [ ] is an object at stage that is the subquotient of N by some partial equivalence defined at stage .The fibration [ ] Eff is small with generic object , but it is not complete.
Although for every Eff-indexed diagram C [ ] there "exists" in the internal sense a limit, we cannot globally choose this limit.This situation is referred to by Hyland et al. [23] as weak completeness as opposed to strong completeness.
In contrast, we may consider the stack completion { } of [ ].An element of { } is given by an object at stage such that there "exists" (in the internal sense) a partial equivalence relation that it is the quotient of -externally, this means that there is a regular epimorphism ˜ such that ˜ * is the subquotient of N by some partial equivalence relation defined at stage ˜ .The stack { } is weakly but not strongly equivalent to [ ]; on the other hand, { } is complete in the strong sense.Finally, we observe that : P P is a weak generic object for the stack completion { }.
If we pull back [ ] to along the inclusion : Asm Eff of assemblies / ¬¬-separated objects in Eff, then we have a complete fibered category * [ ] over Asm which turns out to be (strongly) equivalent to the familiar fibration UFam(PER) Asm of uniform families of PERs over assemblies.In fact, * [ ] is strongly equivalent to * { } -in other words, over assemblies there is no difference between an object that is locally isomorphic to a subquotient of N and an actual subquotient of N.
There is another way to think of the situation described at the end of Example 66, using the observations of Streicher [50] on the relationship between Bénabou's notion of definable class and stackhood: the class of families of subquotients of N is not definable in Eff, but it is definable in Asm.

A new class: acyclic generic objects
Inspired by the crucial realignment property of type theoretic universes [16] that was discussed in Section 3.4(1), we define a new kind of generic object for an arbitrary fibration that restricts in the case of a full subfibration to a universe satisfying realignment.We refer to Gratzer et al. [16] for an explanation of the applications of realignment.In this section, let : ℰ ℬ be a fibration and let ℳ be a class of monomorphisms in ℬ. Convention 68.When ℳ is understood to be the class of all monomorphisms in ℬ, we will simply speak of acyclic generic objects.
All of the examples of acyclic generic objects that we are aware of so far have arisen in the context of the full subfibration spanned by a universe in the sense of Streicher [50], where acyclicity reduces to the realignment property studied in detail by Gratzer et al. [16].
Remark 69.It is reasonable to ask whether there is a "weak" version of acyclicity that pertains to stack completions in the same way that weak generic objects relate to generic objects, e.g. by asking for a cover-cartesian morphism ˜ and an extension of × ˜ → → along × ˜ ˜ .It remains somewhat unclear to this author whether this notion is useful, so we do not include it in our proposal.

Origins of the terminology
The origin of the term "acyclicity" is explained by Shulman [41] and Riehl [39] in essentially the following way.Let ℰ ⊆ P ℬ be a full subfibration equipped with a generic object ; we will write Core(ℰ) for the groupoid core of ℰ, and for any ∈ ℬ we will write y ℬ ℬ for the discrete fibration whose fiber at ∈ ℬ is the set of morphisms .Then we have a canonical morphism of fibered categories y ℬ Core(ℰ) corresponding under the fibered Yoneda lemma [52, §3] to itself.Realignment for ℰ is then the property that y ℬ Core(ℰ) has the following (2-categorical) extension property with respect to any monomorphism : ∈ ℳ, in which we write : y Core(ℰ) for the morphism that we identify with ∈ ℰ under the Above we have exactly a morphism ˆ : extending : along : . The other direction is analogous.
It is clear that an acyclic generic object need not be skeletal; the above discussion confirms that acyclic and skeletal generic objects are in fact incomparable.

Splittings and generic objects
We have shown in Section 2.5 that the correct generalization to non-split fibrations of a split generic object in the sense of Jacobs [24] is what we have proposed to call a generic object, i.e. the weak generic object of op.cit.Thus we conclude that the correct relationship can be established between split -fibrations and -fibrations when re-expressed using our definitions, where ranges over the different kinds of fibered structures (e.g.2-fibration, polymorphic fibration, etc.).For example, the following definition expresses the correct relationship between 2-fibrations and split 2-fibrations: Definition 72.A 2-fibration is a fibration with a generic object , fibered finite products, and simple -products and coproducts.A 2-fibration will be called split if all its structure is split.

In univalent mathematics
The theory of fibered (1-)categories carries over mutatis mutandis to the univalent foundations of mathematics, as shown by Ahrens and Lumsdaine [1].The pertinent notions of generic object are, however, a bit different in a univalent setting.In particular, gaunt generic objects play a more fundamental role in univalent mathematics than in non-univalent mathematics, because the uniqueness of the classifying squares becomes a statement of contractibility rather than strict uniqueness.In non-univalent mathematics, uncontrived instances of gaunt generic objects tend to follow the pattern of subobject classifiers in representing a fibered poset; in univalent mathematics, we also have object classifiers (univalent universes) which are gaunt generic objects for groupoid cores of the full subfibrations they induce.
Just as in non-univalent mathematics, skeletal generic objects are not expected to play a significant structural role although they may appear when considering the deloopings of groups, as in our 1-categorical examples.The most interesting case is that of acyclic generic objects: the acyclicity property is the correct way to present in non-univalent mathematics a generic object that is gaunt in the homotopical sense, as Shulman has recently argued [42].Thus in a homotopical/univalent setting, one expects to work directly with gaunt generic objects rather than acyclic ones.

Concluding remarks
For a number of years, the disorder in the variants of generic object has led to a proliferation of subtle differences in terminology between different papers applying fibered categories to categorical logic, type theory, and the denotational semantics of polymorphic types.Based on the kinds of generic object that occur most naturally or have the most utility, we have proposed a unified terminological scheme for generic objects that we believe will meet the needs of scientists working in these areas.

Remark 23 .
the associated presheaf of categories (Construction 19).When C is an internal category in Set, the externalization [C] is the family fibration Fam(C) described in Construction 3.

Construction 35 (
The canonical splitting of the externalization).The externalization of an internal category is split in a canonical way: given ( , ) ∈ [C] and : , we choose * ( , ) = ( , • ).The cartesian morphism * ( , ) ( , ) is given by the pair ( , id • • ) where id : C 0 C 1 is the generic identity morphism.Construction 36 (Weak, split generic objects in the externalization).The externalization : [C] ℬ has a weak generic object = (C 0 , id C 0 ).Relative to the splitting of [C] from Construction 35, the weak generic object is also a split generic object.

Figure 1 :
Figure 1: An analysis of the comparability of different notions of generic object, where the rightward direction represents (strictly) decreasing strength.
Exercise 26(Intermediate).Let be an ordinary category; show that the fibered category Fam( ) Set is locally small if and only if is locally small.Show that is equivalent to a small category if and only if Fam( ) We have two cartesian maps ℎ 0 , ℎ 1 : { 0 } C 0 via the identity morphism and the other lying over 1 : { * } C 0 via .Since is generic, these two cartesian maps must lie over the same element of C 0 , so we have 0 = 1 .(2a) If C is gaunt, then is strong generic; fix any two cartesian morphisms [26]a weak generic object = { } ∈C 0 .We will relate properties of the category C to corresponding properties of the weak generic object .Definition 39.A category is called skeletal when any to isomorphic objects are equal.Definition 40.A category is called gaunt when any isomorphism in that category is the identity.Comment 41.The term gaunt is used by Barwick and Schommer-Pries[4], nLab [31], Univalent Foundations Program[53].In passing, Johnstone[26]has referred to such categories as stiff.Lemma 42.The category C is (respectively skeletal, gaunt) if and only if is (respectively generic, strong generic).Proof.(1a)IfC is skeletal, then is generic; fix any family { } ∈ and cartesian map : { } ∈ { } ∈C .By Exercise 10, sends each ∈ to an object ∈ C that is isomorphic to ; as C is skeletal, it follows that = and so we conclude that is generic.(1b)Assumeconversely that is generic and fix an isomorphism : 0 1 in C.0 : { * } Proof.If C be gaunt, it is obviously both essentially gaunt and skeletal.Conversely, if C is essentially gaunt and skeletal, given any isomorphism : Lemma 49 that if is strong generic, then C is essentially gaunt in the sense defined below.Later on in Section 4, we will see that the original weak generic object being strong generic corresponds to C being gaunt.Thus we conclude that although the family fibration Fam(C) over Set of a small category C does have a generic object, this generic object cannot be either a split generic object or a strong generic object except in somewhat contrived scenarios.

Table 1 :
A Rosetta stone for the different terminologies for generic objects.
[27,definition and terminology agrees with that of Streicher[50, (5.2)] and Battenfeld[5, Definition 6.3.7],differingonly in that both of the cited works specialize the definition for Bénabou-definable full subfibrations of the fundamental fibration of a topos, i.e. full subfibrations that are stacks.Our definition is also identical to the representability axiom (S2) for classes of small maps in algebraic set theory[27, Definition 1.1].