Theory and Applications of Categories, Vol. 10, No. 6, 2002, pp. 134–147.

COHERENCE FOR FACTORIZATION ALGEBRAS

ROBERT ROSEBRUGH AND R.J. WOOD

ABSTRACT. For the 2-monad ((−)^{2}, I, C) on CAT, with unit I described by identities and multiplication C described by composition, we show that a functor F : K^{2} [figure] K satisfying F I_{K} = 1_{K} admits a unique, normal, pseudo-algebra structure for (−)^{2} if and only if there is a mere natural isomorphism F ·F^{2} [figure: ≃] F ·C_{K}. We show that when this is the case the set of all natural transformations F · F^{2} [figure] F · C_{K} forms a commutative monoid isomorphic to the centre of K.

1. Preliminaries

1.1. When we speak of ‘the 2-monad (−)^{2} on CAT’ we understand the canonical monad that arises by exponentiation of the cocommutative comonoid structure

1 [figure: !] 2 [figure: Δ]2 × 2

on the ordinal 2 in CAT. We write I_{K}:K [figure] K^{2} for the K-component of the unit; it sends an object to its identity arrow. We write C_{K} : (K^{2})^{2} [figure] K^{2} for the K-component of the multiplication; it sends a commutative square to its composite arrow. This monad was very carefully described in [K&T], wherein it was shown that the normal pseudo-algebras for (−)^{2} are equivalent to factorization systems. In [R&W] Coppey’s result [COP] that strict algebras for (−)^{2} are strict factorization systems was rediscovered (in the context of distributive laws). Mindful of the inflection terminology of [K&S] we call a normal pseudo-algebra for (−)^{2} a factorization algebra and call a strict algebra for (−)^{2} a strict factorization algebra. It is convenient in this context to call a mere functor F : K^{2} [figure] K a factorization pre-algebra. In the event that F I_{K} = 1_{K}, we say that F is a normal factorization pre-algebra (although this terminology does not entirely conform with that of [K&S]). (We are also aware that the lax factorization algebras of [R&T] are certain of the oplax algebras for (−)^{2} in the terminology of [K&S] but this presents no difficulty for the terminology employed here.)

1.2. Remark. It was pointed out in [K&T] that the normality equation F I_{K} = 1_{K} imposes no real loss of generality for a (−)^{2}-pseudo-algebra. Given an isomorphism ι: 1_{K} [figure: ≃]F I_{K}, [K&T] explains how to define a new functor F′ : K^{2} [figure] K with F′ [figure: ≃]F and F′I_{K} = 1_{K}. This certainly conforms with practice but it may be worth pointing out that intuitionistically the definition of F′ requires that the identity arrows of K be a complemented subset of the set of all arrows of K. While we will hypothesize normality throughout, we will include methodological remarks about the general case when appropriate.

The authors gratefully acknowledge financial support from the Canadian NSERC. Diagrams typeset using M. Barr’s diagram package, diagxy.tex.

Received by the editors 2001 September 24.

Transmitted by Walter Tholen. Published on 2002 January 27.

2000 Mathematics Subject Classification: 18A32, 18D05.

Key words and phrases: coherence, factorization algebra.

\begin{matrix}c \\ ◯ \end{matrix} Robert Rosebrugh and R.J. Wood, 2002. Permission to copy for private use granted.

1.3. Most of the notation here will be similar to that of [K&T] and [R&W]. In particular, for F :K^{2} [figure] K a factorization pre-algebra and an arrow

[figure: A; B; v; X; A; f; X; Y; u; Y; B; g]

in K^{2} with domain f and codomain g, we write F (u, v) :F (f) [figure] F (g). Since it is (f; u, v; g), not (u, v), which is an arrow of K^{2}, this notation requires care. Also, for any arrow f:X [figure] A in K, we have the following factorization of I_{K}(f) in K^{2} :

[figure: X; A; f; X; X; 1_{X}; X; X; 1_{X}; X; A; f; A; A; 1_{A}; X; A; X; A; f; A; A; 1_{A}]

and if F : K^{2} [figure] K is a normal factorization pre-algebra, we follow [K&T] in writing

X [figure: e_{f}]F (f) [figure: m_{f}]A for the result of applying F to these factors of I_{K}(f). Thus in this context, F (1_{X}, f) = e_{f} and F (f, 1_{A}) = m_{f} (but it is not true that we always have F (1_{S}, f) = e_{f} and F (f, 1_{T}) = m_{f}). When the first square of this section is regarded as an object of (K^{2})^{2} we often write S = (f; u, v; g) and it follows that F F^{2}(S) = F (F (u, v)). If the composite uf = gv is c : X [figure] B then F C_{K}(S) = F (c). For F : K^{2} [figure] K a normal factorization pre-algebra, F is a strict factorization algebra if F F^{2} = F C_{K}, that is if, for all S in (K^{2})^{2}, F (F (u, v)) = F (c).

1.4. For a normal factorization pre-algebra F :K^{2} [figure] K, a factorization algebra structure on F is an isomorphism γ:F F^{2} [figure: ≃]F C_{K} which satisfies:

γI_{K^{2}} = 1_{F}  (1)
γ(I_{K})^{2} = 1_{F}  (2)
γC_{K^{2}} · γ(F^{2})^{2} = γ(C_{K})^{2} · F γ^{2}  (3)

these equations being the specialization of the equations in §2 of [STR] to the case at hand.

1.5. Remark. In the absence of normality, a (−)^{2}-pseudo-algebra structure further requires an isomorphism ι : 1_{K} [figure: ≃]F I_{K} and equations (1) and (2) above must then be replaced by:

γI_{K^{2}} · ιF =  (1′) 1_{F}
γ(I_{K})^{2} · F ι^{2} =  (2′) 1_{F}

2. The main results

For a normal factorization pre-algebra F :K^{2} [figure] K, we will show that existence of a mere isomorphism α:F F^{2} [figure: ≃]F C_{K}, subject to no equations (other than naturality), is equivalent to the existence of a unique factorization algebra structure γ: F F^{2} [figure: ≃]F C_{K}, and that such an α may itself fail to be an algebra structure. Of course this result shows that it is in fact a property of a factorization pre-algebra to be a factorization algebra, as is implicit from the conjunction of Theorems A and B in [K&T]. In general there is no comparison arrow joining F F^{2} and F C_{K} in either direction. But there is both a comparison span and a comparison cospan joining them. The span and cospan form a commutative square that one might call a comparison diamond and which is the subject of our first Lemma.

2.1. Lemma. For F :K^{2} [figure] K a normal factorization pre-algebra and S =

[figure: A; B; v; X; A; f; X; Y; u; Y; B; g]

a typical object of (K^{2})^{2} with C_{K}(S) = X [figure: c] B, the following diagram commutes:

[figure: F (F (u, v)); F (c); F (F (u, v)); F (c); F (e_{c}); F (F (u, v)); F (e_{f}, F (u, 1_{B})); F (e_{c}); F (c); m_{e_{c}}; F (F (u, v)); F (m_{c}); F (F (1_{X}, v), m_{g}); F (F (u, v)); F (c); F (F (u, v)); F (c); F (c); F (m_{c}); e_{m_{c}}]

Proof. Observe first that the diagram of of the statement is meaningfully labelled and that the left-most arrows of it are defined by

[figure: F (e_{c}); F (F (u, v)); X; F (e_{c}); e_{e_{c}}; X; F (f); e_{f}; F (f); F (F (u, v)); e_{F (u,v)}; F (c); F (g); F (u, 1_{B}); F (e_{c}); F (c); m_{e_{c}}; F (e_{c}); F (F (u, v)); F (e_{f}, F (u, 1_{B})); F (F (u, v)); F (g); F (F (u, v)); F (m_{c}); F (F (1_{X}, v), m_{g}); F (f); F (F (u, v)); F (f); F (c); F (1_{X}, v); F (c); F (m_{c}); e_{m_{c}}; F (g); B; m_{g}; F (F (u, v)); F (g); m_{F (u,v)}; F (F (u, v)); F (m_{c}); F (m_{c}); B; m_{m_{c}}]

where, in turn, F (1_{X}, v) and F (u, 1_{B}) are defined by

[figure: F (f); F (c); X; F (f); e_{f}; X; X; 1_{X}; X; F (c); e_{c}; A; B; v; F (f); A; m_{f}; F (f); F (c); F (1_{X}, v); F (c); B; F (c); F (g); F (u, 1_{B}); X; F (c); X; Y; u; Y; F (g); e_{g}; B; B; 1_{B}; F (c); B; m_{c}; F (c); F (g); F (g); B; m_{g}]

From the definition of e_{−} and m_{−}, we have F (1_{X}, v) · e_{f} = e_{c} and m_{g} · F (u, 1_{B}) = m_{c}. Thus by functoriality of F, F (F (1_{X}, v), m_{g}) · F (e_{f}, F (u, 1_{B})) = F (e_{c}, m_{c}). On the other hand, the right-most composite is

e_{m_{c}} · m_{e_{c}} = F (1_{F (c)}, m_{c}) · F (e_{c}, 1_{F (c)}) = F (e_{c}, m_{c})

Our second Lemma is almost a triviality and, like much of what we have to say here, is generalizable in many ways. However, along with obvious variations, this Lemma is quite useful for a number of coherence questions.

2.2. Lemma. For natural transformations

[figure: A; B; F; A; B; G; σ; B; C; S; B; C; T; τ]

if T σ is invertible then τF is determined by τG, in the sense that τF = (T σ)^{−1} · τG · Sσ. Similarly, if Sσ is invertible then τG is determined by τF.

To give an example of the application of Lemma 2.2, suppose that X is a reflective subcategory of B with defining adjunction η, ϵ: A ⊣ I : X [figure] B. Then, if T inverts η, τ is determined by its components of the form τIX. In fact, in situations like this we can say a little more:

2.3. Corollary. For

[figure: B; B; 1_{B}; B; B; G; σ; B; C; S; B; C; T]

with T σ invertible and G well-pointed by σ (meaning that Gσ = σG), precomposition with G provides a bijection

CAT(B, C)(S, T) [figure: (−)G] CAT(B, C)(SG, T G)

Proof. For ω:SG [figure] T G, consider the following squares:

[figure: T; T G; (T σ)^{−1}; S; T; g(ω); S; SG; Sσ; SG; T G; ω; T G; T GG; T σG; SG; T G; g(ω)G; SG; SGG; SσG; SGG; T GG; ωG; T G; T GG; T Gσ; SG; T G; ω; SG; SGG; SGσ; SGG; T GG; ωG]

The first square defines a function g: CAT(B, C)(SG, T G) [figure] CAT(B, C)(S, T) which by Lemma 2.2 admits (−)G as a section. The second square commutes by instantiating an evidently commutative square at G. The third commutes by naturality of ω. Since σG = Gσ, the top arrows of the second and third squares are equal and similarly so are the bottom arrows of the second and third squares. Since T σ is invertible, it follows that g(ω)G = ω which completes the proof that (−)G is a bijection.

2.4. We return now to (−)^{2}. In addition to C_{K}: (K^{2})^{2} [figure] K^{2}[figure]K: I_{K}, it is convenient to name four embeddings K^{2} [figure] (K^{2})^{2} and indicate natural transformations between them as below:

[figure: R_{K}; L_{K}; R_{K}; L_{K}; V_{K}; R_{K}; V_{K}; L_{K}; R_{K}; H_{K}; R_{K}; L_{K}; R_{K}; L_{K}; L_{K}; H_{K}; :; K^{2}; (K^{2})^{2}]

These are defined on an object c: X [figure] B of K^{2} by the following commutative square in (K^{2})^{2}

[figure: B; B; X; B; c; X; X; X; B; c; X; B; c; X; X; X; X; X; B; c; B; B; X; B; c; X; B; c; B; B; X; B; c; X; X; X; B; c; B; B; X; X; X; X; X; B; c; X; X; B; B; B; X; c; B; B; B; B; X; X; X; B; c; X; X; B; B; X; X; B; B; B; X; c; B; B]

it being understood that unlabelled arrows are identities. However, V_{K} = I_{K^{2}}, H_{K} = (I_{K})^{2} and R_{K} ⊣ C_{K} ⊣ L_{K}.

2.5. The unit for R_{K} ⊣ C_{K} and the counit for C_{K} ⊣ L_{K} are identities. For S =

[figure: A; B; v; X; A; f; X; Y; u; Y; B; g]

a typical object of (K^{2})^{2} with C_{K}(S) = X [figure: c]B, the S-component of the counit for R_{K} ⊣ C_{K} and the S-component of the unit for C_{K} ⊣ L_{K} are shown below as a composable pair in (K^{2})^{2}

[figure: B; B; X; B; c; X; B; c; B; B; A; B; v; X; A; f; X; Y; u; Y; B; g; X; B; c; X; X; X; X; X; B; c; X; X; X; Y; u; X; A; f; B; B; X; X; Y; B; g; A; B; v; B; B]

Now for F a normal factorization pre-algebra, let α : F F^{2} [figure] F C_{K} : (K^{2})^{2} [figure] K be any natural transformation. Application of α to the composable pair above gives

[figure: F (c); F (c); 1_{F (c)}; F (e_{c}); F (c); αR_{K}c = αR_{K}C_{K}S; F (e_{c}); F (F (u, v)); F (e_{f}, F (u, 1_{B})); F (F (u, v)); F (c); αS; F (c); F (c); 1_{F (c)}; F (F (u, v)); F (c); F (F (u, v)); F (m_{c}); F (F (1_{X}, v), m_{g}); F (m_{c}); F (c); αL_{K}C_{K}S = αL_{K}c]

as is seen by consulting the definitions of e_{f} and the like in 1.3. For an arbitrary natural transformation β:F C_{K} [figure] F F^{2} we get a diagram similar to that above but with the vertical arrows reversed. From these diagrams several observations follow almost immediately.

2.6. Lemma. For F a normal factorization pre-algebra, any natural transformation α: F F^{2} [figure] F C_{K} is determined by αL_{K} : F F^{2}L_{K} [figure] F : K^{2} [figure] K. Any natural transformation β:F C_{K} [figure] F F^{2} is determined by βR_{K}:F F^{2}R_{K} [figure] F :K^{2} [figure] K. If α is an isomorphism then F (e_{f}, F (u, 1_{B})) and F (F (1_{X}, v), m_{g}) are isomorphisms.

Proof. The right hand square immediately above the statement of the Lemma shows that the hypotheses of Lemma 2.2 are satisfied — with τ the α under consideration and σ the unit for C_{K} ⊣ L_{K}. The second statement follows from a similar consideration while the third also follows from the diagram.

In fact, we can improve Lemma 2.6 by applying Corollary 2.3, with the role of σ:1_{B} [figure] G taken by 1_{(K^{2})^{2}} [figure] L_{K}C_{K}, and noting that C_{K} is cofully faithful (since L_{K} is fully faithful).

2.7. Corollary. For F a normal factorization pre-algebra, precomposition with L_{K} provides a bijection

CAT((K^{2})^{2}, K)(F F^{2}, F C_{K}) [figure: (−)L_{K}] CAT(K^{2}, K)(F F^{2}L_{K}, F)

We now apply our arbitrary α:F F^{2} [figure] F C_{K} to the second diagram in 2.4 resulting in the commutativity of

[figure: F (e_{c}); F (m_{c}); F (e_{c}); F (m_{c}); F (c); F (e_{c}); m_{e_{c}}; F (c); F (m_{c}); e_{m_{c}}; F (e_{c}); F (c); F (1_{X}, m_{c}); F (e_{c}); F (m_{c}); F (e_{c}); F (m_{c}); F (m_{c}); F (c); F (e_{c}, 1_{B}); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (c); F (e_{c}); F (c); αR_{K}c; F (m_{c}); F (c); αL_{K}c; F (c); F (c); αV_{K}c; F (c); F (c); αH_{K}c]

from which further observations follow.

2.8. Lemma. For F a normal factorization pre-algebra and any natural transformation α:F F^{2} [figure] F C_{K}, αR_{K} is determined by αV_{K} which in turn is determined by αL_{K} and also αR_{K} is determined by αH_{K} which in turn is determined by αL_{K}. If α : F F^{2} [figure] F C_{K} is an isomorphism then, for every arrow c in K, m_{e_{c}}, e_{m_{c}}, F (1_{X}, m_{c}) and F (e_{c}, 1_{B}) are isomorphisms and

m_{e_{c}} = F (1_{X}, m_{c}) and e_{m_{c}} = F (e_{c}, 1_{B})

Proof. All aspects of the first sentence follow from the fact that all four arrows in the right-most diamond of the diagram above are identities and thereby enable four applications of Lemma 2.2. From the diagram it is clear that if α is an isomorphism then all m_{e_{c}}, e_{m_{c}}, F (1_{X}, m_{c}) and F (e_{c}, 1_{B}) are isomorphisms. Now from [J&T], merely knowing that the m_{e_{c}} are epimorphisms and the e_{m_{c}} are monomorphisms is enough to ensure that, for each object S = (f; u, v; g) of (K^{2})^{2}, F (u, v) is the unique solution s of the equations

s · e_{f} = e_{g} · u
m_{g} · s = v · m_{f}

Thus commutativity of

[figure: F (e_{c}); F (c); m_{e_{c}}; X; F (e_{c}); e_{e_{c}}; X; X; 1_{X}; X; F (c); e_{c}; F (c); B; m_{c}; F (e_{c}); F (c); m_{e_{c}}; F (e_{c}); F (c); m_{e_{c}}; F (c); B; m_{c}]

shows that m_{e_{c}} = F (1_{X}, m_{c}) and a similar diagram provides e_{m_{c}} = F (e_{c}, 1_{B}).

In fact, from the diagram preceding Lemma 2.8 and the Lemma itself we have:

2.9. Corollary. For F a normal factorization pre-algebra and any natural isomorphism α:F F^{2} [figure] F C_{K}, the following are equivalent:

i) αV_{K} = 1_{F};

ii) αL_{K} = (e_{m_{−}})^{−1};

iii) αH_{K} = 1_{F};

iv) αR_{K} = m_{e_{−}}.

Recall now that V_{K} = I_{K^{2}} and H_{K} = (I_{K})^{2}.

2.10. Theorem. For a normal factorization pre-algebra F :K^{2} [figure] K, if γ:F F^{2} [figure: ≃]F C_{K} is an isomorphism satisfying any of the equivalent conditions of Corollary 2.9 then, for S = (f; u, v; g) in (K^{2})^{2} with C_{K}(S) = X [figure: c] B, γS is given equally by

γ(S) = m_{e_{c}} · (F (e_{f}, F (u, 1_{B})))^{−1}

and by

γ(S) = (e_{m_{c}})^{−1} · F (F (1_{X}, v), m_{g})

Moreover, γ satisfies all of (1),(2) and (3) of 1.4, thereby making (F, γ) a factorization algebra.

Proof. Assume that γ:F F^{2} [figure: ≃]F C_{K} satisfies condition i) of Corollary 2.9. This is (1) of 1.4. The equations for γ(S) follow immediately from the diagrams preceding Lemma 2.6 and Lemma 2.8. For example, we have γ(S) = γL_{K}c · F (F (1_{X}, v), m_{g}) by the righthand rectangle preceding Lemma 2.6, which is equal to (e_{m_{c}})^{−1} · F (F (1_{X}, v), m_{g}) using the back parallelogram in the diagram preceding Lemma 2.8 with γV_{K}c = 1_{F (c)}. Also iii) of Corollary 2.9 is (2) of 1.4. For (3) of 1.4 observe that the relevant configuration is

[figure: F F^{2}(F^{2})^{2}; F C_{K}C_{K^{2}} = F C_{K}(C_{K})^{2}; F F^{2}(F^{2})^{2}; F C_{K}C_{K^{2}} = F C_{K}(C_{K})^{2}; F C_{K}(F^{2})^{2} = F F^{2}C_{K^{2}}; F F^{2}(F^{2})^{2}; γ(F^{2})^{2}; F C_{K}(F^{2})^{2} = F F^{2}C_{K^{2}}; F C_{K}C_{K^{2}} = F C_{K}(C_{K})^{2}; γC_{K^{2}}; F F^{2}(F^{2})^{2}; F F^{2}(C_{K})^{2}; F γ^{2}; F F^{2}(F^{2})^{2}; F C_{K}C_{K^{2}} = F C_{K}(C_{K})^{2}; F F^{2}(F^{2})^{2}; F C_{K}C_{K^{2}} = F C_{K}(C_{K})^{2}; F C_{K}C_{K^{2}} = F C_{K}(C_{K})^{2}; F F^{2}(C_{K})^{2}; γ(C_{K})^{2}; :; ((K^{2})^{2})^{2}; K]

To see that the two composite natural transformations are equal we begin by applying Corollary 2.7 twice. First, the adjunction C_{K^{2}} ⊣ L_{K^{2}} :(K^{2})^{2} [figure] ((K^{2})^{2})^{2} has its unit taken by C_{K^{2}} to an identity because, for any K, the adjunction C_{K} ⊣ L_{K} has its unit taken by C_{K} to an identity. Thus F C_{K}C_{K^{2}} takes the unit of C_{K^{2}} ⊣ L_{K^{2}} to an identity and Corollary 2.7 applies to show that any natural transformation α:F F^{2}(F^{2})^{2} [figure] F C_{K}C_{K^{2}} is uniquely determined by αL_{K^{2}}C_{K^{2}} and hence by αL_{K^{2}} : F F^{2}(F^{2})^{2}L_{K^{2}} [figure] F C_{K}: (K^{2})^{2} [figure] K (since C_{K^{2}} is cofully faithful). Next, we repeat the argument for C_{K} ⊣ L_{K} to show that αL_{K^{2}} is uniquely determined by αL_{K^{2}}L_{K}:F F^{2}(F^{2})^{2}L_{K^{2}}L_{K} [figure] F :K^{2} [figure] K. Consider now

[figure: K^{2}; (K^{2})^{2}; V_{K}; K^{2}; (K^{2})^{2}; L_{K}; σ; (K^{2})^{2}; K; F F^{2}(F^{2})^{2}L_{K^{2}}; (K^{2})^{2}; K; F C_{K}; αL_{K^{2}}]

where σ is the natural transformation V_{K} [figure] L_{K} defined in 2.4. We claim F F^{2}(F^{2})^{2}L_{K^{2}}σ is invertible. To see this, use (F^{2})^{2}L_{K^{2}} = L_{K}F^{2} and compute

F F^{2}L_{K}F^{2}σc = e_{m_{mc}} · e_{m_{c}}

which is an isomorphism, since all e_{m_{f}} are so by Lemma 2.8. Using the second clause of Lemma 2.2 we conclude that αL_{K^{2}}L_{K}, and hence our arbitrary α, is determined by αL_{K^{2}}V_{K}. Since each γV_{K}c is 1_{F (c)} it follows that each (γC_{K^{2}} · γ(F^{2})^{2})L_{K^{2}}V_{K}c and each (γ(C_{K})^{2} · F γ^{2})L_{K^{2}}V_{K}c is also 1_{F (c)} showing that γC_{K^{2}} · γ(F^{2})^{2} = γ(C_{K})^{2} · F γ^{2}.

The theorem shows that if a normal factorization pre-algebra F : K^{2} [figure] K admits a factorization algebra structure, γ then there is no question about what it is. Said otherwise, being a factorization algebra is a property for a normal factorization pre-algebra. We summarize.

2.11. Theorem. For F : K^{2} [figure] K a normal factorization pre-algebra, the following are equivalent:

i) F admits a necessarily unique factorization algebra structure;

ii) there is an isomorphism F F^{2} [figure: ≃]F C_{K};

iii) all m_{e_{f}} and all e_{m_{f}} are isomorphisms.

Proof. Clearly, i) implies ii) is trivial. That ii) implies iii) is contained in Lemma 2.8. Assume iii). It is shown in [K&T] that for E_{F} = {h|m_{h} is invertible} and M_{F} = {h|e_{h} is invertible}, (E_{F} , M_{F}) is a factorization system for K. Consider the first diagram in the proof of Lemma 2.1. Inspection of the top left square shows that F (e_{f}, F (u, 1_{B})) is in E_{F} — since arrows of the form e_{h} are in E_{F} and E_{F} is closed with respect to composition and the cancellation rule. Since m_{g} · F (u, 1_{B}) = m_{c} and arrows of the form m_{h} are in M_{F} and M_{F} is closed with respect to cancellation, F (u, 1_{B}) is in M_{F}. Inspection of the bottom left square now shows that F (e_{f}, F (u, 1_{B})) is in M_{F}. It follows then that F (e_{f}, F (u, 1_{B})), being in E_{F} ∩ M_{F}, is an isomorphism. (Of course it then follows from Lemma 2.1 that F (F (1_{X}, v), m_{g}) is an isomorphism.) Now, for S = (f; u, v; g) with C_{K}S = c, define γS = m_{e_{c}} · (F (e_{f}, F (u, 1_{B})))^{−1}. This provides an isomorphism γ:F F^{2} [figure: ≃]F C_{K}. For c:X [figure] B, it follows that

γI_{K^{2}}c = γV_{K}c = m_{e_{c}} · F (e_{c}, F (1_{X}, 1_{B}))^{−1} = m_{e_{c}} · F (e_{c}, 1_{F (c)})^{−1} = m_{e_{c}} · m_{e_{c}}^{−1} = 1_{F (c)}

so that by Theorem 2.10 (F, γ) is a factorization algebra.

2.12. Remark. Without normality there is a little to change. For example, from (1′) in 1.5 it follows that in Corollary 2.9 we should replace i) by

i′) αV_{K} = (ιF)^{−1}

and continue with similar adjustments — after first redefining e_{−} and m_{−} to absorb ι. We leave the details of this and subsequent modifications that need to be made in the absence of normality to the interested reader.

2.13. It should not be supposed however that in the situation hypothesised by Theorem 2.11 there is at most one isomorphism F F^{2} [figure: ≃]F C_{K}. From the diagram preceding Lemma 2.8 we see using Lemma 2.2 that if F is a factorization algebra then any α:F F^{2} [figure] F C_{K}:(K^{2})^{2} [figure] K is determined equally by αV_{K} = αI_{K^{2}} :F [figure: ≃]F :K^{2} [figure] K, αL_{K}, αH_{K} and αR_{K}. In fact it is convenient to note:

2.14. Lemma. For a factorization algebra F :K^{2} [figure] K, precomposition with V_{K}:K^{2} [figure] (K^{2})^{2} provides a bijection

CAT((K^{2})^{2}, K)(F F^{2}, F C_{K}) [figure: (−)V_{K}] CAT(K^{2}, K)(F, F)

Moreover, both (−)V_{K} and its inverse preserve invertibility.

Proof. Let β:F [figure] F and define v(β) = βC_{K} · (F F^{2}σC_{K})^{−1} · F F^{2}η, where η is the unit for C_{K} ⊣ L_{K} and σ is again the natural transformation V_{K} [figure] L_{K} of 2.4. Note that for S = (f; u, v; g) in (K^{2})^{2} with C_{K}S = c we have F F^{2}σC_{K}S = e_{m_{c}}. For α: F F^{2} [figure] F C_{K} v(αV_{K}) = α has been shown in earlier diagrams. To see that v(β)V_{K} = β it suffices to show that (F F^{2}σC_{K})^{−1}V_{K} ·F F^{2}ηV_{K} = 1_{F}. This follows immediately from ηV_{K} = σ which can be seen by inspection of η as displayed in 2.5 and σ as displayed in 2.4. It is clear that (−)V_{K} preserves invertibility. By Lemma 2.6 we know that F F^{2}η is invertible for a factorization algebra and it then follows from the explicit description of v that v(β) is invertible when β is so.

Now the fully faithful I_{K}:K [figure] K^{2} has both left and right adjoints, given respectively by ‘codomain’= ∂_{1} and ‘domain’= ∂_{0} but neither the unit for ∂_{1} ⊣ I_{K} nor the counit for I_{K} ⊣ ∂_{0} are in general inverted by F so that Lemma 2.2 is not applicable. In fact it is easy to see — and we will use — that F applied to the c-component of the unit for ∂_{1} ⊣ I_{K} is m_{c} and F applied to the c-component of the counit for I_{K} ⊣ ∂_{0} is e_{c}. However:

2.15. Lemma. For a factorization algebra F : K^{2} [figure] K precomposition with I_{K} : K [figure] K^{2} provides a bijection

CAT(K^{2}, K)(F, F) [figure: (−)I_{K}] CAT(K, K)(1_{K}, 1_{K})

which is an isomorphism of monoids. The inverse function is given by F (−)^{2}. Proof. That (−)I_{K} provides a homomorphism of monoids is immediate from the definitions. Consider a β : F [figure] F : K^{2} [figure] K. For any object c : X [figure] B of K^{2} we have the composable pair

I_{K}X = I_{K}∂_{0}c [figure: (1_{X}, c)] c [figure: (c, 1_{B})] I_{K}∂_{1}c = I_{K}B

in K^{2} consisting of the counit for I_{K} ⊣ ∂_{0} and the unit for ∂_{1} ⊣ I_{K}. Application of β:F [figure] F gives

[figure: F (c); F (c); βc; X; F (c); e_{c}; X; X; βI_{K}X; X; F (c); e_{c}; B; B; βI_{K}B; F (c); B; m_{c}; F (c); F (c); βc; F (c); B; m_{c}]

and, again using [J&T], the mere fact that each m_{e_{f}} is an epimorphism and each e_{m_{f}} is a monomorphism ensures that βc = F (βI_{K}X, βI_{K}B). Clearly we have β = F (βI_{K})^{2}. For α:1_{K} [figure] 1_{K} and any arrow c:X [figure] B in K we have

[figure: B; B; αB; X; B; c; X; X; αX; X; B; c]

which can be seen as the c-component of α^{2}. It follows that the c-component of F α^{2} is F (αX, αB) :F (c) [figure] F (c). Hence the X-component of F α^{2}I_{K} is F (αX, αX) :F (1_{X}) [figure] F (1_{X}) which is αX:X [figure] X, showing that F α^{2}I_{K} = α and completing the proof that (−)I_{K} is a bijection. 2.16. Recall that for any 2-category K and object K therein, the set K(K, K)(1_{K}, 1_{K}) is a commutative monoid under (either) composition of transformations (2-cells). It is the familiar centre of K when K is a monoid. We speak simply of the centre of K in the full generality of the last sentence and write ZK for the centre of K. An early, unpublished, but readily available reference is [WD]. The following theorem is an obvious summary of our observations but note carefully the statement — our extension of an isomorphism 1_{K} [figure: ≃]1_{K} : K [figure] K to a particular isomorphism F F^{2} [figure: ≃]F C_{K} : (K^{2})^{2} [figure] K requires the existence of some isomorphism F F^{2} [figure: ≃]F C_{K} (or the equivalent statements of Theorem 2.11).

2.17. Theorem. For F : K^{2} [figure] K a factorization algebra, precomposition with I_{K^{2}}I_{K} : K [figure] (K^{2})^{2} provides a bijection

CAT((K^{2})^{2}, K)(F F^{2}, F C_{K}) [figure: (−)I_{K^{2}}I_{K}] ZK

and both (−)I_{K^{2}}I_{K} and its inverse preserve invertibility.

2.18. Of course one might wonder if the existence of a factorization algebra on a category K forces the set of invertible elements of its centre to be trivial. This is not the case. For example, the category of abelian groups admits several factorization algebras and has as its centre the monoid of integers under multiplication. In fact the set of invertible elements in the centre of a category with a factorization algebra can be arbitrarily large. For if α is an invertible element in the centre of K then α^{2} is an invertible element in the centre of K^{2}, which carries the free strict factorization algebra provided by C_{K}. To finish the argument it suffices to take K to be a commutative group and observe that such K may be arbitrarily large.

We close with what is evidently the core coherence requirement for factorization algebras.

2.19. Theorem. For a factorization algebra F : K^{2} [figure] K, if γ : F F^{2} [figure: ≃]F C_{K} is an isomorphism satisfying

γI_{K^{2}}I_{K} = 1_{1_{K}}

then (F, γ) is the unique factorization algebra structure on F.

References

[COP] L. Coppey. Algèbres de décompositions et précategories. Diagrammes 3, 1980.

[J&T] G. Janelidze and W.Tholen. Functorial factorization, well-pointedness and separability. J. Pure Appl. Algebra, 142:99–130, 1999.

[K&S] G.M. Kelly and R. Street. Review of the elements of 2-categories. In Lecture Notes in Math. 420, 75–103, Springer-Verlag, 1974.

[K&T] M. Korostenski and W.Tholen. Factorization systems as Eilenberg-Moore algebras. J. Pure Appl. Algebra, 85:57–72, 1993.

[R&W] R. Rosebrugh and R.J. Wood. Distributive laws and factorization. J. Pure Appl. Algebra, to appear, 2001.

[R&T] J. Rosický and W.Tholen. Lax factorization algebras. J. Pure Appl. Algebra, to appear, 2001.

[STR] R. Street. Fibrations and Yoneda’s lemma in a 2-category. In Lecture Notes in Math. 420, 104–133, Springer-Verlag, 1974.

[WD] R.J. Wood. The Centre. Unpublished, 1 page, 1977.

Department of Mathematics and Computer Science Mount Allison University Sackville, NB, E4L 1G6 Canada and Department of Mathematics and Statistics Dalhousie University Halifax, NS, B3H 3J5 Canada

Email: rrosebrugh@mta.ca
      rjwood@mathstat.dal.ca

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