nLab reflective subcategory

Idea

A reflective subcategory is a full subcategory

$$C \hookrightarrow D$$

such that objects $d$ and morphisms $f \colon d \to d'$ in $D$ have “reflections” $T d$ and $T f \colon T d \to T d'$ in $C$. Every object in $D$ looks at its own reflection via a morphism $d \to T d$ and the reflection of an object $c \in C$ is equipped with an isomorphism $T c \cong c$.

A canonical example is the inclusion

$$\mathrm{Ab} \hookrightarrow \mathrm{Grp}$$

of the category of abelian groups into the category of groups, whose reflector is the operation of abelianization.

A useful property of reflective subcategories is that the inclusion $C \hookrightarrow D$ creates all limits of $D$ and $C$ has all colimits which $D$ admits.

Definition

Definition

A full subcategory $i : C \hookrightarrow D$ is reflective if the inclusion functor $i$ has a left adjoint $T$:

$$(T \dashv i) : C \stackrel{\stackrel{T}{\leftarrow}}{\hookrightarrow} D \,.$$

The left adjoint is sometimes called the reflector, and a functor which is a reflector (or has a fully faithful right adjoint, which is the same up to equivalence) is called a reflection. Of course, there are dual notions of coreflective subcategory, coreflector, and coreflection.

Remark

A few sources (such as MacLane 1971) do not require a reflective subcategory to be full. However, in light of the fact that non-full subcategories are not invariant under equivalence, consideration of non-full reflective subcategories seems of limited usefulness. The general consensus among category theorists nowadays seems to be that “reflective subcategory” implies fullness. Examples for non-full subcategories and their behaviour can be found in a TAC paper by Adámek and Rosický.

Remark

The components of the unit

$$\array{ & \nearrow &\Downarrow^{\eta}& \searrow^{Id} \\ D &\stackrel{T}{\to}& C &\stackrel{i}{\hookrightarrow} & D }$$

of this adjunction “reflect” each object $d \in D$ into its image $T d$ in the reflective subcategory

$$\eta_d : d \to T d \,.$$

This reflection is sometimes called a localization (due to this Prop. at reflective localization), although sometimes this term is reserved for the case when the functor $T$ is left exact.

Definition

If the reflector $T$ is faithful, the reflection is called a completion.

Characterizations

Proposition

(equivalent characterizations)

Given any pair of adjoint functors

$$(L \dashv R) \;:\; B \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} A$$

the following are equivalent:

1. The right adjoint $R$ is fully faithful. (In this case $B$ is equivalent to its essential image in $A$ under $R$, a full reflective subcategory of $A$.)

2. The counit $\varepsilon : L R \to 1_B$ of the adjunction is a natural isomorphism of functors.

3. There exists some natural isomorphism $L R \to 1_B$.

4. The monad $(R L, R\varepsilon L,\eta)$ associated with the adjunction is idempotent, the right adjoint $R$ is conservative, and the left adjoint $L$ is essentially surjective on objects.

5. If $S$ is the set of morphisms $s$ in $A$ such that $L(s)$ is an isomorphism in $B$, then $L \colon A \to B$ realizes $B$ as the (nonstrict) localization of $A$ with respect to the class $S$.

6. The left adjoint $L$ is dense.

The equivalence of statements (1), (2), (4) and (5) are originally due to (Gabriel-Zisman 67, Prop. 1.3, page 7). The equivalence of (1) and (6) is due to (Ulmer, Theorem 1.13). The equivalence of (2) and (3) is (Johnstone, Lemma A1.1.1).

Proof

The equivalence of (1) and (2) is this proposition. The equivalence of (1) and (4) is this Prop.. For (5) see reflective localization. The equivalence of (1) and (6) can be seen by observing that $lan_L L \cong L lan_L id \cong L R$, which is pointwise, since $lan_L id$ is absolute, and is isomorphic to the identity if and only if $R$ is fully faithful.

To prove that (3) implies (2), the argument is to transfer the comonad structure on $L R$ across the isomorphism to a comonad structure on $1_B$, and observe that for any comonad structure on $1_B$ the counit is inverse to the comultiplication; thus the counit $\varepsilon$ of the original comonad structure on $L R$ must have been invertible. The same argument shows that for a comonad in any 2-category the counit $\varepsilon : L R \to 1_B$ is an isomorphism iff $L R$ is isomorphic to $1_B$.

This is a well-known set of equivalences concerning idempotent monads. The essential point is that a reflective subcategory $i \colon B \to A$ is monadic (Prop. ), i.e., realizes $B$ as the category of algebras for the monad $i r$ on $A$, where $r: A \to B$ is the reflector.

See also the related discussion at reflective sub-(infinity,1)-category.

Special cases

Exact reflective subcategories

If the reflector (which as a left adjoint always preserves all colimits) in addition preserves finite limits, then the embedding is called exact. If the categories are toposes then such embeddings are called geometric embeddings.

In particular, every sheaf topos is an exact reflective subcategory of a category of presheaves

$$Sh(C) \stackrel{\overset{sheafify}{\leftarrow}}{\hookrightarrow} PSh(C) \,.$$

The reflector in that case is the sheafification functor.

Theorem

If $C$ is a reflective subcategory of a cartesian closed category, then it is an exponential ideal if and only if its reflector $D\to C$ preserves finite products.

In particular, $C$ is then also cartesian closed.

This appears for instance as (Johnstone, A4.3.1). See also at reflective subuniverse.

So in particular if $C$ is an exact reflective subcategory of a cartesian closed category $D$, then $C$ is an exponential ideal of $D$.

See Day's reflection theorem for a more general statement and proof.

Complete reflective subcategories

When the unit of the reflector is a monomorphism, a reflective category is often thought of as a full subcategory of complete objects in some sense; the reflector takes each object in the ambient category to its completion. Such reflective subcategories are sometimes called mono-reflective. One similarly has epi-reflective (when the unit is an epimorphism) and bi-reflective (when the unit is both a monomorphism and an epimorphism).

In the last case, note that if the unit is an isomorphism, then the inclusion functor is an equivalence of categories, so nontrivial bireflective subcategories can occur only in non-balanced categories. Also note that ‘bireflective’ here does not mean reflective and coreflective. One sees this term often in discussions of concrete categories (such as topological categories) where really something stronger holds: that the reflector lies over the identity functor on Set. In this case, one can say that we have a subcategory that is reflective over $Set$.

Accessible reflective subcategories

Definition

A reflection

$$\mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} \mathcal{D}$$

is called accessible if $\mathcal{D}$ is an accessible category and the reflector $R\circ L \colon \mathcal{D} \to \mathcal{D}$ is an accessible functor.

Proposition

A reflective subcategory $\mathcal{C} \hookrightarrow \mathcal{D}$ of an accessible category is accessible, def. , precisely if $\mathcal{C}$ is an accessible category.

In this explicit form this appears as (Lurie, prop. 5.5.1.2). From (Adamek-Rosický) the “only if”-direction follows immediately from 2.53 there (saying that an accessibly embedded subcategory of an accessible category is accessible iff it is cone-reflective), while the “if”-direction follows immediately from 2.23 (saying any left or right adjoint between accessible categories is accessible).

Properties

General

A reflective subcategory is always closed under limits which exist in the ambient category (because the full inclusion is monadic, by Prop. , hence creates limits, as noted above), and inherits colimits from the larger category by application of the reflector Riehl, Prop 4.5.15. In particular, if the ambient category is complete and cocomplete then so is the reflective subcategory.

A morphism in a reflective subcategory is monic iff it is monic in the ambient category. A reflective subcategory of a well-powered category is well-powered.

As Eilenberg-Moore category of the idempotent monad

Proposition

Every reflective subcategory inclusion is a monadic functor, exhibiting the reflective subcategory as the Eilenberg-Moore category of modules for its induced idempotent monad. Conversely, the Eilenberg-Moore category of an idempotent monad is a reflective subcategory

A proof is spelled out for instance in Borceux 1994, vol 2, cor. 4.2.4. A formal proof in cubical Agda is given in 1Lab. See also Prop. and see at idempotent monad – Properties – Algebras for an idempotent monad and localization.

Reflective subcategories of locally presentable categories

Both the weak and strong versions of Vopěnka's principle are equivalent to fairly simple statements concerning reflective subcategories of locally presentable categories:

Theorem

The weak Vopěnka's principle is equivalent to the statement:

For $C$ a locally presentable category, every full subcategory $D \hookrightarrow C$ which is closed under limits is a reflective subcategory.

This is AdamekRosicky, theorem 6.28

Theorem

The strong Vopěnka's principle is equivalent to:

For $C$ a locally presentable category, every full subcategory $D \hookrightarrow C$ which is closed under limits is a reflective subcategory; further on, $D$ is then also locally presentable.

(Remark after corollary 6.24 in Adamek-Rosicky book).

Reflective subcategories of cartesian closed categories

In showing that a given category is cartesian closed, the following theorem is often useful (cf. A4.3.1 in the Elephant):

Theorem

If $C$ is cartesian closed, and $D\subseteq C$ is a reflective subcategory, then the reflector $L\colon C\to D$ preserves finite products if and only if $D$ is an exponential ideal (i.e. $Y\in D$ implies $Y^X\in D$ for any $X\in C$). In particular, if $L$ preserves finite products, then $D$ is cartesian closed.

Reflective and coreflective subcategories

Theorem

A subcategory of a category of presheaves $[A^{op}, Set]$ which is both reflective and coreflective is itself a category of presheaves $[B^{op}, Set]$, and the inclusion is induced by a functor $A \to B$.

This is shown in (BashirVelebil).

Property vs structure

Whenever $C$ is a full subcategory of $D$, we can say that objects of $C$ are objects of $D$ with some extra property. But if $C$ is reflective in $D$, then we can turn this around and (by thinking of the left adjoint as a forgetful functor) think of objects of $D$ as objects of $C$ with (if we're lucky) some extra structure or (in any case) some extra stuff.

This can always be made to work by brute force, but sometimes there is something insightful about it. For example, a metric space is a complete metric space equipped with a dense subset. Or, an integral domain is a field equipped with numerator and denominator functions.

Examples

Example

Complete metric spaces are mono-reflective in metric spaces; the reflector is called completion.

Example

The category of sheaves on a site $S$ is a reflective subcategory of the category of presheaves on $S$; the reflector is called sheafification. In fact, categories of sheaves are precisely those accessible reflective subcategories, def. , of presheaf categories for which the reflector is left exact. This makes the inclusion functor precisely a geometric inclusion of toposes.

Example

A category of concrete presheaves inside a category of presheaves on a concrete site is a reflective subcategory.

Example

In a recollement situation, we have several reflectors and coreflectors. We have a reflective and coreflective subcategory $i_*: A' \hookrightarrow A$ with reflector $i^*$ and coreflector $i^!$. The functor $j^*$ is both a reflector for the reflective subcategory $j_*: A'' \hookrightarrow A$, and a coreflector for the coreflective subcategory $j_!: A'' \hookrightarrow A$.

Example

Assuming classical logic, the category Set has exactly three reflective (and replete) subcategories: the full subcategory containing all singleton sets; the full subcategory containing all subsingletons; and $Set$ itself.

In constructive mathematics, there are potentially more reflective subcategories, for instance the subcategory of $j$-sheaves for any Lawvere-Tierney topology on $Set$.

Example

The category of affine schemes is a reflective subcategory of the category of schemes, with the reflector given by $X \mapsto Spec \Gamma(X,\mathcal{O}_X)$.

The generalization of this example to homotopy theory is discussed at function algebras on infinity-stacks. The analogue in noncommutative algebraic geometry is in (Rosenberg 98, prop 4.4.3).

(Counter)Example

The non-full inclusion of unital rings into non-unital rings has a left adjoint (with monic units), whose reflector formally adjoins an identity element. However, we do not call it a reflective subcategory, because the “inclusion” is not full; see remark .

Remark

Notice that for $R \in Ring$ a ring with unit, its reflection $L R$ in the above example is not in general isomorphic to $R$, but is much larger. But an object in a reflective subcategory is necessarily isomorphic to its image under the reflector only if the reflective subcategory is full. While the inclusion $\mathbf{Ring} \hookrightarrow \mathbf{Ring}$‘ does have a left adjoint (as any forgetful functor between varieties of algebras, by the adjoint lifting theorem), this inclusion is not full (an arrow in $\mathbf{Ring}$’ need not preserve the identity).

Example

The subcategory

$$Cat \hookrightarrow sSet$$

of the category of categories into the category of simplical sets is a reflective subcategory Riehl, example 4.5.14 (vi). The reflection is given by the homotopy category functor. This implies that Cat is complete and cocomplete because it inherits all limits and colimits from sSet.

Example

For any Lawvere theory $T$, its category of models is the category

$$Prod(T, Set)$$

of product preserving functors into $Set$ and natural transformations between them. The inclusion

$$Prod(T, Set) \hookrightarrow [T, Set]$$

is a reflective subcategory Buckley, theorem 5.2.1. Therefore, because $[T,Set]$ is complete and cocomplete (limits and colimits are computed pointwise), so is $Prod(T, Set)$. This implies that many familar algebraic categories such as Grp, Mon, Ring, etc. are complete and cocomplete as a special case.

Related concepts

• localization, locally presentable category

• reflective localization

• Quillen reflection, left Bousfield localization

• reflective sub-(infinity,1)-category

• coreflective subcategory

• bireflective subcategory

• adjoint cylinder, describing the situation when the reflector has a further left adjoint

• sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes

References

• Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Springer 1967 (pdf)

• Friedrich Ulmer, Properties of Dense and Relative Adjoint Functors, Journal of Algebra 1968

• Saunders MacLane, §IV.3 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, p. 196 in Basic Category Theory, vol.1 of: Handbook of Categorical Algebra, Cambridge University Press (1994) [doi:10.1017/CBO9780511525858]

• Francis Borceux, Cor. 2.4.2 in: Categories and Structures, Handbook of Categorical Algebra, Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]

• Jiri Adamek, Jiří Rosický, Locally presentable and accessible categories London Mathematical Society Lecture Note Series 189

• Springer eom: reflective subcategory

• C. Cassidy, M. Hébert, G. M. Kelly, Reflective subcategories, localizations and factorizationa systems, J. Austral. Math. Soc. 38:3 (1985) 287–329 doi

• cf. the notion of $Q^\circ$-category in the entry Q-category

• Emily Riehl, Category Theory in Context, p. 142, Courier Dover Publications 2017 (pdf)

• Mitchell Buckley, Lawvere Theories, 2008 pdf

The relation of exponential ideals to reflective subcategories is discussed in section A4.3.1 of

• Peter Johnstone, Sketches of an Elephant

Reflective and coreflective subcategories of presheaf categories are discussed in

• R. Bashir, J. Velebil, Simultaneously reflective and coreflective subcategories of presheaves, Theory and Applications of Categories, Vol 10. No. 16. (2002) (pdf).

Related discussion of reflective sub-(∞,1)-categories is in

• Jacob Lurie, Higher Topos Theory

The example of affine schemes in noncommutative algebraic geometry is in

• Alexander Rosenberg, Noncommutative schemes, Comp. Math. 112, 93–125 (1998)

Formalization in cubical Agda:

• 1lab, Reflective Subcategories

See also:

• Michael Barr, John Kennison, Robert Raphael, On reflective and coreflective hulls, Cahiers Topologie Géométrie Différentielle Catégorique 56 (2015) 162–208 [pdf, pdf]