EQUALIZERS IN THE CATEGORY OF COCOMPLETE COCATEGORIES

BERNHARD KELLER AND OLEKSANDR MANZYUK

Abstract. We prove existence of equalizers in certain categories of cocomplete cocategories. This allows us to complete the proof of the fact that A_{∞}-functor categories arise as internal Hom-objects in the category of differential graded cocomplete augmented cocategories.

1. Introduction

We refer to [9] and to Lyubashenko-Ovsienko [18] for an introduction to A_{∞}-structures and their links to homological algebra.

The notion of an A_{∞}-category appeared in Fukaya’s work on Floer homology [3]. Its relation to mirror symmetry became apparent after Kontsevich’s talk at ICM ’94 [12]. Following Kontsevich, one should consider A_{∞}-categories as models for noncommutative varieties. This approach is being developed by Kontsevich and Soibelman in [15].

For a pair of A_{∞}-categories 𝒜 and ℬ, there is an A_{∞}-category A_{∞}(𝒜, ℬ) whose objects are A_{∞}-functors and whose morphisms are A_{∞}-transformations. These A_{∞}-functor categories have been considered for example by Kontsevich [13], Fukaya [4], Lefèvre-Hasegawa [16], Lyubashenko [17]. They provide models for the internal Hom-functor of the homotopy category of differential graded categories (Drinfeld [2], Toën [20], cf. [11] for a survey), where the internal Hom-functor is not a derived functor. Furthermore, as detailed in [10], A_{∞}-functor categories yield a natural construction of the B_{∞}-structure on the Hochschild complex of an associative algebra (Getzler-Jones [5], Kadeishvili [7], Voronov-Gerstenhaber [21]), which is important for proving Deligne’s conjecture and Tamarkin’s version of Kontsevich’s formality theorem (cf. for example Kontsevich-Soibelman [14], Tamarkin [19], Hinich [6]).

In order to interpret A_{∞}-functor categories as internal Hom-objects, one passes to a suitable category of cocategories following an idea of Lyubashenko [17]. For this suitable category, one can either take the monoidal subcategory generated by the images of graded quivers under the bar construction, as in [17], or the category of all (cocomplete augmented etc.) cocategories. The former approach is developed further in the forthcoming book by Bespalov, Lyubashenko, and Manzyuk [1] using the technique of closed multicategories. The latter approach has been taken by the first author in [10]. He proved in Theorem 5.3 of [10] that the monoidal category of cocomplete augmented cocategories was closed.

1991 Mathematics Subject Classification. 18A30, 18A35.

Key words and phrases. Cocategory, cocomplete cocategory, equalizer.

However, the proof of the theorem was incomplete: it relied on the assumption that the category of cocomplete augmented cocategories has equalizers. In this paper, we close this gap. Note that it suffices to prove existence of equalizers in the category of cocomplete cocategories since it is equivalent to the category of cocomplete augmented cocategories, see Remark 2.3.

Theorem. Suppose k is a field. Then the category of cocomplete k-cocategories admits equalizers. The analogous assertions hold in the graded and in the differential graded settings.

Since the proofs in the three cases are quite similar, we have chosen to present the proof for the ungraded version providing remarks concerning modifications necessary in the other cases. The proof occupies Section 4. It relies on a direct verification. An alternative proof is obtained if one interprets the category of cocomplete cocategories as the category of countable ind-objects of the category of conilpotent cocategories. By Corollary 6.1.17 of [8], it then suffices to prove that the category of conilpotent cocategories has equalizers, which can be checked directly.

Acknowledgment. The first author thanks V. Hinich and the second author for pointing out the gap which is at the origin of this note.

2. Preliminaries

Let k be a commutative ring. A k-quiver 𝒜 consists of a set of objects Ob 𝒜 and of k-modules 𝒜(X, Y), for each pair of objects X, Y ∈ Ob 𝒜. A morphism of k-quivers f : 𝒜 → ℬ consists of a map Ob f : Ob 𝒜 → Ob ℬ, X ↦ f(X), and of k-linear maps

f = f_{X,Y} : 𝒜(X, Y) → ℬ(f(X), f(Y)),

for each pair of objects X, Y ∈ Ob 𝒜. Let 𝒬 denote the category of k-quivers. For a set S, denote by 𝒬/S the subcategory of 𝒬 whose objects are k-quivers 𝒜 such that Ob 𝒜 = S, and whose morphisms are morphisms of k-quivers f: 𝒜 → ℬ such that Ob f = id_{S}. The category 𝒬/S is monoidal. The tensor product of quivers 𝒜 and ℬ is given by

(𝒜 ⊗ ℬ)(X, Z) = ⨁_{Y ∈S} 𝒜(X, Y) ⊗ ℬ(Y, Z), X, Z ∈ S.

The unit object is the discrete quiver kS given by Ob kS = S, kS(X, X) = k and kS(X, Y) = 0 if X ≠ Y, X, Y ∈ S. Recall that a cocategory (𝒞, Δ) is a coassociative coalgebra in the monoidal category 𝒬/ Ob 𝒞. Thus, a cocategory consists of a k-quiver 𝒞 and of a morphism Δ: 𝒞 → 𝒞 ⊗ 𝒞 in 𝒬/ Ob 𝒞, the comultiplication, satisfying the usual coassociativity condition. For X, Y, Z ∈ Ob 𝒞, denote by

Δ_{X,Y,Z} = pr_{X,Y,Z} ∘Δ : 𝒞(X, Z) → 𝒞(X, Y) ⊗ 𝒞(Y, Z)

the components of Δ. Since the k-linear map

(Δ_{X,Y,Z})_{Y ∈Ob 𝒞} : 𝒞(X, Z) → ∏_{Y ∈Ob 𝒞} 𝒞(X, Y) ⊗ 𝒞(Y, Z)

factors as

𝒞(X, Z) ⟶^{Δ} ⨁_{Y ∈Ob 𝒞} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ↪ ∏_{Y ∈Ob 𝒞} 𝒞(X, Y) ⊗ 𝒞(Y, Z),

it follows that, for each t ∈ 𝒞(X, Z), the element Δ_{X,Y,Z}(t) vanishes for all but finitely many Y ∈ Ob 𝒞. The coassociativity is expressed by the following equation:

[𝒞(W, Z) ⟶^{Δ_{W,X,Z}} 𝒞(W, X) ⊗ 𝒞(X, Z) ⟶^{1⊗Δ_{X,Y,Z}} 𝒞(W, X) ⊗ 𝒞(X, Y) ⊗ 𝒞(Y, Z)]

= [𝒞(W, Z) ⟶^{Δ_{W,Y,Z}} 𝒞(W, Y) ⊗ 𝒞(Y, Z) ⟶^{Δ_{W,X,Y} ⊗1} 𝒞(W, X) ⊗ 𝒞(X, Y) ⊗ 𝒞(Y, Z)].

A cocategory homomorphism f: (𝒞, Δ) → (𝒟, Δ) is a morphism of k-quivers f: 𝒞 → 𝒟 compatible with the comultiplication in the sense of the equation

[𝒞 ⟶^{f} 𝒟 ⟶^{Δ} 𝒟 ⊗ 𝒟] = [𝒞 ⟶^{Δ} 𝒞 ⊗ 𝒞 ⟶^{f⊗f} 𝒟 ⊗ 𝒟],

where the morphism f ⊗ f: 𝒞 ⊗ 𝒞 → 𝒟 ⊗ 𝒟 is given by Ob f ⊗ f = Ob f and

(f ⊗ f)_{X,Z} = [ ⨁_{Y ∈Ob 𝒞} 𝒞(X, Y) ⊗ 𝒞(Y, Z) \underrightarrow{⨁_{Y ∈Ob 𝒞} f_{X,Y} ⊗f_{Y,Z}}

⨁_{Y ∈Ob 𝒞} 𝒟(f(X), f(Y)) ⊗ 𝒟(f(Y), f(Z)) ↪ ⨁_{U∈Ob 𝒟} 𝒟(f(X), U) ⊗ 𝒟(U, f(Z))],

for each pair of objects X, Z ∈ Ob 𝒞. Explicitly, for X, Z ∈ Ob 𝒞, U ∈ Ob 𝒟, the following equation holds true:

[𝒞(X, Z) ⟶^{f_{X,Z}} 𝒟(f(X), f(Z)) ⟶^{Δ_{f(X),U,f(Z)}} 𝒟(f(X), U) ⊗ 𝒟(U, f(Z))]

= ∑_{f(Y)=U}^{Y ∈Ob 𝒞}[𝒞(X, Z) ⟶^{Δ_{X,Y,Z}} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{f_{X,Y} ⊗f_{Y,Z}} 𝒟(f(X), U) ⊗ 𝒟(U, f(Z))]. (1)

In particular, the right hand side vanishes if U is not in the image of f.

Let 𝒞 be a cocategory. Let Δ^{(n)} : 𝒞 → 𝒞^{⊗n} denote the comultiplication iterated n − 1 times, so that Δ^{(1)} = id_{𝒞}, Δ^{(2)} = Δ, Δ^{(3)} = (Δ ⊗ 1) ∘ Δ = (1 ⊗ Δ) ∘ Δ, and so on. Denote by

Δ_{X_{0},...,X_{n}}^{(n)} = pr_{X_{0},...,X_{n}} ∘Δ^{(n)} : 𝒞(X_{0}, X_{n}) → 𝒞(X_{0}, X_{1}) ⊗ 𝒞(X_{1}, X_{2}) ⊗ ⋯ ⊗ 𝒞(X_{n−1}, X_{n})

the components of Δ^{(n)}, for X_{0}, …, X_{n} ∈ Ob 𝒞. Suppose f: 𝒞 → 𝒟 is a cocategory homomorphism. By induction on n, it follows that

Δ_{f(X),U_{1},...,U_{n−1},f(Y)}^{(n)}∘f_{X,Y} = ∑_{f(Z_{i})=U_{i}, i=1,...,n−1}^{Z_{1},...,Z_{n−1}∈Ob 𝒞}(f_{X,Z_{1}}⊗f_{Z_{1},Z_{2}}⊗⋯⊗f_{Z_{n−1},Y})∘Δ_{X,Z_{1},...,Z_{n−1},Y}^{(n)} :

𝒞(X, Y) → 𝒟(f(X), U_{1}) ⊗ 𝒟(U_{1}, U_{2}) ⊗ ⋯ ⊗ 𝒟(U_{n−1}, f(Y)), (2)

for an arbitrary collection of objects X, Y ∈ Ob 𝒞, U_{1}, …, U_{n−1} ∈ Ob 𝒟.

A cocategory 𝒞 is cocomplete if, for each pair of objects X, Y ∈ Ob 𝒞,

𝒞(X, Y) = ⋃_{n⩾1}Ker(Δ^{(n)} : 𝒞(X, Y) → 𝒞^{⊗n}(X, Y)).

Equivalently, 𝒞 is cocomplete if for each t ∈ 𝒞(X, Y) there is n ⩾ 1 such that

Δ_{X,Z_{1},...,Z_{n−1},Y}^{(n)} (t) = 0,

for all Z_{1}, …, Z_{n−1} ∈ Ob 𝒞. 2.1. Example. An arbitrary k-quiver 𝒜 gives rise to a cocategory (T^{⩾1}𝒜, Δ), where T^{⩾1}𝒜 = ⨁_{n=1}^{∞} T^{n}𝒜, T^{n}𝒜 = 𝒜^{⊗n} is the n-fold tensor product in 𝒬/ Ob 𝒜, and Δ is the cut comultiplication. Thus,

T^{⩾1}𝒜(X, Y) = ⨁_{Z_{1},...,Z_{n−1}∈Ob 𝒜}^{n⩾1} 𝒜(X, Z_{1}) ⊗ 𝒜(Z_{1}, Z_{2}) ⊗ ⋯ ⊗ 𝒜(Z_{n−1}, Y),

for each pair of objects X, Y ∈ Ob 𝒜, and Δ is given by

Δ(f_{1} ⊗ f_{2} ⊗ ⋯ ⊗ f_{n}) = ∑_{i=1}^{n−1} f_{1} ⊗ ⋯ ⊗ f_{i} ⨂ f_{i+1} ⊗ ⋯ ⊗ f_{n}.

Since Δ(T^{n}𝒜) ⊂ ⨁_{p+q=n}^{p,q>0} T^{p}𝒜 ⊗ T^{q}𝒜, it follows that T^{⩾1}𝒜 is a cocomplete cocategory.

2.2. Remark. The correspondence 𝒜 ↦ T^{⩾1}𝒜 extends to a functor T^{⩾1} : 𝒬 → 𝒬. It is proven in [1, Chapter 8] that the functor T^{⩾1} admits the structure of a comonad, and that the category of T^{⩾1}-coalgebras is isomorphic to the category of cocomplete cocategories. 2.3. Remark. A cocategory 𝒞 is counital if it is equipped with a morphism ε : 𝒞 → k Ob 𝒞 in 𝒬/ Ob 𝒞 such that the two counit equations hold. Note that, for an arbitrary set S, the k-quiver kS admits the natural structure of a counital cocategory, namely the comultiplication is the canonical isomorphism kS ⟶^{∼} kS ⊗ kS in 𝒬/S and the counit is the identity map kS → kS. An augmented cocategory is a counital cocategory endowed with a morphism of counital cocategories η : k Ob 𝒞 → 𝒞 such that Ob η = id_{Ob 𝒞} and ε ∘ η = id_{k Ob 𝒞}. A morphism of augmented cocategories is a cocategory homomorphism compatible with the counit and the augmentation. The category of augmented cocategories is equivalent to the category of cocategories [1, Lemma 8.12]: given a cocategory (𝒜, Δ), there is the natural structure of an augmented cocategory on the k-quiver T^{⩽1}𝒜 = k Ob 𝒜⊕𝒜, where the counit and the augmentation are the projection ε = pr_{0} : T^{⩽1}𝒜 → k Ob 𝒜 and the inclusion η = in_{0} : k Ob 𝒜 → T^{⩽1}𝒜 respectively, and the comultiplication is given by the formulas

Δ|_{k Ob 𝒜} =[k Ob 𝒜 ⟶^{∼} k Ob 𝒜 ⊗ k Ob 𝒜 ⟶^{in_{0} ⊗ in_{0}} T^{⩽1}𝒜 ⊗ T^{⩽1}𝒜],
Δ|_{𝒜} =[𝒜 ⟶^{∼} 𝒜 ⊗ k Ob 𝒜 ⟶^{in_{1} ⊗ in_{0}} T^{⩽1}𝒜 ⊗ T^{⩽1}𝒜]
+[𝒜 ⟶^{Δ} 𝒜 ⊗ 𝒜 ⟶^{in_{1} ⊗ in_{1}} T^{⩽1}𝒜 ⊗ T^{⩽1}𝒜]
+[𝒜 ⟶^{∼} k Ob 𝒜 ⊗ 𝒜 ⟶^{in_{0} ⊗ in_{1}} T^{⩽1}𝒜 ⊗ T^{⩽1}𝒜].

Conversely, given an augmented cocategory (𝒞, Δ, ε, η), the reduced k-quiver \overline{𝒞} = Ker ε becomes a cocategory. The functors 𝒜 ↦ T^{⩽1}𝒜 and 𝒞 ↦ \overline{𝒞} are quasi-inverse equivalences. By definition, an augmented cocategory 𝒞 is cocomplete if its reduction \overline{𝒞} is cocomplete. Thus the category of cocomplete cocategories is equivalent to the category of cocomplete augmented cocategories.

The above definitions admit obvious graded and differential graded variants. For instance, a graded (resp. differential graded) k-quiver 𝒜 consists of a set of objects Ob 𝒜 and of graded k-modules (resp. cochain complexes of k-modules) 𝒜(X, Y), for each pair of objects X, Y ∈ Ob 𝒜. A morphism of graded (resp. differential graded) k-quivers f : 𝒜 → ℬ consists of a map Ob f : Ob 𝒜 → Ob ℬ, X ↦ f(X), and of morphisms of graded k-modules of degree 0 (resp. cochain maps)

f = f_{X,Y} : 𝒜(X, Y) → ℬ(f(X), f(Y)),

for each pair of objects X, Y ∈ Ob 𝒜. The tensor product of quivers with the same set of objects is defined analogously to the considered case, using tensor product of graded k-modules (resp. of cochain complexes). The definitions of cocategory, cocategory homomorphism etc. are modified accordingly.

3. Subcocategory generated by a set of objects

Let 𝒞 be a cocomplete cocategory, ℬ ⊂ 𝒞 a full subquiver, i.e., Ob ℬ ⊂ Ob 𝒞 and ℬ(X, Y) = 𝒞(X, Y), for each pair of objects X, Y ∈ Ob ℬ. Then, in general, ℬ is not a cocategory since the comultiplication

Δ : 𝒞(X, Z) → ⨁_{Y ∈Ob 𝒞} 𝒞(X, Y) ⊗ 𝒞(Y, Z)

does not take values in ⨁_{Y ∈Ob ℬ} 𝒞(X, Y) ⊗ 𝒞(Y, Z) if X, Z ∈ Ob ℬ. However, at least if k is a field, for each subset S ⊂ Ob 𝒞 there exists a maximal cocomplete subcocategory 𝒞_{S} ⊂ 𝒞 such that Ob 𝒞_{S} = S. It is constructed as follows. For a pair of objects X, Y ∈ S, denote

𝒩(X, Y) = ⨁_{∃i: Z_{i}∉S}^{Z_{1},...,Z_{n−1}∈Ob 𝒞, n⩾1} 𝒞(X, Z_{1}) ⊗ 𝒞(Z_{1}, Z_{2}) ⊗ ⋯ ⊗ 𝒞(Z_{n−1}, Y),

and define a k-linear map N = N_{X,Y} : 𝒞(X, Y) → 𝒩(X, Y) by

N = ∑_{∃i: Z_{i}∉S}^{Z_{1},...,Z_{n−1}∈Ob 𝒞, n⩾1} Δ_{X,Z_{1},...,Z_{n−1},Y}^{(n)} .

For each t ∈ 𝒞(X, Y), the sum in the right hand side is finite since 𝒞 is cocomplete. Let 𝒞_{S}(X, Y) = Ker N, so that we have an exact sequence of k-vector spaces:

0 → 𝒞_{S}(X, Y) ⟶^{ι} 𝒞(X, Y) ⟶^{N} 𝒩(X, Y).

Choose a splitting π: 𝒞(X, Y) → 𝒞_{S}(X, Y) such that π ∘ ι = id_{𝒞_{S}(X,Y)}. Suppose that X, Y, Z ∈ S. Then the composite

[𝒞_{S}(X, Z) ⟶^{ι} 𝒞(X, Z) ⟶^{Δ_{X,Y,Z}} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{N⊗1} 𝒩(X, Y) ⊗ 𝒞(Y, Z)]

vanishes. Indeed, by coassociativity, (N ⊗ 1) ∘ Δ_{X,Y,Z} equals

∑_{∃i: Z_{i}∉S}^{Z_{1},...,Z_{n−1}∈Ob 𝒞, n⩾1} (Δ_{X,Z_{1},...,Z_{n−1},Y}^{(n)} ⊗ 1) ∘ Δ_{X,Y,Z} = ∑_{∃i: Z_{i}∉S}^{Z_{1},...,Z_{n−1}∈Ob 𝒞, n⩾1} Δ_{X,Z_{1},...,Z_{n−1},Y,Z}^{(n+1)} :

𝒞(X, Z) → ⨁_{∃i: Z_{i}∉S}^{Z_{1},...,Z_{n−1}∈Ob 𝒞, n⩾1} 𝒞(X, Z_{1}) ⊗ 𝒞(Z_{1}, Z_{2}) ⊗ ⋯ ⊗ 𝒞(Z_{n−1}, Y) ⊗ 𝒞(Y, Z).

It follows from the definition of ι that Δ_{X,U_{1},...,U_{k−1},Y}^{(k)} ∘ ι = 0, for an arbitrary sequence of objects U_{1}, …, U_{k−1} ∈ Ob 𝒞 such that U_{i} ∉ S for some i. Therefore (N ⊗1)∘Δ_{X,Y,Z} ∘ι = 0. Since the sequence

0 → 𝒞_{S}(X, Y) ⊗ 𝒞(Y, Z) ⟶^{ι⊗1} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{N⊗1} 𝒩(X, Y) ⊗ 𝒞(Y, Z)

is exact, the composite Δ_{X,Y,Z} ∘ι factors through 𝒞_{S}(X, Y)⊗𝒞(Y, Z). In other words, there exists a unique k-linear map ϕ: 𝒞_{S}(X, Z) → 𝒞_{S}(X, Y) ⊗ 𝒞(Y, Z) such that (ι ⊗ 1) ∘ ϕ = Δ_{X,Y,Z} ∘ ι. Since ι ⊗ 1 is an embedding split by π ⊗ 1, the map ϕ is necessarily given by the composite

[𝒞_{S}(X, Z) ⟶^{ι} 𝒞(X, Z) ⟶^{Δ_{X,Y,Z}} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{π⊗1} 𝒞_{S}(X, Y) ⊗ 𝒞(Y, Z)].

In particular, the equation (ι ⊗ 1) ∘ ϕ = Δ_{X,Y,Z} ∘ ι takes the form

(ι ∘ π ⊗ 1) ∘ Δ_{X,Y,Z} ∘ ι = Δ_{X,Y,Z} ∘ ι: 𝒞_{S}(X, Z) → 𝒞(X, Y) ⊗ 𝒞(Y, Z).  (3)

Similarly, the following equation holds true:

(1 ⊗ ι ∘ π) ∘ Δ_{X,Y,Z} ∘ ι = Δ_{X,Y,Z} ∘ ι: 𝒞_{S}(X, Z) → 𝒞(X, Y) ⊗ 𝒞(Y, Z).  (4)

Combining these equations yields

(ι ⊗ ι) ∘ (π ⊗ π) ∘ Δ_{X,Y,Z} ∘ ι = Δ_{X,Y,Z} ∘ ι: 𝒞_{S}(X, Z) → 𝒞(X, Y) ⊗ 𝒞(Y, Z).

Define

Δ′_{X,Y,Z} = [𝒞_{S}(X, Z) ⟶^{ι} 𝒞(X, Z) ⟶^{Δ_{X,Y,Z}} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{π⊗π} 𝒞_{S}(X, Y) ⊗ 𝒞_{S}(Y, Z)].

(5)

Then the above equation is equivalent to (ι ⊗ ι) ∘ Δ′_{X,Y,Z} = Δ_{X,Y,Z} ∘ ι. Coassociativity of Δ′ follows from coassociativity of Δ since

(ι ⊗ ι ⊗ ι) ∘ (Δ′ ⊗ 1) ∘ Δ′ = (Δ ⊗ 1) ∘ (ι ⊗ ι) ∘ Δ′ = (Δ ⊗ 1) ∘ Δ ∘ ι = (1 ⊗ Δ) ∘ Δ ∘ ι = (1 ⊗ Δ) ∘ (ι ⊗ ι) ∘ Δ′ = (ι ⊗ ι ⊗ ι) ∘ (1 ⊗ Δ′) ∘ Δ′,

and ι ⊗ ι ⊗ ι is an embedding split by π ⊗ π ⊗ π. Thus, 𝒞_{S} becomes a cocategory. The quiver map ι: 𝒞_{S} → 𝒞 with Ob ι: S ↪ Ob 𝒞 is a cocategory homomorphism. Indeed, it was shown above that equation (1) holds true for X, Y, Z ∈ S. If X, Z ∈ S, Y ∈ Ob 𝒞∖S, then Δ_{X,Y,Z} ∘ ι = 0 by the definition of 𝒞_{S}, therefore equation (1) is satisfied in this case as well. By (2), the equation ι^{⊗n} ∘ Δ_{X,Z_{1},...,Z_{n−1},Y}^{′(n)} = Δ_{X,Z_{1},...,Z_{n−1},Y}^{(n)} ∘ ι holds true for all X, Z_{1}, …, Z_{n−1}, Y ∈ S. This implies that the cocategory 𝒞_{S} is cocomplete: given an element t ∈ 𝒞_{S}(X, Y), there is n ⩾ 1 such that Δ_{X,Z_{1},...,Z_{n−1},Y}^{(n)} ∘ ι(t) = 0, for an arbitrary collection of objects Z_{1}, …, Z_{n−1} ∈ Ob 𝒞. Since ι^{⊗n} is an embedding split by π^{⊗n}, it follows that Δ_{X,Z_{1},...,Z_{n−1},Y}^{′(n)} (t) = 0 for all Z_{1}, …, Z_{n−1} ∈ S.

3.1. Proposition (Universal property of 𝒞_{S}). An arbitrary cocategory homomorphism h: ℬ → 𝒞 with h(Ob ℬ) ⊂ S factors uniquely through 𝒞_{S}.

Proof. It follows from equation (2) that Δ_{h(X),U_{1},...,U_{n−1},h(Y)}^{(n)} ∘ h = 0 if U_{i} ∉ S for some i, thus N ∘ h = 0: ℬ(X, Y) → 𝒩(h(X), h(Y)), for each pair X, Y ∈ Ob ℬ. Therefore, h: ℬ(X, Y) → 𝒞(h(X), h(Y)) factors through 𝒞_{S}(h(X), h(Y)), i.e., there exists a unique linear map \overline{h} = \overline{h}_{X,Y} : ℬ(X, Y) → 𝒞_{S}(h(X), h(Y)) such that ι ∘ \overline{h} = h. The quiver map \overline{h}: ℬ → 𝒞_{S} with Ob \overline{h} = Ob h: Ob ℬ → S is a cocategory homomorphism since

(ι ⊗ ι) ∘ Δ′ ∘ \overline{h} = Δ ∘ ι ∘ \overline{h} = Δ ∘ h = (h ⊗ h) ∘ Δ = (ι ⊗ ι) ∘ (\overline{h} ⊗ \overline{h}) ∘ Δ,

and ι ⊗ ι is an embedding split by π ⊗ π. Uniqueness of \overline{h} is obvious. □

3.2. Remark. The same construction makes sense in graded and differential graded contexts. The proofs transport literally except the following subtlety in the case of differential graded cocategories: in general, the embedding ι : 𝒞_{S}(X, Y) → 𝒞(X, Y) does not admit a splitting which is a cochain map. Nevertheless, the argument can be modified as follows. Choose a splitting π : 𝒞(X, Y) → 𝒞_{S}(X, Y) of graded k-modules. Since ι is a cochain map, i.e., d ∘ ι = ι ∘ d, it follows that the differential in 𝒞_{S}(X, Y) is necessarily given by the composite

d = [𝒞_{S}(X, Y) ⟶^{ι} 𝒞(X, Y) ⟶^{d} 𝒞(X, Y) ⟶^{π} 𝒞_{S}(X, Y)].

In particular, the commutation relation d ∘ ι = ι ∘ d takes the form

ι ∘ π ∘ d ∘ ι = d ∘ ι : 𝒞_{S}(X, Y) → 𝒞(X, Y).

Then the comultiplication Δ′ given by (5) is a cochain map. Indeed,

Δ′ ∘ d = (π ⊗ π) ∘ Δ ∘ ι ∘ π ∘ d ∘ ι = (π ⊗ π) ∘ Δ ∘ d ∘ ι.

On the other hand,

(1 ⊗ d + d ⊗ 1) ∘ Δ′ =(1 ⊗ π ∘ d ∘ ι + π ∘ d ∘ ι ⊗ 1) ∘ (π ⊗ π) ∘ Δ ∘ ι
=(π ⊗ π) ∘ (1 ⊗ d) ∘ (1 ⊗ ι ∘ π) ∘ Δ ∘ ι
+(π ⊗ π) ∘ (d ⊗ 1) ∘ (ι ∘ π ⊗ 1) ∘ Δ ∘ ι
=(π ⊗ π) ∘ (1 ⊗ d + d ⊗ 1) ∘ Δ ∘ ι

due to (3) and (4). Since Δ is a cochain map, it follows that Δ′∘d = (1⊗d+d⊗1)∘Δ′, thus (𝒞_{S}, Δ′) is a differential graded cocategory. The further arguments remain unchanged.

4. Proof of the theorem

Let 𝒞, 𝒟 be cocomplete cocategories, f, g: 𝒞 → 𝒟 cocategory homomorphisms. Denote S = {X ∈ Ob 𝒞|f(X) = g(X)}. Suppose h: ℬ → 𝒞 is a cocategory homomorphism such that f ∘ h = g ∘ h. Then for each W ∈ Ob ℬ, f(h(W)) = g(h(W)), therefore h(Ob ℬ) ⊂ S. By the universal property of the cocategory 𝒞_{S} there exists a unique cocategory homomorphism \overline{h}: ℬ → 𝒞_{S} such that ι ∘ \overline{h} = h. Therefore, it suffices to construct an equalizer of the pair of cocategory homomorphisms f ∘ ι, g ∘ ι: 𝒞_{S} → 𝒟. Thus, we may assume without loss of generality that Ob f = Ob g. Let us construct an equalizer

ℰ ⟶^{e} 𝒞 ⇉_{g}^{f} 𝒟

in the category of cocomplete cocategories. Put Ob ℰ = Ob 𝒞. For X, Y ∈ Ob 𝒞, denote by ℳ(X, Y) the k-vector space

⨁_{\begin{matrix}X=X_{0},X_{1},...,X_{p}∈Ob 𝒞 \\ Y_{0},...,Y_{q−1},Y_{q}=Y ∈Ob 𝒞 \end{matrix}}^{p,q⩾0} 𝒞(X, X_{1})⊗⋯⊗𝒞(X_{p−1}, X_{p})⊗𝒟(f(X_{p}), f(Y_{0}))⊗𝒞(Y_{0}, Y_{1})⊗⋯⊗𝒞(Y_{q−1}, Y).

Define a k-linear map R: 𝒞(X, Y) → ℳ(X, Y) by

R = ∑_{\begin{matrix}X=X_{0},X_{1},...,X_{p}∈Ob 𝒞 \\ Y_{0},...,Y_{q−1},Y_{q}=Y ∈Ob 𝒞 \end{matrix}}^{p,q⩾0}(1^{⊗p} ⊗ (f − g) ⊗ 1^{⊗q}) ∘ Δ_{X,X_{1},...,X_{p},Y_{0},...,Y_{q−1},Y}^{(p+1+q)} .

It is well defined since 𝒞 is cocomplete. Let ℰ(X, Y) = Ker R, so that we have an exact sequence

0 → ℰ(X, Y) ⟶^{e} 𝒞(X, Y) ⟶^{R} ℳ(X, Y).

Choose a splitting p: 𝒞(X, Y) → ℰ(X, Y) such that p ∘ e = id_{ℰ(X,Y)}. Suppose that X, Y, Z ∈ Ob 𝒞. Then the composite

[ℰ(X, Z) ⟶^{e} 𝒞(X, Z) ⟶^{Δ_{X,Y,Z}} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{R⊗1} ℳ(X, Y) ⊗ 𝒞(Y, Z)]

vanishes. Indeed, by coassociativity, the composite (R ⊗ 1) ∘ Δ_{X,Y,Z} equals

∑_{\begin{matrix}X=X_{0},X_{1},...,X_{p}∈Ob 𝒞 \\ Y_{0},...,Y_{q−1},Y_{q}=Y ∈Ob 𝒞 \end{matrix}}^{p,q⩾0}(1^{⊗p} ⊗ (f − g) ⊗ 1^{⊗q+1}) ∘ (Δ_{X,X_{1},...,X_{p},Y_{0},...,Y_{q−1},Y}^{(p+1+q)} ⊗ 1) ∘ Δ_{X,Y,Z}

= ∑_{\begin{matrix}X=X_{0},X_{1},...,X_{p}∈Ob 𝒞 \\ Y_{0},...,Y_{q−1},Y_{q}=Y ∈Ob 𝒞 \end{matrix}}^{p,q⩾0}(1^{⊗p} ⊗ (f − g) ⊗ 1^{⊗q+1}) ∘ Δ_{X,X_{1},...,X_{p},Y_{0},...,Y_{q−1},Y,Z}^{(p+1+q+1)} :

𝒞(X, Z) → ⨁_{\begin{matrix}X=X_{0},X_{1},...,X_{p}∈Ob 𝒞 \\ Y_{0},...,Y_{q−1},Y_{q}=Y ∈Ob 𝒞 \end{matrix}}^{p,q⩾0} 𝒞(X, X_{1}) ⊗ ⋯ ⊗ 𝒞(X_{p−1}, X_{p}) ⊗ 𝒟(f(X_{p}), f(Y_{0}))

⊗ 𝒞(Y_{0}, Y_{1}) ⊗ ⋯ ⊗ 𝒞(Y_{q−1}, Y) ⊗ 𝒞(Y, Z).

It follows from the definition ℰ that (1^{⊗p} ⊗(f −g)⊗1^{⊗q+1})∘Δ_{X,X_{1},...,X_{p},Y_{0},...,Y_{q−1},Y,Z}^{(p+1+q+1)} ∘e = 0, for all X, X_{1}, …, X_{p}, Y_{0}, …, Y_{q−1}, Y, Z ∈ Ob 𝒞, therefore (R ⊗ 1) ∘ Δ_{X,Y,Z} ∘ e = 0. Since the sequence

0 → ℰ(X, Y) ⊗ 𝒞(Y, Z) ⟶^{e⊗1} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{R⊗1} ℳ(X, Y) ⊗ 𝒞(Y, Z)

is exact, it follows that the map Δ_{X,Y,Z} ∘ e factors through ℰ(X, Y) ⊗ 𝒞(Y, Z). In other words, there exists a unique k-linear map ψ: ℰ(X, Z) → ℰ(X, Y) ⊗ 𝒞(Y, Z) such that (e ⊗ 1) ∘ ψ = Δ_{X,Y,Z} ∘ e. Since e ⊗ 1 is an embedding split by p ⊗ 1, the map ψ is necessarily given by the composite

[ℰ(X, Z) ⟶^{e} 𝒞(X, Z) ⟶^{Δ_{X,Y,Z}} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{p⊗1} ℰ(X, Y) ⊗ 𝒞(Y, Z)].

In particular, the equation (e ⊗ 1) ∘ ψ = Δ_{X,Y,Z} ∘ e takes the form

(e ∘ p ⊗ 1) ∘ Δ_{X,Y,Z} ∘ e = Δ_{X,Y,Z} ∘ e: ℰ(X, Z) → 𝒞(X, Y) ⊗ 𝒞(Y, Z).

Similarly, the following equation holds true:

(1 ⊗ e ∘ p) ∘ Δ_{X,Y,Z} ∘ e = Δ_{X,Y,Z} ∘ e: ℰ(X, Z) → 𝒞(X, Y) ⊗ 𝒞(Y, Z).

Combining these equations yields

(e ⊗ e) ∘ (p ⊗ p) ∘ Δ_{X,Y,Z} ∘ e = Δ_{X,Y,Z} ∘ e: ℰ(X, Z) → 𝒞(X, Y) ⊗ 𝒞(Y, Z).

Define

Δ̃_{X,Y,Z} = [ℰ(X, Z) ⟶^{e} 𝒞(X, Z) ⟶^{Δ_{X,Y,Z}} 𝒞(X, Y) ⊗ 𝒞(Y, Z) ⟶^{p⊗p} ℰ(X, Y) ⊗ ℰ(Y, Z)].

As in the case of 𝒞_{S}, one shows that Δ̃ turns ℰ into a cocomplete cocategory, and that e becomes a cocategory homomorphism.

Suppose h: ℬ → 𝒞 is a cocategory homomorphism such that f ∘ h = g ∘ h. Then R ∘ h = 0: ℬ(X, Y) → ℳ(h(X), h(Y)). Indeed, by identity (2),

Δ_{h(X),X_{1},...,X_{p},Y_{0},...,Y_{q−1},h(Y)}^{(p+1+q)} ∘ h

= ∑_{h(V_{0})=Y_{0},...,h(V_{q−1})=Y_{q−1}}^{h(U_{1})=X_{1},...,h(U_{p})=X_{p}}(h_{X,U_{1}} ⊗ ⋯ ⊗ h_{U_{p},V_{0}} ⊗ ⋯ ⊗ h_{V_{q−1},Y}) ∘ Δ_{X,U_{1},...,U_{p},V_{0},...,V_{q−1},Y}^{(p+1+q)} .

It follows that

R ∘ h = ∑_{h(V_{0})=Y_{0},...,h(V_{q−1})=Y_{q−1}}^{h(U_{1})=X_{1},...,h(U_{p})=X_{p}}(h^{⊗p} ⊗ (f − g) ∘ h ⊗ h^{⊗q}) ∘ Δ_{X,U_{1},...,U_{p},V_{0},...,V_{q−1},Y}^{(p+1+q)} = 0,

for X_{1}, …, X_{p}, Y_{0}, …, Y_{q−1} ∈ Ob 𝒞. Therefore the map h: ℬ(X, Y) → 𝒞(h(X), h(Y)) factors through e: ℰ(h(X), h(Y)) → 𝒞(h(X), h(Y)), i.e., h = e ∘ j for some k-linear map j: ℬ(X, Y) → ℰ(h(X), h(Y)). The morphism of k-quivers j: ℬ → ℰ with Ob j = Ob h is a cocategory homomorphisms since

(e ⊗ e) ∘ (j ⊗ j) ∘ Δ = (h ⊗ h) ∘ Δ = Δ ∘ h = Δ ∘ e ∘ j = (e ⊗ e) ∘ Δ ∘ j,

and e ⊗ e is an embedding split by p ⊗ p. Uniqueness of j is obvious since e is an embedding. □

4.1. Remark. The theorem is true for cocomplete graded (resp. differential graded) cocategories as well, with appropriate modifications in the proof similar to those made in Remark 3.2.

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UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France

  E-mail address: keller@math.jussieu.fr

Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, Germany

  E-mail address: manzyuk@mathematik.uni-kl.de