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Classifying space of subcategories and its application

For a collection of subcategories satisfying a fixed set of conditions, for example thick subcategories of a triangulated category, we define a topological space called classifying space of subcategories. We show that this space classifies various prime subcategories in the sense that they bijectively correspond to the closed subsets of the classifying space. Many well-known results of subcategory classification fit into this framework. An example which cannot be classified by a topological space in the above sense is also given.

In our paper, we study this phenomenon in a more general context, in which a collection Φ of subcategories we are interested in forming a complete lattice is considered. Then a space K(Φ) is defined, consisting of points P C such that C ∈ Φ cannot be generated as proper subcategories of the same type (see Definition 3.1 and Definition 2.4). This space K(Φ) however is not in general a topological space in a natural way unless we require the lattice Φ also be distributive. This problem is resolved by restricting our attention into a subset of (generally) prime elements (that is, whenever C ≤ i∈I C i , C ≤ C i for some i, where I is finite or arbitrary accordingly), denoted by K p (Φ) (or K gp (Φ)). Therefore, we obtain our main result as Theorem 4.7, after a brief discussion on connection among those classifying spaces. Interestingly, thanks to lattice duality, there are many similar notions to (generally) prime objects, and if we ignore the condition of point representation, a family of parallel results to Theorem 4.7 such as Theorem 5.4 and Corollary 5.5 are obtained, which can be considered as various criterions of subcategories that are classified by a topological space.
Theorem 2. Let Φ be a complete lattice of subcategories of certain type in a category A. Then the subcategories generated by join primes are classified by the closed subsets of K ∨ p (Φ). The dual statements and their complete versions for the meet primes and irreducibles also hold accordingly.
This provides a framework for many famous results including Kanda's classification of Serre subcategories in [9], Neeman's classification [12] of thick subcategories in D perf (D) and also the classification of localizing subcategories in an noetherian stable homotopy category [7], by showing that each subcategory is g-primely generated, respectively. And we show for the last example as a demonstration in Section 6.
We point out that also as another comparison, in P. Johnstone's book [8] on Stone spaces, he defined a similar notion using lattices, which was used to develop the theory of classifying subcategories in a derived category by J. Kock and W. Pitsch recently in [10].
As other classifications of various subcategories establish a correspondence from either closed or open subsets of a topological space to the target subcategories, we mention in Section 7 an exceptional example of nullity class in an abelian category, which cannot be classified by a topological space (see Definition 2.5) but corresponds to the closed subsets with one extra condition, namely extension-closed, see Theorem 6.8 in [11].

Subcategories and lattices
There is a natural way to equip a collection of subcategories with a structure of complete (distributive) lattice. For details in lattice theory, see [4] and [8] for example. We introduce in this section the concept that subcategories are classified by a topological space in general.
is a partially ordered set with two commutative binary operations defined via the partial order ≤, called join ∨ and meet ∧ (or supremum and infimum, resp.), such that any finte subset of Φ has both join and meet. In particular, the join of the empty set denoted by 0 is called the bottom element, and the meet of the empty set denoted by 1 is called the top element.
We call Φ a distributive lattice if additionally one of the following properties holds for any elements a, b, c in Φ: In fact, that one holds implies the other, see Lemma 1.5 in [8]. A lattice Φ is complete if it has either arbitrary joins or arbitrary meets for any subset of Φ.
There are two typical examples of lattices coming from either a category or a topological space.
Example 2.2. (1) Assume A is an abelian category such that the collection Φ tor of its torsion classes forms a set. Equipped with the inclusion as partial order, the intersection as meet, and generating a torsion class as join, then Φ tor becomes a complete lattice. However, it is not always distributive as we will see in Section 7.
(2) Let X be a topological space and Φ X the set of closed subsets of X. Then Φ X is a complete distributive lattice with the inclusion as partial order, the intersection as meet and the closure of their union as join. Definition 2.3. Let Φ 1 , Φ 2 be partially ordered sets. A homomorphism of partially ordered sets is a map f : Φ 1 → Φ 2 that preserves the partial orders. Such an f becomes isomorphism if it has an inverse homomorphism. For lattices, we require additionally that the map respects both joins and meets, and we define isomorphism of lattices in a similar manner.
It is easy to show that any isomorphism of partially ordered sets is automatically an isomorphism of lattices once the partially ordered sets are also lattices.
The following notion in Definition 2.4 is introduced for our convenience, so that we use it to represent any lattice of subcategories in a given category. For instance, in an abelian category the lattice of Serre subcategories, and in a triangulated category the lattice of localizing subcategories and so on.
Definition 2.4. Let A be a category. The collection Φ A of subcategories of type A (or roughly, subcategories of certain type) in A is the collection of subcategories in which every subcategory satisfies a fixed (finite) set of conditions. The next definition in fact describes a common phenomenon in classification of subcategories.
Definition 2.5. Let A be a category. The collection Φ of subcategories of certain type in A is classified by a topological space X if there is an isomorphism of partially ordered sets f : Φ X ∼ → Φ, where Φ X is the lattice of closed subsets of X.
For example, Gabriel's result [5] in 1962 says that the collection of Serre subcategories in R-mod is classified by the prime spectrum SpecR of the ring R with specialization closed topology. Neeman [12] showed in 1992 that the collection of localizing subcategories in D(R) is classified by SpecR with discrete topology. Another one given by Balmer in 2005 says that the collection of radical thick tensor ideals in a tensor triangulated category A is classified by the Hochster's dual (SpcA) h of Balmer's spectrum, consisting of the prime thick tensor ideals of A, see [1] and [6]. Also, Benson et al. proved [3] in 2011 that the collection of localizing tensor ideals in the homotopy category K(InjkG) is classified by the spectrum SpecH * (G, k) of the cohomology of a finite group G with discrete topology.
We will connect some classical results mentioned above to our frame work in Section 6 once the notion of prime subcategory in a classifying space of subcategories is defined. For convenience, the set theoretical issue is ignored and we consider lattices instead of general collections of subcategories.

Classifying space of subcategories
In this section, we construct for any lattice Φ a topological space K(Φ). The lattice Φ comes from either a category or a topological space as in Example 2.2. We use the capital letters such as C to denote an element in Φ.
Definition 3.1. Let Φ be a lattice. Assign to each element C ∈ Φ a pairing (C, C o ), called a point and denoted by P C , if C = C o , where C o is the supremum of all elements strictly below, namely Notice that the collection of points in Φ may be empty. The collection K(Φ) = {P C | C ∈ Φ} is called the classifying space associated to Φ.
According to the lattices, we have the following properties about the points P C , roughly claiming that every point has a representative. (1) Let Φ A be the lattice of subcategories of type A in a category. Then for each (2) Let Φ X be the lattice of closed subsets of a topological space X. Then for each Proof. We show (2), and (1) follows similarly. Suppose C − C ′ C C ′ ∈Φ C ′ = ∅ and x is an element in this difference. If {x} = C, then the closure has to be one of the C ′ s in Φ properly contained in C. Therefore, Clearly this collection contains the whole space K(Φ) by taking the top element. However, it does not contain the empty set, and we add it artificially. We thus expect the candidate {K(C)} C∈Φ to give a topology on K(Φ).
Recall that a topological space is T 0 (or Kolmogorov ) if for any two distinct points there is an open neighborhood containing one point but not the other. The T 0 -quotient KQ(X) = X/ ∼ of a space X is defined as a quotient space by the equivalence relation that Proof. By definition, i∈I K(C i ) = K( i∈I C i ) for an arbitrary index I and , contradicting the fact that P C is a point. Furthermore, let P C 1 , P C 2 ∈ K(Φ) be distinct points. The complement K(C 1 ) c would give an open neighborhood of P C 2 but not containing P C 1 . Otherwise, P C 2 ∈ K(C 1 ) implies C 2 C 1 , which allows us to choose Suppose Φ X is the lattice of closed sets of a space X. There is a natural map by assigning an equivalence class [x] to P {x} , which is well-defined by our assumption. Clear that φ is injective. Now for any Hence φ is surjective. Finally, it is straightforward to check that such φ is closed and continuous by observing that denotes a closed subset of KQ(X) and p : X → KQ(X) is the projection. It is not difficult to see that for a space X, the classifying space K(Φ X ) can also be defined via the lattice of open subsets of X accordingly, and they turn out to be homeomorphic. Remark 3.4. In K(Φ X ) for any space X, every point P C is represented by a single point x ∈ X by Lemma 3.2. It follows that {K(C)} C∈Φ X automatically gives a topology of closed subsets on K(Φ X ). In particular, each point P C is prime as we will see later. However, it is not true in general for the lattices of subcategories of certain type.
We show that the construction of classifying space K(Φ) respects isomorphisms.
Proof. Statement (2) follows from (1), and (1) • Examples of classifying space of subcategories We end this section by giving three examples of K(Φ) for lattice Φ of subcategories of certain type. First, let Φ r denote the lattice of replete subcategories of a category A, that is, each subcategory is closed under isomorphisms of objects. We assume the ambient category is essentially small for convenience, so that a lattice of subcategories makes sense.
Proposition 3.6. The classifying space K(Φ r ) is homeomorphic to the space of isomorphism classes of objects with discrete topology.
Proof. Take an element C ∈ Φ r and objects x, y ∈ C. We claim that either x ∼ = y or C = C o . In fact, suppose x ≇ y. Then there is a nontrivial decomposition C = x r ∪ ( x r ) c , where x r is the smallest replete subcategory containing x with its complement ( x r ) c in C, and notice that that ( x r ) c is also a replete subcategory. Conversely, each isomorphism class of an objet x ∈ A gives a replete subcategory x r which contains no proper replete subcategories in x r . Hence it represents a point P x r in K(Φ r ). Therefore, the underlying set of K(Φ r ) corresponds to the set of isomorphism classes of objects. Since both x r and its complement x c r in A are replete, and also K( Furthermore, consider in a Krull-Schmidt category A (that is, every object can be written as a coproduct of finite many indecomposables), the lattice Φ of replete subcategories that are also closed under retracts and finite coproducts.
Proposition 3.7. The classifying space K(Φ ) has points corresponding to the isomorphism classes of indecomposable objects.
Proof. By Lemma 3.2, choose a representative x ∈ A for a point P C . If there is a decomposition say x = x 1 x 2 , then either x 1 = x or x 2 = x , where a represents the smallest subcategory in Φ containing a. In fact, if it is not the case, then x cannot represent a point in K(Φ ). Therefore, each representative x of P C can be chosen to be indecomposable. The conclusion then follows from the fact that x contains no proper subcategories of the same type, for any indecomposable x.
Lastly, consider the lattice Φ s of Serre subcategories in an abelian category A. An object x is monoform if it contains no subquotient of x as a subobject of x, other than the trivial cases. Denote by x s the smallest Serre subcategory containing x. See [9] for detail. Proof. First we show that the representative x of every point P C = P x s can be chosen to be monoform. In fact, suppose x is not monoform, then by Proposition 2.2 in [9] it has a monoform subobject x 1 such that either x s = x 1 s or x s = x/x 1 s . Since otherwise x s = x 1 s ∨ x/x 1 s implies that x s does not represent a point in K(Φ s ). If x s = x 1 s or x s = x/x 1 s such that x/x 1 is monoform, then we are done. If x s = x/x 1 s but x/x 1 is not monoform, then we can replace x by x/x 1 and repeat the procedure again, say a subobject x 2 of x with a monomorphism x 1 ֒→ x 2 . Thus we can obtain a chain of subobjects of x which has to stop after finitely many stages, since A is noetherian. Then x n = x n+1 = · · · and x s = x/x 1 s = · · · = x/x n s . Since the last term x/x n contains no proper subobject, it has to be monoform. The converse statement is not necessarily true because a monoform object may not represent a point. For example, in the abelian category A of finite dimensional representations of type A 2 : • over any field, there are three indecomposables which are also monoform, namely P 1 , P 2 and S 2 with a short exact sequence 0 → P 1 → P 2 → S 2 → 0. However, this gives a nontrivial decomposition of the Serre subcategory A = P 2 s = P 1 s ∨ S 2 s .

Generally prime objects and classification
By introducing the concept of (generally) prime elements of a lattice Φ of subcategories, we obtain a classification of these subcategories, requiring no distributivity of Φ. We also compare various classifying spaces of subcategories in this section. Definition 4.1. Let A be a category and Φ a lattice of subcategories in A of certain type. A subcategory C ∈ Φ or a point P C ∈ K(Φ) is called prime (generally prime or simply g-prime, resp.) if whenever C ≤ C 1 ∨ C 2 with C i ∈ Φ we have either C ≤ C 1 or C ≤ C 2 (C ≤ i∈I C i implies C ≤ C i for some i ∈ I with I an arbitrary index set, resp.).
In particular, if P C is represented by a single object x ∈ A, that the category C is prime is equivalent to that whenever x ∈ C 1 ∨ C 2 we have either x ∈ C 1 or x ∈ C 2 . We denote by K p (Φ) = {P C ∈ K(Φ) | C is prime} the set of prime points. Now as before choose a collection of subsets {K p (C)} C∈Φ in which Proposition 4.2. Let A be a category and Φ a lattice of subcategories in A of certain type. Then the collection {K p (C)} C∈Φ forms a topology of closed subsets of K p (Φ), which is T 0 .
Proof. Since ∅, A ∈ Φ, we have K p (∅) = ∅ and K p (A) = K p (Φ). It is clear that K p (C i ) = K p ( C i ) since one can take the prime points of both sides from the equality K(C i ) = K( C i ). We need to show that In fact, take the primes of both sides from K(C 1 ) ∨ K(C 2 ) ≤ K(C 1 ∨ C 2 ) to obtain one containment. For the converse, take P C ∈ K p (C 1 ∨ C 2 ). Then C ≤ C 1 ∨ C 2 so that C ≤ C 1 or C ≤ C 2 by primeness of C. The same argument of Proposition 3.3 applies to the second assertion. Similar to Lemma 3.5 we also have Lemma 4.3. Let f : Φ 1 → Φ 2 be an isomorphism of lattices. Then it induces a homeomorphism f p : We define K gp (Φ) similarly as the set of generally prime points and show without difficulty that it has a topology of closed subsets given by Example 4.5. The whole category is usually prime but not g-prime. Consider the derived category A = D b f g (Q) of bounded chain complexes over the rationals Q with finitely generated homologies, and Φ the lattice of nullity classes. Since the generators are of the form Σ i Q for some i ∈ Z and every nullity class becomes Σ i Q n ∩ A with Σ i Q n the nullity class generated in D(Q), the category A as a nullity class is prime because the nullity classes are truncated from below. However, it is not g-prime since we have with every Σ i Q n ∩ A properly contained in A. See [16] for detail.
Next we give some observations on their connection among various classifying spaces for the lattice of . Proposition 4.6. Let Φ be a lattice from either a topological space X or subcategories of certain type.
(1) If X has specialization-closed topology, then K gp (Φ X ) = K(Φ X ); (2) For any well-ordered index set I, if union i∈I C i of any chain Proof. (1) Suppose an arbitrary union of closed subsets of X is closed, and P C ∈ K(Φ X ). By Lemma 3.2, C = {x} for some x ∈ X. Thus if C = i∈I C i = i∈I C i with C i closed, we have x ∈ C i so that C ≤ C i for some i.
(2) Assume C ∈ Φ is prime and C ≤ i∈I C i with C i ∈ Φ. Then we form a chain C ′ i := j∈I,j≤i C i , so that i∈I C i = i∈I C ′ i = i∈I C ′ i by assumption. Suppose C has a representative x and let i ∈ I be the least element such that x ∈ C ′ i . Then we claim x ∈ C i . Indeed, if i has a successor, then For those i which has no successor, we have in which j∈I,j<i C ′ j ∈ Φ again by assumption. Hence the primeness of C implies either x ∈ j∈I,j<i C ′ j or x ∈ C i . However, that x ∈ j∈I,j<i C ′ j would imply x ∈ C ′ j for some j < i, contradicting the minimality of i. Therefore, x ∈ C i as well.
Let Φ be a complete lattice of certain subcategories of a category A. Notice that every generally prime subcategory necessarily represents a point in K(Φ) by definition, since otherwise C = C o = C ′ C,C ′ ∈Φ C ′ implies C ≤ C ′ C for some C ′ ∈ Φ by g-primeness, which is absurd. We call a subcategory of A is generated by (g-)primes or (g-)primely generated if it can be written as a join of subset of (g-)prime objects.
Theorem 4.7. Assume that Φ is a complete lattice of subcategories of certain type in A. Then there is a bijection Proof. Define θ as θ(K gp (C)) = P C ′ ∈Kgp(C) Notice that every g-prime C ∈ Φ represents a point in K(Φ). Therefore, θ(K gp (C)) = C = D.
Then by comparing the index sets, the fact C ′ ≤ C = D implies Hence θ is well-defined because D = D. For the inverse map, set ξ(C) = K gp (C). The order preserving property follows immediately from the definitions of θ and ξ.
Also for every K gp (C) with C ∈ Φ, where the last equality holds because C ≤ C implies K gp ( C) ⊆ K gp (C), while that ev- Theorem 4.7 can be restated as the following, in terms of Definition 2.5.
Corollary 4.8. For a complete lattice Φ of subcategories of a certain type in a category A, these subcategories are classified by K gp (Φ) if they are g-primely generated.

Irreducible vs. prime, and functoriality
In this section, we briefly discuss in a lattice the irreducible and the prime elements in various versions. In fact, the two notions coincide if the lattice is distributive shown in Lemma 5.2. Then we show that our construction K gp (Φ) of a complete lattice Φ is functorial in a non-obvious way, shedding light on the point-free construction of Johnstone [8] presented in Section II.1.3, where the point in Φ is described as a homomorphism p : Φ → {0, 1} of lattices with {0, 1} the lattice consisting of the bottom and the top elements.
By dualization of a lattice (Φ, ≤, ∨, ∧), we refer to the lattice (Φ, ≥, ∧, ∨) with the same underlying set but with reversed order and join, meet interchanged. In other words, we turn the graph of Φ upside down.
Definition 5.1. Let Φ be a (complete) lattice and C, C i ∈ Φ for i ∈ I. Then ( Their dual notions in Φ, (completely) meet irreducible and (completely) meet prime elements, are defined respectively.
The notion of completely join prime element is identified with that of generally prime element in Section 4. Also notice that completely join (meet resp.) irreducible implies finite join (meet resp.) irreducible, and similarly for primeness.
Lemma 5.2. Let Φ be a lattice and C ∈ Φ. Then (1) if C is join prime, then it is join irreducible; (2) if additionally Φ is distributive, then that C is join irreducible implies it is join prime. The statements of complete lattices with the dual notions involved are also true.
Proof. We only show for the finite versions and the complete cases are similar.
The problem of functoriality becomes clear if we use the point-free description, see Section II.1.3 in Johnstone [8]. Consider Notice that for this collection of elements in Φ, we do not require they represent points in the sense of Definition 3.1, although in some cases (such as generally prime), it is superfluous.
Again we need a formal empty set to be included in this collection to give a topology. For a homomorphism f : Φ → Ψ of complete lattices, define K ∧ p (f ) : where ↓ (C) denotes the principal ideal generated by C meaning that it contains all elements below C, for every meet prime C in Ψ.
Proposition 5.3. For a complete distributive lattice Φ, the collection {K ∧ p (C) | C ∈ Φ} forms a topology of closed subsets of K ∧ p (Φ), and K ∧ p is a contravariant functor.
Corollary 5.5. Let Φ be a complete lattice of subcategories of certain type in a category A. Then the subcategories generated by (complete resp.) join irreducibles are classified by the closed subsets of K ∨ i (Φ) (K ∨ ci (Φ) resp.). The dual statements for the meet primes and irreducibles also hold accordingly.

Comparing classical results
There are several examples that fit into the context of Theorem 4.7 in the sense that each subcategory is g-primely generated. The cases when we already know the generators would be easier to handle; for example, Serre subcategories in a noetherian abelian category, thick subcategories in D perf (R) and localizing subcategories in a derived category D(R) of a commutative noetherian ring R, or more generally, localizing subcategories in a stable homotopy category.
The proofs are in fact restated from the known results, but they are reorganized for our purpose to show that the subcategories are g-primely generated, so that Theorem 4.7 applies to give a classification by our classifying spaces. To avoid repetition we only include the last example as a demonstration. As usual, we use the brackets − to denote a subcategory of certain type generated by a collection of objects.
• Localizing subcategories in a stable homotopy category A stable homotopy category A is a triangulated category with a compatible closed symmetric monoidal structure, in which arbitrary coproducts exist, every cohomology functor is representable and there is a set of strongly dualizable objects generating A. See [7] for detail. The following lemma gives a description of localizing subcategories in D(R), which has a parallel result in a more general setting of noetherian stable homotopy category. Lemma 6.1. Let T ⊆ SpecR be an arbitrary subset. Then we can characterize the localizing subcategory generated by the residue field k(p) parametrized by T as where K T = p∈T c k(p) is a coproduct over the compliment T c .
Recall from Section 6 of [7] that a noetherian stable homotopy category is a monogenic stable homotopy category (i.e. the generating set consists of only the unit object which is also compact) with a single compact generator S such that the graded commutative ring π * (S) = [S, S] * is noetherian, denoted by R. The derived category of a commutative noetherian ring gives such an example. For any prime p of R, denote K(p) = S p ∧S/p, where S p is the localization at p and S/p is the Koszul complex associated to p. Lemma 6.2. Let A be a noetherian stable homotopy category such that every monochromatic category K(p) is minimal among nonzero localizing subcategories of A. Then every localizing subcategory has the form Proof. This is in fact Corollary 6.3.4 in [7]. Since K(p) ∧ K T = 0 for any p ∈ T c by Proposition 6.1.7 in [7], we have K(p) | p ∈ T ⊆ {X ∈ A | X ∧ K T = 0}. Conversely, suppose X ∈ A such that X ∧ K T = 0. Notice that X ∧ K T = 0 if and only if X ∧ K(p) = 0 for every p ∈ T c . It follows that X ∈ X ∧ K(p) | p ∈ SpecR by Proposition 6.3.2 in [7], so that X ∈ X ∧ K(p) | p ∈ T ⊆ K(p) | p ∈ T since every localizing subcategory is an ideal. Proposition 6.3. Let A be a noetherian stable homotopy category such that every monochromatic category K(p) is minimal among nonzero localizing subcategories of A. Then every localizing subcategory is g-primely generated.
Proof. Thanks to Lemma 6.2, we assume that K(q) | q ∈ T i for i ∈ I are localizing subcategories. Then K(p) ∈ i∈I K(q) | q ∈ T i implies K(p) ∈ i∈I K(q) | q ∈ T i = K(q) | q ∈ i∈I T i = {X ∈ A | X ∧ K i∈I T i = 0}. Hence K(p) ∧ K i∈I T i = 0, so that p ∈ i∈I T i and p ∈ T i for some i ∈ I. In particular, K(p) ∈ K(q) | q ∈ T i for some i.

An example
Nevertheless, we do have examples of subcategories that are not g-primely generated. Consider the quiver A 2 : 1 •←− 2 • and the category A of finite dimensional representations of A 2 over a field k. For convenience, we denote a = P 1 , b = P 2 and c = S 2 .
Let Φ be the lattice of nullity classes (subcategories that are closed under quotients and extensions) in A. We can specify Φ, K(Φ), K p (Φ), K gp (Φ) and their topology K p (C) explicitly thanks to the Auslander-Reiten quiver of A 2 Thus Φ = { 0 , a , c , b, c , A} and K(Φ) = {P 0 , P a , P c , P b,c }, so that K gp (Φ) = K p (Φ) = {P 0 , P a , P c }, with 5 closed subsets ∅, {P 0 }, {P 0 , P a }, {P 0 , P c } and K gp (Φ). This example demonstrates that nullity class cannot be classified by a topological space but only those primely generated ones. Notice that P b,c is not prime since b, c ≤ a ∨ c . However, nullity class is classified by closed subsets with extra conditions in a different topological space. For example, in the spectrum SpecA = {a, b, c} of premonoforms, they correspond to the closed and extension-closed subsets, see Theorem 6.8 in [11].
Finally, it is interesting to point out that the lattice Φ does not come from a topological space by computing its point space ptΦ (see the definition of point space in Chapter II.1 of [8]), on which there is no valid topology. As a matter of fact, by depicting the diagram of the lattice Φ A and a general theory of lattices (see e.g. Chapter 4 of [4]), we conclude that the lattice Φ is not even distributive since it contains a pentagon, see Theorem 4.10 (ii) in [4].