CHAPTER 5: TOPOLOGICAL FUNCTORS

1 Factorization structures

Definition (5.1.1). Let C be a category.

(i) If E is a class of C-morphisms which is closed under composition with isomorphisms and M is a conglomerate of sources in C which is closed under composition with isomorphisms, then the pair (E, M) is called a factorization structure on C provided that the following are satisfied:

(a) For each source (X, (f_{i})_{i∈I}) there exists e: X → Y in E and (Y, (m_{i})_{i∈I}) in M such that f_{i} = m_{i} ∘ e for each i ∈ I; briefly each source has (E, M)-factorization.

(b) For any two C-morphisms f and e and any two sources (Y, (m_{i})_{i∈I}) and (Z, (f_{i})_{i∈I}) in C such that e ∈ E (Y, (m_{i})_{i∈I}) ∈ M and f_{i} ∘ e = m_{i} ∘ f for each i ∈ I, there exists a unique C-morphism g: Z → Y such that the diagram

\xymatrix{X \ar[r]^{e} \ar[d]_{f} & Z \ar[d]^{f_{i}} \ar@{-->}[dl]_{g} \\ Y \ar[r]_{m_{i}} & X_{i}}

commutes for each i ∈ I; briefly: C satisfies the (E, M)-diagonalization property.

(ii) C is called (E, M)-category provided that (E, M) is a factorization structure on C.

Let C be a topological category or C = Haus. E consists of all extremal epimorphisms and M consists of all mono-sources in C. Then (E, M) is a factorization structure on C.

Theorem (5.1.4). Let C be a category and (E, M) a factorization structure on C. Then the following are satisfied:

(i) (E, M)-factorizations are uniquely determined (up to isomorphisms).

(ii) E ∩ M is the class of all C-isomorphisms.

(iii) Every extremal source in C belongs to M.

(iv) If f, g and h are C-morphisms such that h = g ∘ f, then the following are satisfied:

(a) If h ∈ E and f is a C-epimorphism, then g ∈ E.

(b) f ∈ E and g ∈ E imply h ∈ E, i.e. E is closed under composition.

(v) If (X, (f_{i})_{i∈I}) is a source in C and (X, (g_{j})_{j∈J}), (Z_{j}, (k_{j_{i}})_{i∈I_{j}})_{j∈J} is a factorization of (X, (f_{i})_{i∈I}), then the following hold:

(a) (X, (f_{i})_{i∈I}) ∈ M implies (X, (g_{j})_{j∈J}) ∈ M.

(b) (X, (g_{j})_{j∈J}) ∈ M and (Z_{j}, (k_{j_{i}})_{i∈I_{j}}) ∈ M for each j ∈ J imply (X, (f_{i})_{i∈I}) ∈ M.

(vi) If (X, (f_{i})_{i∈I}) is a source in C and there is some J ⊂ I such that (X, (f_{j})_{j∈J}) ∈ M, then (X, (f_{i})_{i∈I}) ∈ M.

(vii) E and M determine each other by the diagonalization property.

2 Definition and properties of topological functors

Definition (5.2.1). Let C be a category supplied with a factorization structure (E, M), let A be any category and let T: A → C be a functor.

(i) A source (A, (f_{i} : A → A_{i})_{i∈I}) in A is called T -initial provided that for each source (B, (g_{i} : B → A_{i})_{i∈I}) in A and each C-morphism f: T (A) → T (B) such that T (f_{i})∘f = T (g_{i}) for each i ∈ I, there exists a unique A-morphism \overline{f}: B → A with T (\overline{f}) = f and f_{i} ∘ \overline{f} = g_{i} for each i ∈ I.

(ii) A source (A, (f_{i} : A → A_{i})_{i∈I}) in A T -lifts a source (X, (g_{i} : X → T (A_{i}))_{i∈I}) in C provided that there exists an isomorphism h: X → T (A) in C with T (f_{i}) ∘ h = g_{i} for each i ∈ I.

(iii) T is called (E, M)-topological provided that for each family (A_{i})_{i∈I} of A-objects and each source (X, (m_{i} : X → T (A_{i}))_{i∈I}) in M, there exists a T -initial source (A, (f_{i} : A → A_{i})_{i∈I}) in A which T -lifts (X, (m_{i})_{i∈I}).

(iv) T is called absolutely topological provided that T is topological for any factorization structure (E, M) on C.

Theorem (5.2.4). Let C be a category supplied with a factorization structure (E, M) and let T: A → C be an (E, M)-topological functor. Then T is faithful.

Theorem (5.2.5). Let C be an (E, M)-category and T: A → C an (E, M)-topological functor. If E_{T} denotes the class of all morphisms f in A with T (f) ∈ E and M_{T} denotes the conglomerate of all T -initial sources (A, (f_{i})_{i∈I}) in A with (T (A), (T (f_{i}))_{i∈I}) ∈ M, then A is an (E_{T} , M_{T})-category.

Theorem (5.2.8). Let T: A → C be an (E, M)-topological functor, let D: I → A be a diagram and let (L, (l_{i} : L → D(i))_{i∈|I}|) be a source in A. Then the following are equivalent:

(i) (L, (l_{i})_{i∈|I|}) is a limit of D.

(ii) (L, (l_{i})_{i∈|I|}) is T -initial and (T (L), (T (l_{i}))_{i∈|I|}) is a limit of T ∘ D.

Corollary (5.2.9). If T: A → C is an (E, M)-topological functor and C is complete, then A is complete.

Proposition (5.2.10). Let T: A → C be an (E, M)-topological functor. Then for each family (A_{i})_{i∈I} of A-objects and each sink ((f_{i}: T (A_{i}) → X)_{i∈I}, X) in C, there exists a sink ((t_{i}: A_{i} → A)_{i∈I}, A) in A and a morphism e: X → T (A) in E such that T (t_{i}) = e ∘ f_{i} for each i ∈ I and such that the following condition is satisfied: (F) For each sink ((g_{i}: A_{i} → B), B) in A and each morphism g: X → T (B) with T (g_{i}) = g ∘ f_{i} for each i ∈ I, there exists a morphism k: A → B with T (k) ∘ e = g.

Theorem (5.2.11). Let T: A → C be an (E, M)-topological functor and let D: I → A be a diagram such that T ∘ D has a colimit. Then D has a colimit.

Corollary (5.2.12). Let T: A → C be an (E, M)-topological functor. If C is cocomplete, then A is cocomplete.

3 Initially structured categories

A functor F: A → B is called amnestic provided that any A-isomorphism f is an A-identity iff F (f) is a B-identity.

A functor F: A → B is called transportable provided that for each A-object A, each B-object B and each isomorphism q: B → F (A), there exists a unique A-object C and isomorphism \overline{q}: C → A with F (\overline{q}) = q.

An object X in a category C is called terminal provided thet for every object Y in C, the set of morphisms [Y, X]_{C} is a singleton.

Definition (5.3.1). A pair (A, T) is called an initially structured category provided that A is a category and T: A → Set is a functor which is amnestic and transportable such that the following hold:

(i) T is (epi, mono-source)-topological.

(ii) T has small fibres, i.e. for each X ∈ Set {A ∈ |A| : T (A) = X} is a set.

(iii) There is precisely one object P in A (up to isomorphism) such that T (P) is a terminal separator in Set, i.e. T (P) is a singleton.

Remark (5.3.3). Obviously the condition (i) can be replaced by: Every source (X, (f_{i}: X → T (A_{i}))_{i∈I}) is Set has an (epi, mono-source) factorization

\xymatrix{X \ar[rr]^{f_{i}} \ar[dr]_{e} &  & T (A_{i}) \\  & T (B) \ar@{-->}[ur]^{T (g_{i})}}

Theorem (5.3.4). Let (A, T) be an initially structured category. Then the following are satisfied:

(i) T is faithful.

(ii) A source (A, (f_{i}: A → D_{i})_{i∈I}) in A is a limit of a diagram D: I → A with |I| = I if and only if if this source is T -initial and (T (A), (T (f_{i}): T (A) → T (D_{i}))_{i∈I}) is limit of T ∘ D.

(iii) For any sink ((f_{i}: D_{i} → A)_{i∈I,A}) in A and an epimorphism e: X → T (A) with e∘f_{i} = T (a_{i}) for each i ∈ I such that the following condition (F) is satisfied: For each sink ((b_{i} : D_{i} → B)_{i∈I}, B) in A and each morphism d: X → T (B) there exists a (unique) morphism c: A → B such that T (c) ∘ e = d.

(iv) A is complete and cocomplete. Proposition (5.3.5). Let (A, T) be an initially structured category. Any sink ((f_{i}: A_{i} → C)_{i∈I}, C) in A has

(i) a factorization

\xymatrix{A_{i} \ar[rr]^{f_{i}} \ar[dr]_{a_{i}} &  & C \\  & A \ar[ur]^{c}}

such that T (c) is an isomorphism and ((a_{i}: A_{i} → A)_{i∈I}, A) is T -final (i.e. for any sink ((b_{i}: A_{i} → B)_{i∈I}, B) and any morphism f: T (A) → T (B) with f ∘ T (a_{i}) = T (b_{i}) for each i ∈ I, there exists a unique morphism \overline{f}: A → B with T (\overline{f}) = f and \overline{f} ∘ a_{i} = b_{i} for each i ∈ I), and

(ii) a factorization f_{i} = c ∘ a_{i} where c is a monomorphism and ((a_{i}: A_{i} → A)_{i∈I}, A) is a

T -final epi-sink. Proposition (5.3.6). Let (A, T) be an initially structured category. Then the following are satisfied:

(i) The object P in A given by IS_{3} (definition of initially structured category, (iii)) is terminal and a separator in A.

(ii) If X and Y are A-objects and g: T (X) → T (Y) is a constant morphism, then there

exists a unique A-m0orphism f: X → Y with T (f) = g. Definition (5.3.7). Let (A, T) be an initially structured category. An A-morphism f: A → B is called a

(i) T -embedding provided that (A, f: A → B) is a T -initial mono-source. (ii) T -quotient map provided that (A, f: A → B) is a T -final epi-sink. Proposition (5.3.8). Let (A, T) be an initially structured category. Then the following are satisfied:

(i) f is an A-monomorphism if and only if T (f) is a monomorphism in Set. (ii) f is an A-epimorphism whenever T (f) is an epimorphism in Set. (iii) Every extremal monomorphism in A is a T -embedding. Proposition (5.3.10). Let (A, T) be an initially structured category. Then the following are satisfied:

(i) If ((f_{i}: A_{i} → A)_{i∈I}, A) is a T -final epi-sink in A, then ((T (f_{i}): T (A_{i}) → T (A))_{i∈I}, T (A)) is an epi-sink in Set.

(ii) A sink in A is an extremal epi-sink if and only if it is a T -final epi-sink.

Proposition (5.3.11). For every A-morphism f: A → B in an initially structured category (A, T) the following are equivalent:

(i) f is a T -quotient map. (ii) f is a regular epimorphism. (iii) f is an extremal epimorphism. Proposition (5.3.13). Every initially structured category (A, T) is well-powered. Proposition (5.3.15). Let (A, T) be an initially structured category. Then A is a category with a factorization structure (E, M) where E consists of all extremal epimorphisms and M of all mono-sources. Theorem (5.3.16). Every E-reflective (full and isomorphism-closed) subcategory of an initially structured category (A, T) is initially structured provided that E consists of all Amorphisms f for which T (f) is an epimorphism in Set. Remark (5.3.17). Every extremal epireflective (resp. epireflective) full and isomorphismclosed subcategory of an initially structure (resp. topological) category is initially structured. Proposition (5.3.18). For each T -final epi-sink (f_{i}: A_{i} → A)_{i∈I} in an initially structured category, there is a set J ⊂ I such that (f_{j}: A_{j} → A)_{j∈J} is likewise a T -final epi-sink. Proposition (5.3.19). Let (A, T) and (A′, T′) be initially structured categories and let F: A → A′ be a functor preserving colimits. If (f_{i}: A_{i} → A) is a T -final epi-sink in A, then (T (f_{i}): T (A_{i}) → T (A)) is a T′-final epi-sink in A′. Theorem (5.3.20). Let (A, T) be an initially structured category. A is cartesian closed if and only if A × − preserves T -final epi-sinks for each A ∈ |A|. Theorem (5.3.22). Every extremal epireflective (full and isomorphism-closed) subcategory B of an initially structured cartesian closed category (A, T) is cartesian closed. Theorem (5.3.24). For a concrete category C (in the sense of 1.1.1) the following are equivalent:

(i) C is initially structured. (ii) C is an epireflective (full and isomorphism-closed) subcategory of a topological category. (iii) C is an extremal epireflective (full and isomorphism-closed) subcategory of a topological

category.