When is Existential Quantification Conservative?

(or Lan N : P(A) -→ P(C))

given by the coend formula

(see [2] for notation), is conservative (that is, reflects isomorphisms) for a given V-functor N : A -→ C

with A a small V-category. In this note we establish a “simple” sufficient condition for this to hold; namely, that A should also have a natural V-opcategory structure (with mild assumptions on V).

If a small V-category A is equipped with V-natural transformations

is conservative for a given functor

is a regular mono (that is, the kernel of some pair of maps) and that the composite of regular monos in V is again a regular mono, and the functor

The rest of this section is concerned with the proof of this statement. Note: We shall we use the term “opcat” to refer to any application using a V-natural transformation of the form

derived from the V-natural transformation δ by use of the Yoneda expansion of a given V-functor f . In fact, we can suppose that A is merely a Frobenius category in the sense that the given family of maps

is only V-natural in a and b (and not necessarily in c) and that the family

in the following calculations, where x replaces x , etc.

To show that ∃ N is conservative, it suffices to show that the unit of the V-adjunction ∃ N ⊣ [N, 1] is a regular mono (see [1] or [5], for example, and the references therein to W. Tholen). To do this, we now consider the following diagram in V.

x x q q q q q q q q q q q q q q q q f b

where the isomorphisms are by the Yoneda expansion of f , and “can” denotes the canonical “interchange” maps. Since x N ⊗ 1 is a regular mono (by the hypotheses on V and N ), so also is its pullback along the right-hand composite map. Hence, since opcat :

is a coretraction, with left inverse the composite

is the composite of a coretraction (which is always a regular mono) and a regular mono (the pullback of x N ⊗ 1), so is a regular mono (by hypothesis on V). Thus, the adjunction unit we are looking at, which is the composite

is a regular monomorphism, as required.

Similarly, given a V-functor N : A -→ E with A small and E a V-cocomplete V-category, we have the standard V-adjunction

where Y (e)(a) = E(N a, e) describes the N -Yoneda functor Y , and

describes its left adjoint. Hence, with corresponding hypotheses on A, N , and V as before, we can replace the right-hand side of the diagram displayed in §2 by the following composite (*):

where the isomorphism comes from the Yoneda-lemma expansion

for N . Then, by the same argument as before, we obtain a regular mono in V from the commuting diagram

is a regular mono in V for each X in V. Consequently, if this additional condition ( †) holds on N and V, then the left-adjoint V-functor

The result of §2 can then be recovered from this latter result by putting E = [C, V], and taking the new N : A -→ E to be the composite of the Vfunctor N op : A op -→ C op (this N from §2) with the Yoneda embedding

noting that A is a V-opcategory iff A op is also a V-opcategory. The condition ( †) holds for the new N since we have the isomorphisms

by the Yoneda lemma applied twice.

The following result is related to that of the earlier §2, but is much simpler. Suppose that regular monomorphisms (that is, kernels in V) are closed under composition in V and also that n is a regular mono in V if mn and m are regular monos in V. Then, provided both coproduct Σ and tensoring X ⊗ -preserve regular monos in V, we have that

is conservative if regular epimorphisms (that is, cokernels) split in the functor category [A op ⊗ A, V] and each component

To establish this, one simply notes that the canonical regular epimorphism splits naturally in S ∈ [A op ⊗ A, V], so that a N ⊗ 1 is a regular monomorphism by the hypotheses on N and V. This implies that the unit components

are regular monomorphisms, as required for ∃ N to be conservative. Note: In practice, it is often the case that, under the given hypotheses on [A op ⊗ A, V], each monomorphism of the form

splits naturally in [A op , V], in which case the result is obvious and generalises the familiar fact that ∃ N is fully faithful if N is.

Neither the condition in §2 and §3 that A should have an opcategory structure, nor the condition in this section that regular epimorphisms should split in [A op ⊗ A, V], is in any way necessary for the process of left Kan extension along the Yoneda embedding of A op into [A, V], to be conservative. One notable example is the (left) Cayley functor

which is given by the coend formula ∃ P (f ) = a f (a) ⊗ P (a, -, -), and is not fully faithful in general. This functor is conservative for each (small) V-promonoidal category (A, P, J) defined over any complete and cocomplete base category V.

Finally we note that both the sufficient conditions mentioned immediately above are closely related to the splitting properties of regular epimorphi

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