Theory and Applications of Categories, Vol. 14, No. 12, 2005, pp. 286–293.

GROUPOID ENRICHED CATEGORIES AND NATURAL SYSTEMS

TEIMURAZ PIRASHVILI

Abstract. We generalize the Baues-Jibladze descent theorem to a large class of groupoid enriched categories.

1. Introduction

A natural system D on a category C is a covariant functor on the category of factorizations FC of C (also called twisted arrow category of C, see [Mac Lane 1971]). Cohomology of C with coefficients in a natural system was first defined in [Baues & Wirsching 1985] and plays important rôle in many areas of algebra and topology [Baues 1989, Baues 2003, Jibladze & Pirashvili 1991, Jibladze & Pirashvili 2005, Pirashvili 1990]. One of the main points for the applications is the fact that the third cohomology group H^{3}(C; D) classifies the so called linear track extensions of C by D [Pirashvili 1988, Pirashvili 1990]. Recently in [Baues & Jibladze 2002] it was proved that linear track extensions are essentially the same as groupoid enriched categories such that automorphism groups of all 1-arrows are abelian (=abelian track categories, see below). The proof of this important result relies on the fact that in any abelian track category 𝒯 , automorphism groups Aut_{𝒯} (f) of 1arrows can be “descended” to the homotopy category 𝒯_{≃}, i. e. they only depend on the isomorphism class of f in a nice way – see Theorem 2.4 in [Baues & Jibladze 2002].

The aim of this work is to generalize this descent result for a large class of non-abelian natural systems equipped with certain type of descent data.

2. Preliminaries

A groupoid is called abelian if the automorphism group of each object is an abelian group. We will use the following notation for 2-categories. Composition of 1-arrows will be denoted by juxtaposition; for 2-arrows we will use additive notation, so composition is + and identity 2-arrows are denoted by 0. The hom-category for objects A, B of a 2-category will be denoted by ⟦A, B⟧.

There are several categories associated with a 2-category 𝒯 . The category 𝒯_{0} has the same objects as 𝒯 , while morphisms in 𝒯_{0} are 1-arrows of 𝒯 . The category 𝒯_{1} has the same objects as 𝒯_{0}. The morphisms A → B in 𝒯_{1} are 2-arrows α : f ⇒ f_{1} where f, f_{1} : A → B are 1-arrows in 𝒯 . The composition in 𝒯_{1} is given by (β : x ⇒ x_{1})(α : f ⇒ f_{1}) := (βα : xf ⇒ x_{1}f_{1}), where

Received by the editors 2005-05-12 and, in revised form, 2005-09-03.

Transmitted by Jean-Louis Loday. Published on 2005-09-08.

2000 Mathematics Subject Classification: 18D05.

Key words and phrases: Track category, linear track extension, natural system.

\begin{matrix}c \\ ◯ \end{matrix} Teimuraz Pirashvili, 2005. Permission to copy for private use granted.

βα = βf_{1} + xα = x_{1}α + βf.

One furthermore has the source and target functors

\xymatrix{𝒯_{1} \ar[r]^{s} \ar[r]_{t} & 𝒯_{0}} ,

where s(α : f ⇒ f_{1}) = f and t(α : f ⇒ f_{1}) = f_{1}, the “identity” functor i : 𝒯_{0} → 𝒯_{1} assigning to an 1-arrow f the triple 0_{f} : f ⇒ f. Moreover, consider the pullback diagram

(∗)  \xymatrix{𝒯_{1} ×_{𝒯_{0}} 𝒯_{1} \ar[r]^{p_{2}} \ar[d]_{p_{1}} & 𝒯_{1} \ar[d]^{t} \\ 𝒯_{1} \ar[r]^{s} & 𝒯_{0}};

there is also the “composition” functor m : 𝒯_{1} ×_{𝒯_{0}} 𝒯_{1} → 𝒯_{1} sending (α : f ⇒ f_{1}, α′ : f_{2} ⇒ f) to α + α′ : f_{2} ⇒ f_{1}. Note that these functors satisfy the identities sp_{1} = tp_{2}, sm = sp_{2}, tm = tp_{1} and si = ti = id_{𝒯_{0}} . Sometimes we will also simply write 𝒯_{1} ⇉ 𝒯_{0} to indicate a 2-category 𝒯 .

A track category 𝒯 is a category enriched in groupoids, i. e. is the same as a 2category all of whose 2-arrows are invertible. If the groupoids ⟦A, B⟧ are abelian for all A, B ∈ Ob𝒯 , then 𝒯 is called an abelian track category. For track categories we might occasionally talk about maps instead of 1-arrows and homotopies or tracks instead of 2-arrows. If there is a homotopy α : f ⇒ g between maps f, g ∈ Ob(⟦A, B⟧), we will say that f and g are homotopic and write f ≃ g. Since the homotopy relation is a natural equivalence relation on morphisms of 𝒯_{0}, it determines the homotopy category 𝒯_{≃} = 𝒯_{0}/ ≃. Objects of 𝒯_{≃} are once again objects in Ob(𝒯 ), while morphisms of 𝒯_{≃} are homotopy classes of morphisms in 𝒯_{0}. For objects A and B we let [A, B] denote the set of morphisms from A to B in the category 𝒯_{≃}. Thus

[A, B] = ⟦A, B⟧ / ≃ .

Usually we let q : 𝒯_{0} → 𝒯_{≃} denote the quotient functor. Sometimes for a 1-arrow f in 𝒯 we will denote q(f) by [f]. A map f : A → B is a homotopy equivalence if there exists a map g : B → A and tracks fg ≃ 1 and gf ≃ 1. This is the case if and only if q(f) is an isomorphism in the homotopy category 𝒯_{≃}. In this case A and B are called homotopy equivalent objects.

A track functor F : 𝒯 → 𝒯 ′ between track categories is by definition a groupoid enriched functor. Let C be a category. Then the category FC of factorizations in C is defined as follows. Objects of FC are morphisms f : A → B in C and morphisms (a, b) : f → g in FC are commutative diagrams

\xymatrix{A \ar[d]^{f} & A′ \ar[d]^{g} \ar[l]_{a} \\ B \ar[r]_{b} & B′}

in the category C. A natural system on C with values in a category 𝒞 is a covariant functor D : FC → 𝒞. We write D(f) = D_{f}. If a : C → D, f : A → C and g : D → B are morphisms in C, then the morphism D_{f} → D_{af} induced by the morphism (1_{A}, a) : f → af in FC will be denoted by a_{∗}, while the morphism D_{g} → D_{ga} induced by (a, 1_{B}) : g → ga will be denoted by a^{∗}.

A morphism of natural systems is just a natural transformation. For a functor q : C′ → C, any natural system D on C gives a natural system D ∘ (Fq) on C′ which we will denote q^{∗}(D).

For us a crucial observation is that any 2-category gives rise to a natural system. Indeed let ℬ be a 2-category. There is a natural system End_{ℬ} of monoids on ℬ_{0} (i. e. a functor Fℬ_{0} → Monoids) which assigns to an 1-arrow f : A → B the monoid of all 2-arrows f ⇒ f in ℬ. Indeed for g : B → B′, h : A′ → A morphisms in ℬ_{0} we already defined the induced homomorphisms:

(ε ↦ g_{∗}ε = gε) : Hom_{ℬ} (f, f) → Hom_{ℬ} (gf, gf),
(ε ↦ h^{∗}ε = εh) : Hom_{ℬ} (f, f) → Hom_{ℬ} (fh, fh).

For a track category 𝒯 , clearly End_{𝒯} = Aut_{𝒯} takes values in the category of groups.

3. 𝒯 -natural systems

To state our main result we need to introduce some more notions.

3.1. Definition. Consider a track category 𝒯 . A 𝒯 -natural system with values in a category 𝒞 is a natural system D : F𝒯_{0} → 𝒞 on 𝒯_{0} together with a family of morphisms

∇_{ξ} : D_{f} → D_{g}

in the category 𝒞, one for each track ξ : f ⇒ g in 𝒯 , such that the following conditions are satisfied:

i) ∇_{0_{f}} = id_{D_{f}} for all 1-arrows f in 𝒯 .

ii) For ξ : f ⇒ g, η : g ⇒ h one has ∇_{η+ξ} = ∇_{η} ∘ ∇_{ξ}.

iii) For a diagram

[figure: •; •; f; •; g_{1}; g; ξ; •; h]

the following diagram

\xymatrix{D_{fg} \ar[d]_{∇_{fξ}} & D_{g} \ar[l]_{f_{∗}} \ar[d]_{∇_{ξ}} \ar[r]^{h^{∗}} & D_{gh} \ar[d]_{∇_{ξh}} \\ D_{fg_{1}} & D_{g_{1}} \ar[l]_{f_{∗}} \ar[r]^{h^{∗}} & D_{g_{1}h}}

commutes.

iv) For a diagram

[figure: •; •; f_{1}; f; ξ; •; g; •; h_{1}; h; η]

the diagram

\xymatrix{D_{fg} \ar[dd]_{∇_{ξg}} &  & D_{gh} \ar[dd]^{∇_{gη}} \\  & D_{g} \ar[ul]_{f_{∗}} \ar[dl]^{_{f_{1}_{∗}}} \ar[ur]^{h^{∗}} \ar[dr]_{_{h_{1}_{∗}}} \\ D_{f_{1}g} &  & D_{gh_{1}}}

commutes.

A morphism Φ : (D, ∇) → (D′, ∇′) of 𝒯 -natural systems is a natural transformation Φ between the functors D, D′ : F𝒯_{0} → 𝒞, such that the diagram

\xymatrix{D ∘ Fs \ar[r]^{ΦFs} \ar[d]_{∇} & D′ ∘ Fs \ar[d]^{∇′} \\ D ∘ Ft \ar[r]^{ΦFt} & D′ ∘ Ft}

commutes. We denote by 𝒯 -Nat the category of 𝒯 -natural systems.

Let G : 𝒯 ′ → 𝒯 be a track functor. For any 𝒯 -natural system (D, ∇) one defines a 𝒯 ′-natural system G^{∗}(D, ∇) = (D ∘ FG, ∇G), where for ξ′ : f′ ⇒ g′ in 𝒯 ′, (∇G)_{ξ′} : D_{Gf′} → D_{Gg′} is defined to be ∇_{Gξ′}. In this way one obtains a functor

G^{∗} : 𝒯 -Nat → 𝒯 ′-Nat.

3.2. Example. For a track category 𝒯 , the group-valued natural system Aut_{𝒯} is equipped with a canonical structure of a 𝒯 -natural system given by

∇_{ξ}(a) = ξ + a − ξ.

Let D be a natural system on 𝒯_{≃}. Then q^{∗}D is a natural system on 𝒯_{0} given by (q^{∗}D)_{f} = D_{q(f)}. Here q : 𝒯_{0} → 𝒯_{≃} is the canonical projection. Define the structure of a 𝒯 -natural system on q^{∗}D by ∇ = id : D ∘ Fq ∘ Fs = D ∘ Fq ∘ Ft. In this way one obtains the functor q^{∗} : Nat(𝒯_{≃}) → 𝒯 -Nat. Our Theorem 4.1 claims that the functor q^{∗} is a full embedding. Actually we also identify the essential image of the functor q^{∗}. We need the following definition.

3.3. Definition. A 𝒯 -natural system (D, ∇) is called inert if ∇_{ε} = id_{f} for all ε : f ⇒ f.

Inert 𝒯 -natural systems form a full subcategory of the category of 𝒯 -natural systems, which is denoted by 𝒯 -Inert. It is clear that the image of the functor q^{∗} lies in 𝒯 -Inert. It is also clear that Aut_{𝒯} equipped with the canonical 𝒯 -natural system structure defined in Example 3.2 is inert if and only if 𝒯 is an abelian track category.

Let us observe that for any track functor G : 𝒯 ′ → 𝒯 restriction of the functor G^{∗} : 𝒯 -Nat → 𝒯 ′-Nat yields the functor G^{∗} : 𝒯 -Inert → 𝒯 ′-Inert.

4. The main result

4.1. Theorem. Let 𝒯 be a track category. Then q^{∗} : Nat(𝒯_{≃}) → 𝒯 -Inert is an equivalence of categories. Furthermore, for any track functor G : 𝒯 ′ → 𝒯 the diagram

\xymatrix{Nat(𝒯_{≃}) \ar[r]^{q^{∗}} \ar[d]^{G_{≃}^{∗}} & 𝒯 -Inert \ar[d]^{G^{∗}} \\ Nat(𝒯′_{≃}) \ar[r]^{_{q′_{∗}}} & 𝒯 ′-Inert}

commutes.

Proof. Let E and E′ be natural systems on 𝒯_{≃} and let Φ : q^{∗}E → q^{∗}E′ be a morphism of 𝒯 -natural systems. We claim that if f and g are homotopic maps in 𝒯_{0} (and therefore qf = qg), then the homomorphisms Φ_{f} : E_{qf} → E′_{qf} and Φ_{g} : E_{qg} → E′_{qg} are the same. Indeed, we can choose a track ξ : f ⇒ g. Then we have the following commutative diagram:

\xymatrix{(q^{∗}E)_{f} \ar[r]^{∇_{ξ}} \ar[d]_{Φ_{f}} & (q^{∗}E)_{g} \ar[d]^{Φ_{g}} \\ (q^{∗}E′)_{f} \ar[r]_{∇′_{ξ}} & (q^{∗}E′)_{g}}

By definition of the 𝒯 -natural system structure on q^{∗}E and q^{∗}E′ the morphisms ∇_{ξ} and ∇′_{ξ} are the identity morphisms, hence the claim. This shows that the functor q^{∗} is full and faithful.

It remains to show that for any inert 𝒯 -natural system (D, ∇) there exists a natural system E on 𝒯_{≃} and an isomorphism Δ : D → q^{∗}E of 𝒯 -natural systems. First of all one observes that if ξ, η : f ⇒ g are tracks, then ∇_{ξ} = ∇_{η} : D_{f} → D_{g}. Indeed, thanks to the property ii) of Definition 3.1 we have

∇_{ξ} = ∇_{ξ−η+η} = ∇_{ξ−η}∇_{η} = ∇_{η},

because ξ − η : g ⇒ g and D is inert. Therefore for qf = qg there is a well defined homomorphism ∇_{f,g} : D_{f} → D_{g} induced by any track f ⇒ g. Then the relation ii) of Definition 3.1 shows that ∇_{g,h}∇_{f,g} = ∇_{f,h} for any composable 1-arrows f, g, h. By harmless abuse of notation we will just write ∇ instead of ∇_{f,g} in what follows.

Since the functor q : 𝒯_{0} → 𝒯_{≃} is identity on objects and full, we can choose for any arrow a in 𝒯_{≃} a map u(a) in 𝒯_{0} such that qu(a) = a. Moreover for any map f in 𝒯_{0} we can choose a track δ(f) : f ⇒ u(qf). Now we put

E_{a} := D_{u(a)} and Δ_{f} := ∇ = ∇_{f,u(qf)} = ∇_{δ(f)} : D_{f} → D_{u(qf)} = E_{qf}.

For a diagram ⟵^{c}⟵^{a}⟵^{b} in the category 𝒯_{≃} we define the homomorphism c_{∗} : E_{a} → E_{ca} to be the following composite:

E_{a} = D_{u(a)} ⟶^{u(c)_{∗}} D_{u(c)u(a)} ⟶^{∇} D_{u(ca)} = E_{ca}.

Similarly we define the homomorphisms b^{∗} : E_{a} → E_{ab} to be the following composites:

E_{a} = D_{u(a)} ⟶^{u(b)^{∗}} D_{u(a)u(b)} ⟶^{∇} D_{u(ab)} = E_{ab}.

It follows from the property iii) of Definition 3.1 that for any diagram ⟵^{c_{1}}⟵^{c}⟵^{a} in the category 𝒯_{≃} we have the following commutative diagram:

\xymatrix{D_{u(a)} \ar[d]_{u(c)_{∗}} \ar[dr]^{c_{∗}} \\ D_{u(c)u(a)} \ar[d]_{u(c_{1})_{∗}} \ar[r]^{∇} & D_{u(ca)} \ar[d]_{u(c_{1})_{∗}} \ar[dr]^{_{c_{1}_{∗}}} \\ D_{u(c_{1})u(c)u(a)} \ar[r]_{∇} & D_{u(c_{1})u(ca)} \ar[r]_{∇} & D_{u(c_{1}ca)}}

Thus c_{1}(c_{∗}) = ∇(u(c_{1})(u(c)_{∗})). On the other hand by definition we have the commutative diagram:

\xymatrix{D_{u(a)} \ar[d]_{u(c_{1}c)_{∗}} \ar[dr]^{(c_{1}c)_{∗}} \\ D_{u(c_{1}c)u(a)} \ar[r]^{∇} & D_{u(c_{1}ca)}}

It follows from the property iv) of Definition 3.1 that one has also the following commutative diagram

\xymatrix{D_{u(a)} \ar[r]^{(u(c_{1})u(c))_{∗}} \ar[d]_{u(c_{1}c)_{∗}} & D_{u(c_{1})u(c)u(a)} \ar[d]_{∇} \ar[dl]^{∇} \\ D_{u(c_{1}c)u(a)} \ar[r]_{∇} & D_{u(c_{1}ca)}}

Therefore

(c_{1}c)_{∗} = ∇(u(c_{1}c)_{∗}) = ∇(∇(u(c_{1})(u(c)_{∗}))) = c_{1}(c_{∗}).

Similarly (b_{1}b)^{∗} = (b_{1}^{∗})b and E is a well-defined natural system on 𝒯_{≃}. It remains to show that Δ : D → q^{∗}E is a natural transformation of functors defined on F𝒯_{0}. To this end, one observes that for any composable morphisms g, f in the category 𝒯_{0} we have the following commutative diagram

\xymatrix{D_{f} \ar[d]^{g_{∗}} \ar[r]^{∇} & D_{u(qf)} \ar[d]^{g_{∗}} \ar[dr]^{u(qg)_{∗}} \\ D_{gf} \ar[r]^{∇} \ar[rrr]_{∇} & D_{gu(qf)} \ar[r]^{∇} & D_{u(qg)u(qf)} \ar[r]^{∇} & D_{uq(gf)}}

This means that the following diagram also commutes:

\xymatrix{D_{f} \ar[d]^{g_{∗}} \ar[r]^{Δ_{f}} & E_{qf} \ar[d]^{(qg)_{∗}} \\ D_{gf} \ar[r]_{Δ_{gf}} & E_{q(gf)}}

Similarly the diagram

\xymatrix{D_{g} \ar[d]^{f^{∗}} \ar[r]^{Δ_{g}} & E_{qg} \ar[d]^{(qf)^{∗}} \\ D_{gf} \ar[r]_{Δ_{gf}} & E_{q(gf)}}

also commutes and therefore Δ is indeed a natural transformation.

Now let 𝒯 be an abelian track category, so that Aut_{𝒯} is a natural system on 𝒯_{0} with values in the category of abelian groups. According to Example 3.2 it is equipped with the canonical structure of a 𝒯 -natural system, which is moreover inert, because 𝒯 is abelian. Thus one can use Theorem 4.1 to conclude that there is a natural system D defined on 𝒯_{≃} and an isomorphism of 𝒯 -natural systems τ : Aut_{𝒯} → q^{∗}D defined on 𝒯_{0}. This was the main result of [Baues & Jibladze 2002].

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A. Razmadze Mathematical Institute M. Alexidze st. 1 Tbilisi 0193 Georgia

Email: pira@rmi.acnet.ge

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