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Separators

Separators

Idea

An object SS (or family 𝒮\mathcal{S} of objects) in a category 𝒞\mathcal{C} is called a separator or generator if generalized elements with domain SS (or domain from 𝒮\mathcal{S}) are sufficient to distinguish morphisms in 𝒞\mathcal{C}.

The dual notion is that of a coseparator.

Caution on terminology

The term ‘generator’ is slightly more ambiguous because of the use of ‘generators’ in generators and relations. That said, there is a link between these two senses provided by theorem (q.v.).

Definitions

Definition

An object S𝒞S \in \mathcal{C} of a category 𝒞\mathcal{C} is called a separator or a generator or a separating object or a generating object, or is said to separate morphisms if:

  • for every pair of parallel morphisms f,g:XYf,g \colon X \to Y in 𝒞\mathcal{C}, if fe=gef\circ e = g\circ e for every morphism e:SXe\colon S \to X, then f=gf = g.

Assuming that 𝒞\mathcal{C} is locally small category, we have equivalently that SS is a separator if the hom functor Hom(S,):𝒞Hom(S,-) \colon \mathcal{C} \to Set is faithful.

More generally:

Definition

A family 𝒮=(S a) (a:A)\mathcal{S} = (S_a)_{(a\colon A)} of objects of a category 𝒞\mathcal{C} is a separating family or a generating family if:

  • for every pair of parallel morphisms f,g:XYf,g \colon X \to Y in 𝒞\mathcal{C}, if fe=gef \circ e = g \circ e for every e:S aXe \colon S_a \to X sourced in the family, then f=gf = g.

Assuming again that 𝒞\mathcal{C} is locally small, we have equivalently that 𝒮\mathcal{S} is a separating family if the family of hom functors Hom(U a,):𝒞Hom(U_a,-) \colon \mathcal{C} \to Set is jointly faithful?.

Since repetition is irrelevant in a separating family, we may also speak of a separating class instead of a separating family.

Definition

A separating set is a small separating class.

In fibered categories

The notion of separating family can be generalized from categories to fibered categories in such a way that the family fibration? of a category C\mathbf{C} has a separating family if and only if C\mathbf{C} has a small separating family.

Definition

A separating family in a fibered category P:EBP:\mathbf{E}\to \mathbf{B} is an object SES\in \mathbf{E} such that for every parallel pair f,g:ABf,g:A\to B in EE with fgf\neq g and P(f)=P(g)P(f) = P(g) there exist arrows c:XSc: X\to S and h:XAh:X\to A (constituting a span) such that cc is PP-cartesian, and fhghf h \neq g h .

See Definition B2.4.1 in the Elephant.

Examples and applications

Strengthened separators

Motivating theorem

If CC is locally small and has all small coproducts, then a set-indexed family (S a) (a:A)(S_a)_{(a\colon A)} is separating if and only if, for every XCX\in C, the canonical morphism

ε X: a:A,f:S aXS aX \varepsilon_X\colon \coprod_{a\colon A, f\colon S_a \to X} S_a \longrightarrow X

is an epimorphism.

This theorem explains a likely origin of the term “generator” or “generating family”. For example, in linear algebra, one says that a set of morphisms f a:S aXf_a: S_a \to X spans or generates XX if the induced map S aX\oplus S_a \to X maps epimorphically onto XX.

More generally:

Definition

If \mathcal{E} is a subclass of epimorphisms, we say that (S a) (a:A)(S_a)_{(a\colon A)} is an \mathcal{E}-separator or \mathcal{E}-generator if each morphism ε X\varepsilon_X (as above) is in \mathcal{E}.

The weakest commonly-seen strengthened notion is that of extremal separator, i.e. separator where all maps ε X\varepsilon_X are extremal epimorphisms. The notion of extremal separator admits an equivalent reformulation not referencing coproducts:

Proposition

If CC is locally small and has all small coproducts, then a set-indexed family (S i) (i:I)(S_i)_{(i\colon I)} is an extremal separator if and only if the functors C(S i,):CSetC(S_i,-):C\to\mathrm{Set} are jointly faithful and jointly conservative.

Proof

Assume first that the family (S i) (i:I)(S_i)_{(i\colon I)} is an extremal separator. The functors C(S i,):CSetC(S_i,-):C\to\mathrm{Set} are jointly faithful for every separator. To see that they are also jointly conservative, let f:ABf:A\to B such that all C(S i,f)C(S_i,f) are bijective. Then ε B\varepsilon_B factors through ff since all its components do, which implies that ff is an extremal epi since ε B\varepsilon_B is one by assumption. It remains to show that ff is a monomorphism. For this, let u,v:XAu,v:X\to A such that fu=fvf u = f v. Then we have fuh=fvhf u h = f v h for all iIi\in I and h:S iXh:S_i\to X, which implies uh=vhu h = v h since the C(S i,f)C(S_i,f) are bijective, and we conclude that u=vu=v since (S i) (i:I)(S_i)_{(i\colon I)} is separating.

Conversely, assume that the functors C(S i,)C(S_i,-) are jointly faithful and jointly conservative. Given ACA\in C, joint faithfulness implies that ε A\varepsilon_A is epic. To see that it is extremally so, assume a factorization ε A=mg\varepsilon_A = m g with mm monic. We have to show that mm is an isomorphism, and for this it is sufficient to show that all C(S i,m)C(S_i,m) are bijections. Injectivity is clear since mm is monic, and surjectivity follows since every h:S iAh:S_i\to A factors through ε A\varepsilon_A.

The concepts “strong separator” and “regular separator” corresponding to the notions of strong epimorphism and regular epimorphism do not admit such a reformulation, but the following result shows that they are equivalent to extremal separators in reasonable categories.

Proposition

Assume that CC is locally small and has all small coproducts.

  1. If CC is balanced, then every separator is extremal.

  2. If CC has pullbacks, then every extremal separator is strong.

  3. If CC is regular, then every strong separator is regular.

The converse implications do always hold.

Proof

This is a direct consequence of the facts that

  1. in a balanced category every epi is extremal,

  2. in a category with pullbacks, every extremal epi is strong, and

  3. in a regular category every strong epi is regular.

Remarks
  1. Proposition gives rise to a notion of extremal separator that makes sense independently of the existence of coproducts. In fact claim 1 of the preceding result holds in this more general setting, since every faithful functor out of a balanced category is conservative.

  2. Most of the literature uses the term “strong separator” (or strong generator) for what we call an extremal separator. Adamek and Rosicky (Section 0.6) also comment on this mismatch, writing “It would be more reasonable, but unfortunately less standard, to call [a strong generator] an extremal generator”. However, item 2 of the preceding result shows that this discrepancy disappears and the terms coincide in presence of pullbacks (and coproducts).

  3. In the Elephant, Johnstone uses “separator” in the same sense as we do, and writes “generator” for extremal separators, in the more general sense not assuming coproducts. Since he always assumes finite limits, he can use a simplified criterion only requiring joint conservativity of the hom-functors (since a conservative functor F:CDF:C\to D is automatically faithful whenever CC has equalizers and FF preserves them).

Dense separators

Finally, the strongest kind of separator commonly seen is that of dense separator.

Definition

A dense separator in a category CC is a family (S i) (i:I)(S_i)_{(i\colon I)} of objects such that the generated full subcategory is dense.

Every dense separator is an extremal separator, and it is also strong and regular whenever those words make sense, i.e. CC is locally small and has small coproducts. If CC furthermore has pullbacks and the coproducts are pullback-stable, then every regular separator is dense (see Borceux I, Proposition 4.5.6). To see that the pullback-stability condition is necessary, consider the category of abelian groups. Here, the free group on one generator is a regular, but not a dense separator.

Giraud’s axioms

Giraud's axioms characterize Grothendieck toposes as locally small regular categories with effective equivalence relations and disjoint and pullback-stable coproducts admitting a small separator. The previously stated and cited results show that in fact every such separator is dense (the effectivity and disjointness assumptions don’t play a role for this conclusion).

References

Last revised on October 15, 2019 at 17:55:04. See the history of this page for a list of all contributions to it.