nLab reflective subcategory



Category theory

Notions of subcategory

Modalities, Closure and Reflection



A reflective subcategory is a full subcategory

CD C \hookrightarrow D

such that objects dd and morphisms f:ddf \colon d \to d' in DD have “reflections” TdT d and Tf:TdTdT f \colon T d \to T d' in CC. Every object in DD looks at its own reflection via a morphism dTdd \to T d and the reflection of an object cCc \in C is equipped with an isomorphism TccT c \cong c.

A canonical example is the inclusion

AbGrp \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 CDC \hookrightarrow D creates all limits of DD and CC has all colimits which DD admits.



A full subcategory i:CDi : C \hookrightarrow D is reflective if the inclusion functor ii has a left adjoint TT:

(Ti):CTD. (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.


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.


The components of the unit

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

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

η d:dTd. \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 TT is left exact.


If the reflector TT is faithful, the reflection is called a completion.



(equivalent characterizations)

Given any pair of adjoint functors

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

the following are equivalent:

  1. The right adjoint RR is fully faithful. (In this case BB is equivalent to its essential image in AA under RR, a full reflective subcategory of AA.)

  2. The counit ε:LR1 B\varepsilon : L R \to 1_B of the adjunction is a natural isomorphism of functors.

  3. There exists some natural isomorphism LR1 BL R \to 1_B.

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

  5. If SS is the set of morphisms ss in AA such that L(s)L(s) is an isomorphism in BB, then L:ABL \colon A \to B realizes BB as the (nonstrict) localization of AA with respect to the class SS.

  6. The left adjoint LL 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).


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 LLLlan LidLRlan_L L \cong L lan_L id \cong L R, which is pointwise, since lan Lidlan_L id is absolute, and is isomorphic to the identity if and only if RR is fully faithful.

To prove that (3) implies (2), the argument is to transfer the comonad structure on LRL R across the isomorphism to a comonad structure on 1 B1_B, and observe that for any comonad structure on 1 B1_B the counit is inverse to the comultiplication; thus the counit ε\varepsilon of the original comonad structure on LRL R must have been invertible. The same argument shows that for a comonad in any 2-category the counit ε:LR1 B\varepsilon : L R \to 1_B is an isomorphism iff LRL R is isomorphic to 1 B1_B.

This is a well-known set of equivalences concerning idempotent monads. The essential point is that a reflective subcategory i:BAi \colon B \to A is monadic (Prop. ), i.e., realizes BB as the category of algebras for the monad iri r on AA, where r:ABr: 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)sheafifyPSh(C). Sh(C) \stackrel{\overset{sheafify}{\leftarrow}}{\hookrightarrow} PSh(C) \,.

The reflector in that case is the sheafification functor.


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

In particular, CC is then also cartesian closed.

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

So in particular if CC is an exact reflective subcategory of a cartesian closed category DD, then CC is an exponential ideal of DD.

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 SetSet.

Accessible reflective subcategories


A reflection

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

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


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

In this explicit form this appears as (Lurie, prop. 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).



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


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:


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

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

This is AdamekRosicky, theorem 6.28


The strong Vopěnka's principle is equivalent to:

For CC a locally presentable category, every full subcategory DCD \hookrightarrow C which is closed under limits is a reflective subcategory; further on, DD 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):


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

Reflective and coreflective subcategories


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

This is shown in (BashirVelebil).

Property vs structure

Whenever CC is a full subcategory of DD, we can say that objects of CC are objects of DD with some extra property. But if CC is reflective in DD, then we can turn this around and (by thinking of the left adjoint as a forgetful functor) think of objects of DD as objects of CC 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.



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


The category of sheaves on a site SS is a reflective subcategory of the category of presheaves on SS; 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.


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


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


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 SetSet itself.

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


The category of affine schemes is a reflective subcategory of the category of schemes, with the reflector given by XSpecΓ(X,𝒪 X)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).


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 .


Notice that for RRingR \in Ring a ring with unit, its reflection LRL R in the above example is not in general isomorphic to RR, 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 RingRing\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 Ring\mathbf{Ring}’ need not preserve the identity).


The subcategory

CatsSet 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.


For any Lawvere theory TT, its category of models is the category

Prod(T,Set)Prod(T, Set)

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

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

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


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

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

The example of affine schemes in noncommutative algebraic geometry is in

Formalization in cubical Agda:

See also:

Last revised on March 31, 2024 at 09:58:03. See the history of this page for a list of all contributions to it.