Theory and Applications of Categories, Vol. 30, No. 28, 2015, pp. 985–1000.

SKEW-MONOIDAL REFLECTION AND LIFTING THEOREMS

In memory of Brian Day

STEPHEN LACK AND ROSS STREET

Abstract. This paper extends the Day Reflection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore coalgebras for a comonad.

1. Introduction

In the first part of the paper, we squeeze some more results out of Brian Day’s PhD thesis [2]. The question with which the thesis began was how to extend monoidal structures along dense functors, all at the level of enriched categories. Brian separated the general problem into two special cases. The first case concerned extending along a Yoneda embedding, which led to promonoidal categories and Day convolution [3]. The second case involved extending along a reflection into a full subcategory: the Day Reflection Theorem [4].

While the thesis was about monoidal categories, we can, without even modifying the biggest diagrams, adapt the results to skew monoidal categories. Elsewhere [5, 8] we have discussed convolution. Here we will provide the skew version of the Day Reflection Theorem [4]. The beauty of this variant is further evidence that the direction choices involved in the skew notion are important for organizing, and adding depth to, certain mathematical phenomena.

In the second part of the present paper, the skew warpings of [5] are slightly generalized to involve a skew action; they can in turn be seen as a special case of the skew warpings of [6]. Under certain natural conditions these warpings can be lifted to the category of Eilenberg-Moore coalgebras for a comonad. In particular, this applies to lift skew monoidal structures. For idempotent comonads, we compare the result with our skew reflection theorem.

Both authors gratefully acknowledge the support of the Australian Research Council Discovery Grant DP130101969; Lack acknowledges with equal gratitude the support of an Australian Research Council Future Fellowship

Received by the editors 2015-03-10 and, in revised form, 2015-07-09.

Transmitted by R.J. Wood. Published on 2015-07-12, this revision 2016-09-12.

2010 Mathematics Subject Classification: 18D10; 18A40, 18D15, 18D20.

Key words and phrases: skew monoidal category, reflective subcategory, warping, comonad.

\begin{matrix}c \\ ◯ \end{matrix} Stephen Lack and Ross Street, 2015. Permission to copy for private use granted.

2. Skew monoidal reflection

Recall from [9, 5, 8] the notion of (left) skew monoidal structure on a category 𝒳 . It involves a functor ⊗ : 𝒳 × 𝒳 ⟶ 𝒳 , an object I ∈ 𝒳 , and natural families of (not necessarily invertible) morphisms

α_{A,B,C} : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C),  λ_{A} : I ⊗ A → A,  ρ_{A} : I → A ⊗ I,

satisfying five coherence conditions. It was shown in [1] that these five conditions are independent.

Recall, also from these references, that an opmonoidal structure on a functor L: 𝒳 → 𝒜 consists of a natural family of morphisms

ψ_{X,Y} : L(X ⊗ Y) → LX ¯⊗LY

and a morphism ψ_{0} : LI → Ī satisfying three axioms. We say the opmonoidal functor is normal when ψ_{0} is invertible. We say the opmonoidal functor is strong when ψ_{0} and all ψ_{X,Y} are invertible. However, in this paper, a limited amount of such strength, in which only certain components of ψ are invertible, will be important.

Suppose (𝒳 , ⊗, I, α, λ, ρ) and (𝒜 , ¯⊗, Ī, ᾱ, λ̄, ρ̄) are skew monoidal categories.

2.1. Theorem. Suppose L ⊣ N: 𝒜 → 𝒳 is an adjunction with unit η: 1_{𝒳} ⇒ NL and invertible counit ε: LN ⇒ 1_{𝒜} . Suppose 𝒳 is skew monoidal. There exists a skew monoidal structure on 𝒜 for which L: 𝒳 → 𝒜 is normal opmonoidal with each ψ_{X,NB} invertible if and only if, for all X ∈ 𝒳 and B ∈ 𝒜, the morphism

L(η_{X} ⊗ 1_{NB}): L(X ⊗ NB) → L(NLX ⊗ NB)  (2.1)

is invertible. In that case, the skew monoidal structure on 𝒜 is unique up to isomorphism.

Proof. Suppose 𝒜 has a skew monoidal structure ( ¯⊗, Ī, ᾱ, λ̄, ρ̄) for which L is normal opmonoidal with the ψ_{X,NB} invertible. We have the commutative square

\xymatrix{LX ¯⊗LNB \ar[r]^{Lη_{X} ¯⊗1} \ar[d]_{ψ^{−1}} & LNLX ¯⊗LNB \ar[d]^{ψ^{−1}} \\ L(X ⊗ NB) \ar[r]^{L(η_{X}⊗1)} & L(NLX ⊗ NB)}

in which the vertical arrows are invertible. The top arrow is invertible with inverse ε_{LX} ¯⊗1. So the bottom arrow is invertible.

Conversely, suppose each L(η_{X} ⊗1_{NB}) is invertible. Wishing L to become opmonoidal with the limited strength, we are forced (up to isomorphism) to put

A ¯⊗B = L(NA ⊗ NB) and Ī = LI ,

and to define the constraints ᾱ, λ̄, ρ̄ by commutativity in the following diagrams.

\xymatrix{L((NA ⊗ NB) ⊗ NC) \ar[r]^{L(η⊗1)} \ar[d]_{Lα} & L(NL(NA ⊗ NB) ⊗ NC) \ar[d]^{ᾱ} \\ L(NA ⊗ (NA ⊗ NC)) \ar[r]_{L(1⊗η)} & L(NA ⊗ NL(NB ⊗ NC))}

\xymatrix{L(I ⊗ NA) \ar[r]^{L(η_{I}⊗1)} \ar[d]_{Lλ} & L(NLI ⊗ NA) \ar[d]^{λ̄} & LNA \ar[r]^{ε_{A}} \ar[d]_{Lρ} & A \ar[d]^{ρ̄} \\ LNA \ar[r]_{ε_{A}} & A & L(NA ⊗ I) \ar[r]_{L(1⊗η_{I})} & L(NA ⊗ NLI)}

The definitions make sense because the top arrows of the squares are invertible (while the bottom arrows may not be). Now we need to verify the five axioms. The proofs all proceed by preceding the desired diagram of barred morphisms by suitable invertible morphisms involving only ε_{A}, Lη_{X}, η_{NA}, or L(η_{X} ⊗ 1_{NB}), then manipulating until one can make use of the corresponding unbarred diagram.

The biggest diagram for this is the proof of the pentagon for ᾱ. Fortunately, the proof in Brian Day’s thesis [2] of the corresponding result for closed monoidal categories has the necessary Diagram 4.1.3 on page 94 written without any inverse isomorphisms, so saves us rewriting it here. (The notation is a little different with ψ in place of N and with some of the simplifications we also use below.)

It remains to verify the other four axioms. The simplest of these is

λ̄_{LI} ρ̄_{LI} = λ̄_{LI} ρ̄_{LI}ε_{LI}Lη_{I}
= λ̄_{LI}L(1 ⊗ η_{I})Lρ_{NLI}Lη_{I}
= λ̄_{LI}L(1 ⊗ η_{I})L(η_{I} ⊗ I)Lρ_{I}
= λ̄_{LI}L(η_{I} ⊗ I)L(1 ⊗ η_{I})Lρ_{I}
= ε_{LI}Lλ_{NLI}L(1 ⊗ η_{I})Lρ_{I}
= ε_{LI}Lη_{I}Lλ_{I}Lρ_{I}
= 1_{LI}L(λ_{I}ρ_{I})
= 1_{LI} .

For the other three, to simplify the notation (but to perhaps complicate the reading), we write as if N were an inclusion of a full subcategory, choose L so that the counit is an identity, and write XY for X ⊗ Y. Then we have

λ̄_{B ¯⊗C} ᾱ_{LI,B,C}L(η_{(LI) ¯⊗B}1_{C})L((η_{I}1_{B})1_{C}) = λ̄_{B ¯⊗C}L(1η_{BC})Lα_{LI,B,C}L((η_{I}1_{B})1_{C})
= λ̄_{B ¯⊗C}L(1_{LI}η_{BC})L(η_{I}1_{BC})Lα_{I,B,C}
= λ̄_{B ¯⊗C}L(η_{I}1_{BC})L(1_{I}η_{BC})Lα_{I,B,C}
= Lλ_{BC}Lα_{I,B,C}
= L(λ_{B}1_{C})
= (λ̄_{B} ¯⊗1_{C})L(η_{(LI)B}1_{C})L((η_{I}1_{B})1_{C})

yielding the axiom λ̄_{B ¯⊗C} ᾱ_{LI,B,C} = λ̄_{B} ¯⊗1_{C} on right cancellation.

For the proof of the axiom (1_{A} ¯⊗λ̄_{C})ᾱ_{A,LI,C}(ρ̄_{A} ¯⊗1_{C}) = 1_{A ¯⊗C}, we can look at Diagram 4.1.2 on page 93 of [2]. The required commutativities are all there once we reverse the direction of the right unit constraint which Day calls r instead of ρ.

For the final axiom, we have

ᾱ_{A,B,LI} ρ̄_{A ¯⊗B} = ᾱ_{A,B,LI}L(η_{AB}1_{LI})L(1_{AB}η_{I})Lρ_{AB}
= L(1_{A}η_{BLI})Lα_{A,B,LI}L(1_{AB}η_{I})Lρ_{AB}
= L(1_{A}η_{BLI})L(1_{A}(1_{B}η_{I}))Lα_{A,B,I}Lρ_{AB}
= L(1_{A}η_{BLI})L(1_{A}(1_{B}η_{I}))L(1_{A}ρ_{B})
= 1_{A} ¯⊗ρ̄_{B} .

The desired opmonoidal structure on L is defined by ψ_{0} = 1: LI → Ī and ψ_{X,Y} = L(η_{X} ⊗ η_{Y}): L(X ⊗ Y) → L(NLX ⊗ NLY). The three axioms for opmonoidality are easily checked and we have each ψ_{X,NB} = L(1_{NLX} ⊗ η_{NB})L(η_{X} ⊗ 1_{NB}) invertible.

3. A reflective lemma

In this section we state a standard result in a form required for later reference. For the sake of completeness, we include a proof.

Assume we have an adjunction L ⊣ N: 𝒜 → 𝒳 with unit η: 1_{𝒳} ⇒ NL and counit ε: LN ⇒ 1_{𝒜} . Assume N is fully faithful; that is, equivalently, the counit ε is invertible.

3.1. Lemma. For Z ∈ 𝒳 , the following conditions are equivalent:

(i) there exists A ∈ 𝒜 and Z ≅ NA;

(ii) for all X ∈ 𝒳 , the function 𝒳 (η_{X}, 1): 𝒳 (NLX, Z) → 𝒳 (X, Z) is surjective;

(iii) the morphism η_{Z}: Z → NLZ is a coretraction (split monomorphism);

(iv) the morphism η_{Z}: Z → NLZ is invertible;

(v) for all X ∈ 𝒳 , the function 𝒳 (η_{X}, 1): 𝒳 (NLX, Z) → 𝒳 (X, Z) is invertible.

Proof. (i) ⇒ (ii)

\xymatrix{𝒳 (X, Z) \ar[r]^{≅} \ar[d]_{1} & 𝒳 (X, NA) \ar[r]^{≅} & 𝒜 (LX, A) \ar[d]^{N} \\ 𝒳 (X, Z) & 𝒳 (NLX, Z) \ar[l]_{𝒳 (η_{X},1)} & 𝒳 (NLX, NA) \ar[l]_{≅}}

(ii) ⇒ (iii) Take X = Z and obtain ν: NLZ → Z with 𝒳 (η_{Z}, 1)ν = 1_{Z}.

(iii) ⇒ (iv) If νη_{Z} = 1 then (η_{Z}ν)η_{Z} = 1η_{Z}, so, by the universal property of η_{Z}, we have η_{Z}ν = 1.

(iv) ⇒ (v) The non-horizontal arrows in the commutative diagram

\xymatrix{𝒳 (NLX, Z) \ar[d]_{𝒳 (1,η_{Z})} \ar[rr]^{𝒳 (η_{X},1)} &  & 𝒳 (X, Z) \ar[d]^{𝒳 (1,η_{Z})} \\ 𝒳 (NLX, NLZ) \ar[rr]^{𝒳 (η_{X},1)} &  & 𝒳 (X, NLZ) \\  & 𝒜 (LX, LZ) \ar@{-->}[ul]^{N} \ar@{-->}[ur]_{≅}}

are all invertible, so the horizontal arrows are invertible too.

(v) ⇒ (i) Clearly (v) ⇒ (ii) and we already have (ii) ⇒ (iii) ⇒ (iv), so take A = LZ and the invertible η_{Z}.

4. Skew closed reflection

The Reflection Theorem [4] also deals with closed structure.

If, for objects Y and Z the functor 𝒳 (− ⊗ Y, Z) is representable, say via a natural isomorphism

𝒳 (X ⊗ Y, Z) ≅ 𝒳 (X, [Y, Z]),

we call the representing object [Y, Z] a left internal hom. Recall from Section 8 of [8] that if this exists for all Z, so that −⊗Y has a right adjoint, then 𝒳 becomes left skew closed.

4.1. Theorem. Suppose L ⊣ N: 𝒜 → 𝒳 is an adjunction with unit η: 1_{𝒳} ⇒ NL and invertible counit ε: LN ⇒ 1_{𝒜} . Suppose 𝒳 is skew monoidal and left internal homs of the form [NB, NC] exist for all B, C ∈ 𝒜. The morphisms (5.7) are invertible for all X ∈ 𝒳 and B ∈ 𝒜 if and only if the morphisms

η_{[NB,NC]} : [NB, NC] → NL[NB, NC]  (4.2)

are invertible for all B, C ∈ 𝒜. In that case, the skew monoidal structure abiding on 𝒜, as seen from Theorem 2.1, is left closed. Also, the functor N is strong left closed.

Proof. Consider the following commutative diagram.

\xymatrix{𝒜 (L(NLX ⊗ NB), C) \ar[r]^{𝒜 (L(η⊗1),1)} \ar[d]_{≅} & 𝒜 (L(X ⊗ NB), C) \ar[d]^{≅} \\ 𝒳 (NLX ⊗ NB, NC) \ar[d]_{≅} & 𝒳 (X ⊗ NB, NC) \ar[d]^{≅} \\ 𝒳 (NLX, [NB, NC]) \ar[r]_{𝒳 (η_{X},1)} & 𝒳 (X, [NB, NC])}

Invertibility of the arrows (5.7) is equivalent to the invertibility of the top horizontal arrows. This is equivalent to invertibility of the bottom horizontal arrows. By Lemma 3.1, this is equivalent to invertibility of the arrows (4.2).

For the penultimate sentence of the Theorem, we now have the natural isomorphisms:

𝒜 (A ¯⊗B, C) ≅ 𝒳 (NA ⊗ NB, NC)
≅ 𝒳 (NA, [NB, NC])
≅ 𝒳 (NA, NL[NB, NC])
≅ 𝒜 (A, L[NB, NC])

yielding the left internal hom [B, C] = L[NB, NC] for 𝒜. For the last sentence, we have N[B, C] = NL[NB, NC] ≅ [NB, NC].

Our notation for a right adjoint to X ⊗ − is

𝒳 (X ⊗ Y, Z) ≅ 𝒳 (Y, ⟨X, Z⟩) .

The right internal hom ⟨X, Z⟩ may exist for only certain objects Z. In general, the existence of right homs in a left skew monoidal category does not give a left or right skew closed structure. When they do exist, we can reinterpret a stronger form of the invertibility condition (5.7) of Theorem 2.1.

4.2. Theorem. Suppose L ⊣ N: 𝒜 → 𝒳 is an adjunction with unit η: 1_{𝒳} ⇒ NL and invertible counit ε: LN ⇒ 1_{𝒜} . Suppose 𝒳 is skew monoidal, and left internal homs of the form [Y, NC] and right internal homs of the form ⟨X, NC⟩ exist. The invertibility of one of the following three natural transformations implies invertibility of the other two:

L(η_{X} ⊗ 1_{Y}): L(X ⊗ Y) → L(NLX ⊗ Y) ;  (4.3)

η_{[Y,NC]} : [Y, NC] → NL[Y, NC] ;  (4.4)

⟨η_{X}, NC⟩: ⟨NLX, NC⟩ → ⟨X, NC⟩ .  (4.5)

Proof. Consider the commutative diagram (4.6). Invertibility of any one of the horizontal families in the diagram implies that of the other two. Invertibility of the arrows (4.3) is equivalent to the invertibility of the top horizontal family. By Lemma 3.1, invertibility of the middle horizontal family is equivalent to invertibility of the arrows (4.2). By the Yoneda Lemma, invertibility of the bottom horizontal family is equivalent to invertibility of the arrows (4.5).

\xymatrix{𝒜 (L(NLX ⊗ Y), C) \ar[r]^{𝒜 (L(η⊗1),1)} \ar[d]_{≅} & 𝒜 (L(X ⊗ Y), C) \ar[d]^{≅} \\ 𝒳 (NLX ⊗ Y, NC) \ar[d]_{≅} & 𝒳 (X ⊗ Y, NC) \ar[d]^{≅} \\ 𝒳 (NLX, [Y, NC]) \ar[r]^{𝒳 (η_{X},1)} \ar[d]_{≅} & 𝒳 (X, [Y, NC]) \ar[d]_{≅} \\ 𝒳 (Y, ⟨NLX, NC⟩) \ar[r]^{𝒳 (1,⟨η_{X},1⟩)} & 𝒳 (Y, ⟨X, NC⟩)}  (4.6)

5. An example

This is an example of the opposite (dual) of Theorem 2.1 which we enunciate explicitly as Proposition 5.1 below. Instead of a reflection we have a coreflection. To keep using left skew monoidal categories we also reverse the tensor product. For a monoidal functor R: 𝒳 → 𝒜, we denote the structural morphisms by

φ_{0} : I → RI and φ_{X,Y} : RX ⊗ RY → R(X ⊗ Y) .

5.1. Proposition. Suppose R ⊢ N: 𝒜 → 𝒳 is an adjunction with counit ε: NR ⇒ 1_{𝒳} and invertible unit η: 1_{𝒜} ⇒ RN. Suppose 𝒳 is left skew monoidal. There exists a left skew monoidal structure on 𝒜 for which R: 𝒳 → 𝒜 is normal monoidal each φ_{NA,Y} invertible if and only if, for all A ∈ 𝒜 and Y ∈ 𝒳 , the morphism

R(NA ⊗ ε_{Y}): R(NA ⊗ NRY) → R(NA ⊗ Y)  (5.7)

is invertible.

Consider an injective function μ: U → O. For an object A of the slice category Set/U, we write A_{u} for the fibre over u ∈ U. We have an adjunction

R ⊢ N: Set/U → Set/O

defined by (NA)_{i} = ∑_{μ(u)=i} A_{u} and (RX)_{u} = X_{μ(u)} with invertible unit. The ith component of the counit ε_{X}: NRX → X is the function ∑_{μ(u)=i} X_{μ(u)} → X_{i} which is the identity of X_{i} when i is in the image of μ.

Let 𝒞 be a category with ob𝒞 = O. Then Set/O becomes left skew monoidal on defining the tensor X ⊗ Y by

(X ⊗ Y)_{j} = ∑_{i} X_{i} × 𝒞 (i, j) × Y_{j}

and the (skew) unit I by I_{j} = 1. The associativity constraint α: (X⊗Y)⊗Z → X⊗(Y ⊗Z) is defined by the component functions

∑_{i,j} X_{i} × 𝒞 (i, j) × Y_{j} × 𝒞 (j, k) × Z_{k} → ∑_{i,j} X_{i} × 𝒞 (i, k) × Y_{j} × 𝒞 (j, k) × Z_{k}

induced by the functions

𝒞 (i, j) × 𝒞 (j, k) → 𝒞 (i, k) × 𝒞 (j, k)

taking (a: i → j, b: j → k) to (b ∘ a: i → k, b: j → k). Define λ_{Y} : I ⊗ Y → Y to have j-component ∑_{i} 𝒞 (i, j) × Y_{j} → Y_{j} whose restriction to the ith injection is the second projection onto Y_{j}. Define ρ_{X}: X → X ⊗ I to have j-component X_{j} → ∑_{i} X_{i} × 𝒞 (i, j) equal to the composite of X_{j} → X_{j} × 𝒞 (j, j), x ↦ (x, 1_{j}), with the jth injection.

This provides an example of Proposition 5.1. In fact, it satisfies the stronger condition of the dual to Theorem 4.2. To see that

R(X ⊗ ε_{Y}): R(X ⊗ NRY) → R(X ⊗ Y)

is invertible, since N is fully faithful, we need to prove

G(X ⊗ ε_{Y}): G(X ⊗ GY) → G(X ⊗ Y)

is invertible where G = NR is the idempotent comonad generated by the reflection. Notice that (GX) = U_{j} × X_{j} where U_{j} is the fibre of μ over j ∈ O. Since μ is injective, U_{j} ≅ U_{j} ⊗ U_{j}, so

G(X ⊗ GY)_{j} = U_{j} × (X ⊗ GY)_{j}
= U_{j} × ∑_{i} X_{i} × 𝒞 (i, j) × (GY)_{j}
= ∑_{i} U_{j} × X_{i} × 𝒞 (i, j) × U_{j} × Y_{j}
≅ ∑_{i} U_{j} × X_{i} × 𝒞 (i, j) × Y_{j}
= U_{j} × (X ⊗ Y)_{j}
= G(X ⊗ Y)_{j} .

The resultant left skew structure on Set/U has tensor product

(A ¯⊗B)_{v} = R(NA ⊗ NB)_{v}
= (NA ⊗ NB)_{μ(v)}
= ∑_{i} (NA)_{i} × 𝒞 (i, μ(v)) × (NB)_{μ(v)}
≅ ∑_{u} A_{u} × 𝒞 (μ(u), μ(v)) × B_{v} .

Of course we can see that this is merely the left skew structure on Set/U arising from the category whose objects are the elements u ∈ U and whose morphisms u → v are morphisms μ(u) → μ(v) in 𝒞; that is, the category arising as the full image of the functor μ: U → 𝒞.

As an easy exercise the reader might like to calculate the monoidal structure

RX ¯⊗RY → R(X ⊗ Y)

on R and check that these components are not invertible in general while, of course, they are for X = NA.

6. Skew warpings riding a skew action

We slightly generalize the notion of skew warping defined in [5] to involve an action. This is actually a special case of skew warping on a two-object skew bicategory in the sense of [6].

Let 𝒞 denote a left skew monoidal category. A left skew action of 𝒞 on a category 𝒜 is an opmonoidal functor

𝒞 ⟶ [𝒜 , 𝒜] , X ↦ X ⋆ −  (6.8)

where the skew monoidal (in fact strict monoidal) tensor product on the endofunctor category [𝒜 , 𝒜] is composition. The opmonoidal structure on (6.8) consists of natural families

α_{X,Y,A}: (X ⊗ Y) ⋆ A ⟶ X ⋆ (Y ⋆ A) and λ_{A}: I ⋆ A ⟶ A  (6.9)

subject to the three axioms (6.10), (6.11), (6.12).

\xymatrix{((X ⊗ Y) ⊗ Z) ⋆ A \ar[rr]^{α} \ar[d]_{α⋆1} &  & (X ⊗ Y) ⋆ (Z ⋆ A) \ar[d]^{α} \\ (X ⊗ (Y ⊗ Z)) ⋆ A \ar[r]_{α} & X ⋆ ((Y ⊗ Z) ⋆ A) \ar[r]_{1⋆α} & X ⋆ (Y ⋆ (Z ⋆ A))}  (6.10)

\xymatrix{(I ⊗ Y) ⋆ A \ar[dr]_{λ⋆1} \ar[rr]^{α} &  & I ⋆ (Y ⋆ A) \ar[dl]^{λ} \\  & Y ⋆ A}  (6.11)

\xymatrix{(X ⊗ I) ⋆ A \ar[r]^{α} & X ⋆ (I ⋆ A) \ar[d]^{1⋆λ} \\ X ⋆ A \ar[r]_{1} \ar[u]^{ρ⋆1} & X ⋆ A}  (6.12)

A category 𝒜 equipped with a skew action of 𝒞 is called a skew 𝒞 -actegory. A skew left warping riding the skew action of 𝒞 on 𝒜 consists of the following data:

(a) a functor T : 𝒜 ⟶ 𝒞;

(b) an object K of 𝒜;

(c) a natural family of morphisms v_{A,B} : T (T A ⋆ B) ⟶ T A ⊗ T B in 𝒞;

(d) a morphism v_{0} : T K ⟶ I; and,

(e) a natural family of morphisms k_{A} : A ⟶ T A ⋆ K; such that the following five diagrams commute.

\xymatrix{T (T A ⋆ B) ⊗ T C \ar[rr]^{v_{A,B}⊗1} &  & (T A ⊗ T B) ⊗ T C \ar[d]^{α_{T A,T B,T C}} \\ T (T (T A ⋆ B) ⋆ C) \ar[u]^{v_{T A⋆B,C}} \ar[d]_{T (v_{A,B}⋆1)} &  & T A ⊗ (T B ⊗ T C) \\ T ((T A ⊗ T B) ⋆ C) \ar[dr]_{T α_{T A,T B,C}} &  & T A ⊗ T (T B ⋆ C) \ar[u]_{1⊗v_{B,C}} \\  & T (T A ⋆ (T B ⋆ C)) \ar@{-->}[ur]_{v_{A,T B⋆C}}}  (6.13)

\xymatrix{ & T K ⊗ T B \ar[dr]^{v_{0}⊗1_{T B}} \\ T (T K ⋆ B) \ar[ur]^{v_{K,B}} \ar[d]_{T (v_{0}⋆1_{B})} &  & I ⊗ T B \ar[d]^{λ_{T B}} \\ T (I ⊗ B) \ar[rr]_{T λ_{B}} &  & T B}  (6.14)

\xymatrix{T (T A ⋆ K) \ar[r]^{v_{A,K}} & T A ⊗ T K \ar[d]^{1⊗v_{0}} \\ T A \ar[u]^{T k_{A}} \ar[r]_{ρ_{T A}} & T A ⊗ I}  (6.15)

\xymatrix{T (T A ⋆ B) ⋆ K \ar[r]^{v_{A,B}⋆1_{K}} & (T A ⊗ T B) ⋆ K \ar[d]^{α_{T A,T B,K}} \\ T A ⋆ B \ar[u]^{k_{T A⋆B}} \ar[r]_{1_{T A}⋆k_{B}} & T A ⋆ (T B ⋆ K)}  (6.16)

\xymatrix{T K ⋆ K \ar[r]^{v_{0}⋆1_{K}} & I ⋆ K \ar[d]^{λ_{K}} \\ K \ar[r]_{1_{K}} \ar[u]^{k_{K}} & K}  (6.17)

6.1. Example. A skew warping on a skew monoidal category (in the sense of [5]) is just the case where 𝒜 = 𝒞 with tensor as action.

Just as in Proposition 3.6 of [5], we obtain a skew monoidal structure from a skew warping.

6.2. Proposition. A skew left warping riding a left skew action of a left skew monoidal category 𝒞 on a category 𝒜 determines left skew monoidal structure on 𝒜 as follows:

(a) tensor product functor A ¯⊗B = T A ⋆ B;

(b) unit K;

(c) associativity constraint

T (T A ⋆ B) ⋆ C ⟶^{v_{A,B}⋆1_{C}} (T A ⊗ T B) ⋆ C ⟶^{α_{T A,T B,C}} T A ⋆ (T B ⋆ C) ;

(d) left unit constraint

T K ⋆ B ⟶^{v_{0}⋆1_{B}} I ⋆ B ⟶^{λ_{B}} B ;

(e) right unit constraint

A ⟶^{k_{A}} T A ⋆ K .

There is an opmonoidal functor (T, v_{0}, v_{A,B}) : (𝒜 , ¯⊗, K) ⟶ (𝒞 , ⊗, I).

6.3. Example. Skew warpings are more basic than skew monoidal structures in the following sense. Just pretend, for the moment, that we do not know what a skew monoidal (or even monoidal) category is, except that we would like endofunctor categories to be examples. For any category 𝒜, the endofunctor category 𝒞 = [𝒜 , 𝒜] acts on 𝒜 by evaluation; as a functor (6.8), the action is the identity. A left skew warping riding this action could be taken as the definition of a left skew monoidal structure on 𝒜.

7. Comonads on skew actegories

For a left skew monoidal category 𝒞, let Cat^{𝒞} denote the 2-category whose objects are left skew 𝒞 -actegories as defined in Section 6. A morphism is a functor F: 𝒜 → ℬ equipped with a natural family of morphisms

γ_{X,A}: X ⋆ F A ⟶ F (X ⋆ A)  (7.18)

such that (7.19) and (7.20) commute.

\xymatrix{(X ⊗ Y) ⋆ F A \ar[rr]^{γ} \ar[d]_{α} &  & F ((X ⊗ Y) ⋆ A) \ar[d]^{F α} \\ X ⋆ (Y ⋆ F A) \ar[r]_{1⋆γ} & X ⋆ F (Y ⋆ A) \ar[r]_{γ} & F (X ⋆ (Y ⋆ A))}  (7.19)

\xymatrix{I ⋆ F A \ar[r]^{γ} \ar[dr]_{λ} & F (I ⋆ A) \ar[d]^{F λ} \\  & F A}  (7.20)

Such a morphism is called strong when each γ_{X,A} is invertible. A 2-cell ξ: (F, γ) ⇒ (G, γ) in Cat^{𝒞} is a natural transformation ξ: F ⇒ G such that (7.21) commutes.

\xymatrix{X ⋆ F A \ar[r]^{γ_{X,A}} \ar[d]_{1⋆ξ_{A}} & F (X ⋆ A) \ar[d]^{ξ_{X⋆A}} \\ X ⋆ GA \ar[r]_{γ_{X,A}} & G(X ⋆ A)}  (7.21)

As usual with actions, there is another way to view the 2-category Cat^{𝒞} . Regard 𝒞 as the homcategory of a 1-object skew bicategory Σ𝒞 in the sense of Section 3 of [6]. A left skew 𝒞 -actegory is an oplax functor 𝒜 : Σ𝒞 → Cat. A morphism (F, γ): 𝒜 → ℬ in Cat^{𝒞} can be identified with a lax natural transformation between the oplax functors. The 2-cells are the modifications.

We are interested in comonads (𝒜 , G, γ, δ, ε) in the 2-category Cat^{𝒞} . These are objects of the 2-category Mnd_{∗}(Cat^{𝒞} ) as defined in [7]. Alternatively, they are oplax functors Σ𝒞 → Mnd_{∗}(Cat). For later reference, apart from the conditions for being a comonad on 𝒜 and the conditions (7.19) and (7.20), we require commutativity of (7.22).

\xymatrix{X ⋆ GA \ar[rr]^{γ} \ar[d]_{1⋆δ} &  & G(X ⋆ A) \ar[d]^{δ} & X ⋆ GA \ar[r]^{γ} \ar[dr]_{1⋆ε} & G(X ⋆ A) \ar[d]^{ε} \\ X ⋆ G^{2}A \ar[r]_{γ} & G(X ⋆ GA) \ar[r]_{Gγ} & G^{2}(X ⋆ A) &  & X ⋆ A}  (7.22)

The Eilenberg-Moore coalgebra construction (𝒜 , G, δ, ε) ↦ 𝒜^{G} is the 2-functor right adjoint to the 2-functor Cat → Mnd_{∗}(Cat) taking each category to that category equipped with its identity comonad.

7.1. Proposition. For each comonad (𝒜 , G, γ, δ, ε) in the 2-category Cat^{𝒞} , the Eilenberg-Moore coalgebra category 𝒜^{G} becomes a left skew 𝒞 -actegory with skew action

X ⋆ (A ⟶^{a} GA) = (X ⋆ A, X ⋆ A ⟶^{X⋆a} X ⋆ GA ⟶^{γ_{X,A}} G(X ⋆ A)) .

This provides the Eilenberg-Moore construction in the 2-category Cat^{𝒞} (in the sense of [7]).

Proof. Compose the oplax functor Σ𝒞 → Mnd_{∗}(Cat) corresponding to (𝒜 , G, γ, δ, ε) with the Eilenberg-Moore 2-functor Mnd_{∗}(Cat) → Cat.

Let U: 𝒜^{G} → 𝒜 denote the underlying functor (A, a) ↦ A.

7.2. Proposition. Suppose (T, K, v, v_{0}, k) is a skew left warping riding the 𝒞 -actegory 𝒜. Suppose (𝒜 , G, γ, δ, ε) is a comonad in the 2-category Cat^{𝒞} for which all morphisms of the form γ_{T A,K} and γ_{T A,T B⋆K} are invertible. Then (T U, (GK, δ_{K}), v, v′_{0}, k′) is a skew left warping riding the 𝒞 -actegory 𝒜^{G} of Proposition 7.1, where v′_{0} = v_{0} ∘ T ε_{K} and k′_{(A,a)} = γ_{T A,K}^{−1} ∘ Gk_{A} ∘ a.

Proof. First we need to see that k′_{(A,a)} : (A, a) → (T A ⋆ GK, γ_{T A,GK} ∘ (1 ⋆ δ_{K})) is a Gcoalgebra morphism. This uses the first diagram of (7.22), naturality of δ with respect to k_{A}, and the coassociativity of the coaction a: A → GA.

It remains to verify the five axioms (6.13), (6.14), (6.15), (6.16), (6.17). Since only v is involved in (6.13), it follows from axiom (6.13) for the original skew warping. For (6.14), we have the diagram

\xymatrix{T (T GK ⋆ B) \ar[r]^{T (T ε_{K}⋆1)} \ar[d]_{v_{GK,B}} & T (T K ⋆ B) \ar[rr]^{T (v_{0}⋆1)} \ar[d]^{v_{K,B}} &  & T (I ⊗ B) \ar[d]^{T λ_{B}} \\ T GK ⊗ T B \ar[r]_{T ε_{K}⊗1} & T K ⊗ T B \ar[r]_{v_{0}⊗1} & I ⊗ T B \ar[r]_{λ_{T B}} & T B}

which uses naturality of v and axiom (6.14) for the original skew warping. The next diagram proves (6.15).

\xymatrix{T A \ar[r]^{T a} \ar[dd]_{1} & T GA \ar[ddl]^{T ε_{A}} \ar[r]^{T Gk_{A}} & T G(T A ⋆ K) \ar[r]^{T γ^{−1}} \ar[d]_{T ε_{T A⋆K}} & T (T A ⋆ GK) \ar[d]^{v_{A,GK}} \ar[dl]^{T (1⋆ε_{K})} \\  &  & T (T A ⋆ K) \ar@{-->}[dr]^{v_{A,K}} & T A ⊗ T GK \ar[d]^{1⊗T ε_{K}} \\ T A \ar[rr]_{ρ_{T A}} \ar[urr]^{T k_{A}} &  & T A ⊗ I & T A ⊗ T K \ar[l]^{1⊗v_{0}}}

Precomposing the next diagram with 1 ⋆ b: T A ⋆ B → T A ⋆ GB proves (6.16). Take note here of which components of γ are required to be invertible.

\xymatrix{T A ⋆ GB \ar[d]_{1⋆Gk_{B}} \ar[r]^{γ} & G(T A ⋆ B) \ar[r]^{Gk_{T A⋆B}} \ar[dddl]^{G(1⋆k_{B})} & G(T (T A ⋆ B) ⋆ K) \ar[d]^{γ^{−1}} \ar[dl]_{1} \\ T A ⋆ G(T B ⋆ K) \ar[dd]_{γ} & G(T (T A ⋆ B) ⋆ K) \ar[d]^{G(v_{T A,B}⋆1)} & T (T A ⋆ B) ⋆ GK \ar[d]^{v_{A,B}⋆1} \ar[l]_{γ} \\  & G((T A ⊗ T B) ⋆ K) \ar[dl]_{Gα} & (T A ⊗ T B) ⋆ GK \ar[d]^{α} \ar[l]_{γ} \\ G(T A ⋆ (T B ⋆ K)) \ar[r]_{γ^{−1}} & T A ⋆ G(T B ⋆ K) \ar[r]_{1⋆γ^{−1}} & T A ⋆ (T B ⋆ GK)}

Then

\xymatrix{GK \ar[r]^{δ_{K}} \ar[dr]_{1} & G^{2}K \ar[r]^{Gk_{GK}} \ar[d]^{Gε_{K}} & G(T GK ⋆ K) \ar[r]^{γ^{−1}} \ar[d]^{G(T ε_{K}⋆1)} & T GK ⋆ GK \ar[d]^{T ε_{K}⋆1} \\  & GK \ar[r]^{Gk_{K}} \ar[ddr]_{1} & G(T K ⋆ K) \ar[r]^{γ^{−1}} \ar[d]^{G(v_{0}⋆1)} & T K ⋆ GK \ar[d]^{v_{0}⋆1} \\  &  & G(I ⋆ K) \ar[d]^{Gλ_{K}} & I ⋆ GK \ar[d]^{λ_{GK}} \ar[l]^{γ} \\  &  & GK \ar[r]_{1} & GK}

yields (6.17), which completes the proof.

7.3. Corollary. Under the hypotheses of Proposition 7.2, the functor U: 𝒜^{G} → 𝒜 preserves the tensor products obtained from the skew warpings via Proposition 6.2 and becomes opmonoidal when equipped with the unit constraint ε_{I}: GI → I.

7.4. Corollary. Since 𝒞 is an object of Cat^{𝒞} with its own tensor product as skew action, and since it supports the identity skew warping, for any comonad (𝒞 , G, γ, δ, ε) in the 2-category Cat^{𝒞} , Corollary 7.3 applies to give a skew monoidal structure on 𝒞^{G} with U: 𝒞^{G} → 𝒞 opmonoidal.

7.5. Remark. If the comonad of Corollary 7.4 is idempotent and (G, γ) is strong in Cat^{𝒞} then U: 𝒞^{G} → 𝒞 is a coreflection and the dual of Theorem 2.1 applies. The same skew monoidal structure on 𝒞^{G} is obtained as in Corollary 7.4. The point is that the diagram (7.23) commutes by G applied to the right-hand diagram of (7.22) and a counit property of the comonad. So Theorem 2.1 appears to be a stronger result than Corollary 7.4 in the idempotent comonad case.

\xymatrix{G(X ⊗ GY) \ar[r]^{G(1⊗ε_{Y} )} \ar[d]_{Gγ_{X,GY}} & G(X ⊗ Y) \ar[d]^{δ_{X⊗Y}} \\ GG(X ⊗ Y) \ar[r]_{1} \ar@{-->}[ur]_{Gε_{X⊗Y}} & GG(X ⊗ Y)}  (7.23)

8. The example of Section 5 without injectivity

Let 𝒞 be a category with object set O and morphism set E, and let ξ: U → O be a function (not necessarily injective). Composition with ξ induces a comonadic functor N = ξ_{!} : Set/U → Set/O; write R = ξ^{∗} for the right adjoint, given by pullback. The comonad G = NR = ξ_{!}ξ^{∗} is given by − ×_{O} U.

The category structure on 𝒞 induces a skew monoidal structure on Set/O, with tensor product X ⊗ Y given by:

(X ⊗ Y)_{j} = ∑_{i} X_{i} × 𝒞 (i, j) × Y_{j}

and so X ⊗ − is given by X ×_{O} E ×_{O} −. The unit I is the terminal object 1: O → O.

From the formulas for G and X ⊗ − involving products in Set/O, it is clear that we have natural isomorphisms γ_{X,Y} : X ⊗ GY ≅ G(X ⊗ Y), compatible with the comonad structure, in the sense that the diagrams (7.22) commute. Almost as easy is compatibility with the associativity map and left unit constraint in the sense of diagrams (7.19) and (7.20).

So we have a category 𝒞 with object-set O, giving rise to the skew monoidal category Set/O, and the comonad G = ξ_{!}ξ^{∗} on Set/O as required by Corollary 7.4. This gives rise to a skew monoidal structure on Set/U, with unit ξ^{∗}I; in other words with unit I′ equal to the terminal object 1: U → U. It is clear from the construction that this tensor product preserves colimits in each variable. So from the general theory, it must correspond to some category 𝒜 with object-set U.

Since ξ^{∗}: Set/U → Set/O is opmonoidal, ξ is the object part of a functor F: 𝒜 → 𝒞. Since ξ^{∗} preserves the tensor, the functor F is fully faithful.

Thus 𝒜 must in fact be obtained from ξ: U → 𝒞 via the factorization into a bijectiveon-objects functor followed by a fully faithful functor.

References

[1] Jim Andrianopoulos, Remarks on units of skew monoidal categories, arXiv:1505.02048v1. 986

[2] Brian J. Day, “Construction of Biclosed Categories” (PhD Thesis, University of New

    South Wales, 1970) http://www.math.mq.edu.au/~street/DayPhD.pdf. 985, 987,

[3] Brian J. Day, On closed categories of functors, Lecture Notes in Mathematics 137 (Springer-Verlag, 1970)1–38. 985

[4] Brian J. Day, A reflection theorem for closed categories, J. Pure Appl. Algebra 2 (1972) 1–11. 985, 989

[5] Stephen Lack and Ross Street, Skew monoidales, skew warpings and quantum categories, Theory and Applications of Categories 26 (2012) 385–402. 985, 986, 993, 995

[6] Stephen Lack and Ross Street, On monads and warpings, Cahiers de topologie et géométrie différentielle catégoriques 60 (2014) 244–266. 985, 993, 996

[7] Ross Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149–168. 996, 997

[8] Ross Street, Skew-closed categories, Journal of Pure and Applied Algebra 217 (June 2013) 973–988. 985, 986, 989

[9] Kornel Szlachányi, Skew-monoidal categories and bialgebroids, Advances in Math. 231 (2012) 1694–1730. 986

Department of Mathematics, Macquarie University NSW 2109, Australia

Email: steve.lack@mq.edu.au, ross.street@mq.edu.au
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