On lifting of biadjoints and lax algebras

By the biadjoint triangle theorem, given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then this lifting is also right biadjoint provided that $\mathfrak{A} $ has codescent objects. In this paper, we give general results on lifting of biadjoints. As a consequence, we get a \textit{biadjoint triangle theorem} which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. More precisely, we prove that, denoting by $\ell :\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} \to\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg}_\ell $ the inclusion, if $R: \mathfrak{A}\to\mathfrak{B} $ is right biadjoint and has a lifting $J: \mathfrak{A} \to \mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $, then $\ell\circ J$ is right biadjoint as well provided that $\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such theorem, we study the descent objects and the lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence. In particular, we give the construction of the left $2$-adjoint to the inclusion of the strict algebras into the lax algebras.


Introduction
This paper has three main theorems. One of them (Theorem 2.3) is about lifting of biadjoints: a generalization of Theorem 4.4 of [27]. The others (Theorem 5.2 and Theorem 5.3) are consequences of the former on lifting biadjoints to the 2-category of lax algebras. These results can be seen as part of what is called two-dimensional universal algebra, or, more precisely, two-dimensional monad theory: for an idea of the scope of this field (with applications), see for instance [5,33,34,19,22,13,7,27,28].
There are several theorems about lifting of adjunctions in the literature [1,6,16,38,44], including, for instance, adjoint triangle theorems [9,2]. Although some of these results can be proved for enriched categories or more general contexts [32,27], they often are not enough to deal with problems within 2-dimensional category theory. The reason is that these problems involve concepts that are not of (strict/usual) Cat-enriched category theory nature, as it is explained in [24,5].
For example, in 2-dimensional category theory, the enriched notion of monad, the 2monad, gives rise to the 2-category of (strict/enriched) algebras, but it also gives rise to the 2-category of pseudoalgebras and the 2-category of lax algebras. The last two types of 2-categories of algebras (and full sub-2-categories of them) are usually of the most interest despite the fact that they are not "strict" notions.
In short, most of the aspects of 2-dimensional universal algebra are not covered by the usual Cat-enriched category theory of [17,10] or by the formal theory of monads of [35]. Actually, in the context of pseudomonad theory, the appropriate analogue of the formal theory of monads is the formal theory (and definition) of pseudomonads of [29,21]. In this direction, the problem of lifting biadjunctions is the appropriate analogue of the problem of lifting adjunctions.
Some results on lifting of biadjunctions are consequences of the biadjoint triangle theorems proved in [27]. One of these consequences is the following: let T be a pseudomonad on a 2-category B. Assume that R : A Ñ B, J : A Ñ Ps-T -Alg are pseudofunctors such that we have the pseudonatural equivalence below. If R is right biadjoint then J is right biadjoint as well provided that A has some needed codescent objects.
One simple application of this result is, for instance, within the 2-monadic approach to coherence [27]: roughly, the 2-monadic approach to coherence is the study of biadjunctions and 2-adjunctions between the many types of 2-categories of algebras rising from a given 2-monad. This allows us to prove "general coherence results" [33,5,22] which encompass many coherence results -such as the strict replacement of monoidal categories, the strict/flexible replacement of bicategories [22,23], the strict/flexible replacement of pseudofunctors [5] and so on [12].
If T 1 is a 2-monad, the result described above gives the construction of the left biadjoint to the inclusion T 1 -Alg s Ñ Ps-T 1 -Alg subject to the existence of some codescent objects in T 1 -Alg s . The strict version of the biadjoint triangle theorem of [27] shows when we can get a genuine left 2-adjoint to this inclusion (and also studies when the unit is a pseudonatural equivalence), getting the coherence results of [22] w.r.t. pseudoalgebras. In this paper, we prove Theorem 2.3 which is a generalization of Theorem 4.3 of [27] on biadjoint triangles. Our result allows us to study lifting of biadjunctions to lax algebras. Hence, we prove the analogue of the result described above for lax algebras. More precisely, let T be a pseudomonad on a 2-category B and let ℓ : Lax-T -Alg Ñ Lax-T -Alg ℓ be the locally full inclusion of the 2-category of lax T -algebras and T -pseudomorphisms into the 2-category of lax T -algebras and lax T -morphisms. Assuming that is a pseudonatural equivalence in which R is right biadjoint, we prove that J is right biadjoint as well, provided that A has some needed codescent objects. Moreover, ℓ˝J is right biadjoint if and only if A has lax codescent objects of some special diagrams. Still, we study when we can get strict left 2-adjoints to J and ℓ˝J, provided that J is a 2-functor.
As an immediate application, we also prove general coherence theorems related to the work of [22]: we get the construction of the left biadjoints of the inclusions T is a pseudomonad and T 1 -Alg s , Ps-T -Alg have some needed lax codescent objects. We start in Section 1 establishing our setting: we recall basic results and definitions, such as weighted bicolimits and computads. In Section 2, we give our main theorems on lifting of biadjoints: these are simple but pretty general results establishing basic techniques to prove theorems on lifting of biadjoints. These techniques apply to the context of [27] but also apply to the study of other biadjoint triangles, such as our main application -which is the lifting of biadjoints to the 2-category of lax algebras.
Then, we restrict our attention to 2-dimensional monad theory: in order to do so, we present the weighted bicolimits called lax codescent objects and codescent objects in Section 3. Our approach to deal with descent objects is more general than the approach of [40,27,28], since it allows us to study descent objects of more general diagrams. Thanks to this approach, in Section 4, after defining pseudomonads and lax algebras, we show how we can get the category of pseudomorphisms between two lax algebras as a descent object at Proposition 4.5. This result also shows how we can get the category of lax morphisms between two lax algebras as a lax descent object.
In Section 5, we prove our main results on lax algebras: Theorem 5.2 and Theorem 5.3. They are direct consequences of the results of Section 2 and Section 4, but we also give explicit calculations of the weighted bicolimits/weighted 2-colimits needed in A to get the left biadjoints/left 2-adjoints. We finish the paper in Section 6 giving straightforward applications of our results within the context of the 2-monadic approach to coherence explained above.
This work was realized in the course of my PhD studies at University of Coimbra. I wish to thank my supervisor Maria Manuel Clementino for her support, attention and useful feedback.

Preliminaries
In this section, we recall some basic results related to our setting, which is the tricategory 2-CAT of 2-categories, pseudofunctors, pseudonatural transformations and modifications. Most of what we need was originally presented in [3,37,39,40]. Also, for elements of enriched category theory, see [17]. We use the notation established in Section 2 of [27] for pseudofunctors, pseudonatural transformations and modifications.
We start with considerations about size. Let cat " int(Set) be the cartesian closed category of small categories. Also, assume that Cat, CAT are cartesian closed categories of categories in two different universes such that cat is an internal category of the subcategory of discrete categories of Cat, while Cat is itself an internal category of the subcategory of discrete categories of CAT. Since these three categories of categories are complete and cartesian closed, they are enriched over themselves and they are cocomplete and complete in the enriched sense.
Henceforth, Cat-category is a Cat-enriched category such that its collection of objects is a discrete category of CAT. Thereby, we have that Cat-categories can be seen as internal categories of CAT such that their categories of objects are discrete. In other words, there is a full inclusion Cat-CAT Ñ int(CAT) in which Cat-CAT denotes the category of Cat-categories. Moreover, since there is a forgetful functor int(CAT) Ñ CAT, there is a forgetful functor Cat-CAT Ñ CAT.
So, we adopt the following terminology: Firstly, a 2-category is a Cat-category. Secondly, a possibly (locally) large 2-category is an internal category of CAT such that its category of objects is discrete. Finally, a small 2-category is a 2-category which can be seen as an internal category of cat.
Let W : S Ñ Cat, W 1 : S op Ñ Cat and D : S Ñ A be 2-functors with small domains. If it exists, we denote the weighted limit of D with weight W by tW, Du. Dually, we denote by W 1˚D the weighted colimit provided that it exists.
1.1. Remark. Consider the category, denoted in this remark by S ist with two objects and two parallel arrows between them. We can define the weight in which 2 is the category with two objects and only one morphism between them and I is the terminal category. The colimits with this weight are called coinserters (see [18]).
The bicategorical Yoneda Lemma says that there is a pseudonatural equivalence rS, Cats P S pSpa,´q, Dq » Da given by the evaluation at the identity, in which rS, Cats P S is the possibly large 2-category of pseudofunctors, pseudonatural transformations and modifications S Ñ Cat. As a consequence, the Yoneda embedding Y S op : S op Ñ rS, Cats P S is locally an equivalence (i.e. it induces equivalences between the hom-categories). If W : S Ñ Cat, D : S Ñ A are pseudofunctors with a small domain, recall that the weighted bilimit, when it exists, is an object tW, Du bi of A endowed with a pseudonatural equivalence (in X) ApX, tW, Du bi q » rS, Cats P S pW, ApX, D´qq.
The dual concept is that of weighted bicolimit: if W 1 : S op Ñ Cat, D : S Ñ A are pseudofunctors, the weighted bicolimit W 1˚b i D is the weighted bilimit tW 1 , D op u bi in A op . That is to say, it is an object W 1˚b i D of A endowed with a pseudonatural equivalence (in X) ApW 1˚b i D, Xq » rS op , Cats P S pW 1 , ApD´, Xqq. By the bicategorical Yoneda Lemma, tW, Du bi , W 1˚b i D are unique up to equivalence, if they exist.

1.2.
Remark. If W and D are 2-functors, tW, Du bi and tW, Du may exist, without being equivalent to each other. This problem is related to the notion of flexible presheaves/weights (see [4]): whenever W is flexible, these two types of limits are equivalent, if they exist.

Remark.
Recall that a biadjunction pE % R, ρ, ε, v, wq has an associated pseudonatural equivalence χ : Bp´, R´q » ApE´,´q, in which If L, U are 2-functors, we say that L is left 2-adjoint to U whenever there is a biadjunction pL % U, η, ε, s, tq in which s, t are identities and η, ε are 2-natural transformations. In this case, we say that pL % U, η, εq is a 2-adjunction.

On computads.
We employ the concept of computad, introduced in [37], to define the 2-categories 9 ∆ ℓ , 9 ∆, ∆ ℓ in Section 3. For this reason, we give a short introduction to computads in this subsection.
Herein a graph G " pd 1 , d 0 q is a pair of functors d 0 , d 1 : G 1 Ñ G 0 between discrete categories of CAT. In this case, G 0 is called the collection of objects and, for each pair of objects pa, bq of G 0 , d´1 0 paq X d´1 1 pbq " Gpa, bq is the collection of arrows between a and b. A graph morphism T between G, G 1 is a function T : G 0 Ñ G 1 0 endowed with a function T pa,bq : Gpa, bq Ñ G 1 pTa, Tbq for each pair pa, bq of objects in G 0 . That is to say, a graph morphism T " pT 1 , T 0 q is a natural transformation between graphs. The category of graphs is denoted by GRPH.
We also define the full subcategories of GRPH, denoted by Grph and grph: the objects of Grph are graphs in the subcategory of discrete categories of Cat and the objects of grph, called small graphs, are graphs in the subcategory of discrete categories of cat. The forgetful functors CAT Ñ GRPH, Cat Ñ Grph and cat Ñ grph have left adjoints.
We denote by F : GRPH Ñ CAT the functor left adjoint to CAT Ñ GRPH and F the monad on GRPH induced by this adjunction. If G " pd 1 , d 0 q is an object of GRPH, F G is the coinserter of this diagram pd 1 , d 0 q.
Recall that F G, called the category freely generated by G, can be seen as the category with the same objects of G but the arrows between two objects a, b are the paths between a, b (including the empty path): composition is defined by juxtaposition of paths.

Definition. [Computad]
A computad c is a graph c G endowed with a graph cpa, bq such that cpa, bq 0 "`F c G˘p a, bq for each pair pa, bq of objects of c G .

Remark.
A small computad is a computad c such that the graphs c G and cpa, bq are small for every pair pa, bq of objects of c G . Such a computad can be entirely described by a diagram A morphism T between computads c, c 1 is a graph morphism T G : c G Ñ c 1 G endowed with a graph morphism T pa,bq : cpa, bq Ñ cpT G a, T G bq for each pair of objects pa, bq in c G such that T pa,bq 0 coincides with F pT G q pa,bq . The category of computads is denoted by CMP. We can define a forgetful functor U : Cat-CAT Ñ CMP in which pUAq G is the underlying graph of the underlying category of A. Recall that, for each pair of objects pa, bq of pUAq G , an object f of pUAq pa, bq is a path between a and b. Then the composition defines a map˝: pUAq pa, bq Ñ Apa, bq and we can define the arrows of the graphs pUAq pa, bq as follows: pUAq pa, bqpf, gq :" Apa, bqp˝pf q,˝pgqq.
The left reflection of a small computad c along U is denoted by Lc and called the 2category freely generated by c. The underlying category of Lc is F c G and its 2-dimensional structure is constructed below.
The diagram of Remark 1.7 induces the graph morphisms ppB 0 , idq, idq and ppB 1 , idq, idq above between a graph denoted by c´and Fc G . Using the multiplication of the monad F, these morphisms induce two morphisms Fc´Ñ Fc G . These two morphisms define in particular the graph c´below and F c´defines the 2-dimensional structure of Lc.
Defining all compositions by juxtaposition, we have a sesquicategory (see [41]). We define Lc to be the 2-category obtained from the quotient of this sesquicategory, forcing the interchange laws.

Remark. Let
Preord the category of preordered sets. We have an inclusion Preord Ñ Cat which is right adjoint. This adjunction induces a 2-adjunction between Preord-CAT and Cat-CAT.
If c is a computad, the locally preordered 2-category freely generated by c is the image of Lc by the left 2-adjoint functor Cat-CAT Ñ Preord-CAT.

Lifting of biadjoints
In this section, we assume that a small weight W : S Ñ Cat, a right biadjoint pseudofunctor R : A Ñ B and a pseudofunctor J : A Ñ C are given. We investigate whether J is right biadjoint.
We establish Theorem 2.3 and its immediate corollary on biadjoint triangles. We omit the proof of Lemma 2.2, since it is analogous to the proof of Lemma 2.1.

Lemma.
Assume that, for each object y of C, there are pseudofunctors D y : SˆA Ñ Cat, A y : S op Ñ A such that D y » ApA y´,´q and tW, D y p´, Aqu bi » Cpy, JAq for each object A of A. The pseudofunctor J is right biadjoint if and only if, for every object y of C, the weighted bicolimit W˚b i A y exists in A. In this case, J is right biadjoint to G, defined by Gy " W˚b i A y .
Thereby, an object Gy of A is the weighted bicolimit W˚b i A y if and only if there is a pseudonatural equivalence (in A) ApGy, Aq » tW, ApA y´, Aqu bi » Cpy, JAq. That is to say, an object Gy of A is the weighted bicolimit W˚b i A y if and only if Gy is a birepresentation of Cpy, J´q.

Lemma.
Assume that J, W are 2-functors and, for each object y of C, there are 2-functors D y : SˆA Ñ Cat, A y : S op Ñ A such that there is a 2-natural isomorphism D y -ApA y´,´q and tW, D y p´, Aqu -Cpy, JAq for every object A of A. The 2-functor J is right 2-adjoint if and only if, for every object y of C, the weighted colimit W˚A y exists in A. In this case, J is right 2-adjoint to G, defined by Gy " W˚A y .
Let D : SˆA Ñ Cat be a pseudofunctor. We denote by |D| : S 0ˆA Ñ Cat the restriction of D in which S 0 is the discrete 2-category of the objects of S. Also, herein we say that |D| can be factorized through R˚:" Bp´, R´q if there are a pseudofunctor D 1 : S 0 Ñ B op and a pseudonatural equivalence |D| » R˚˝pD 1ˆI d A q.
2.3. Theorem. Assume that, for each object y of C, there is a pseudofunctor D y : SÂ Ñ Cat such that |D y | can be factorized through R˚and tW, D y p´, Aqu bi » Cpy, JAq for every object A of A. In this setting, for each object y of C there are a pseudofunctor A y : S op Ñ A and a pseudonatural equivalence D y » ApA y´,´q .
As a consequence, the pseudofunctor J is right biadjoint if and only if, for every object y of C, the weighted bicolimit W˚b i A y exists in A. In this case, J is right biadjoint to G, defined by Gy " W˚b i A y .
Proof. Indeed, if E : B Ñ A is left biadjoint to R, then there is a pseudonatural equivalence R˚» ApE´,´q. Therefore, by the hypotheses, for each object Y of C, there is a pseudofunctor D 1 y : S 0 Ñ B op such that |D y | » R˚˝pD 1 yˆI d A q » ApED 1 y´,´q . From the bicategorical Yoneda lemma, it follows that we can choose a pseudofunctor A y : S op Ñ A which is an extension of ED 1 y such that ApA y´,´q » D y . The consequence follows from Lemma 2.1.

Corollary.
[Biadjoint Triangle] Assume that V : C 1 Ñ C is a pseudofunctor and is a commutative triangle of pseudofunctors satisfying the following: for each object y of C, there is a pseudofunctor D y : SˆC 1 Ñ Cat such that |D y | can be factorized through Uå nd tW, D y p´, xqu bi » Cpy, Vxq for each object x of C 1 . In this setting, for each object y of C, there is a pseudofunctor A y : S op Ñ A such that D y p´, J´q » ApA y´,´q .
As a consequence, the pseudofunctor V˝J is right biadjoint if and only if, for every object y of C, the weighted bicolimit W˚b i A y exists in A. In this case, V˝J is right biadjoint to G, defined by Gy " W˚b i A y .
Proof. We prove that D y :" D y p´, J´q satisfies the hypotheses of Theorem 2.3. We have that, for each object y of C and each object A of A, tW, D y p´, Aqu bi » Cpy, VJAq.
Also, for each object y of C, there is a pseudofunctor D 1 y : S 0 Ñ B op such that Corollary 5.10 of [27] is a direct consequence of the last corollary and Proposition 5.7 of [27]. In particular, if T is a pseudomonad on B and U : Ps-T -Alg Ñ B is the forgetful 2-functor, Proposition 5.5 of [27] shows that the category of pseudomorphisms between two pseudoalgebras is given by a descent object (which is a type of weighted bilimit) of a diagram satisfying the hypotheses of Corollary 2.4. Therefore, assuming the existence of codescent objects in A, J has a left biadjoint.
In Section 4, we define the 2-category of lax algebras of a pseudomonad T . There, we also show Proposition 4.5 which is precisely the analogue and a generalization of Proposition 5.5 of [27]: the category of lax morphisms and the category of pseudomorphisms between lax algebras are given by appropriate types of weighted bilimits. Then, we can apply Corollary 2.4 to get our desired result on lifting of biadjoints to the 2-category of lax algebras: Theorem 5.2. Next section, we define and study the weighted bilimits appropriate to our problem, called lax descent objects and descent objects.
To finish this section, we get a trivial consequence of Corollary 2.4: 2.5. Corollary. If RJ " U are pseudofunctors in which R is right biadjoint and U is locally an equivalence, then J is right biadjoint as well. Actually, if E is left biadjoint to R, Gy :" EUy defines the pseudofunctor left biadjoint to J.
In page 177 of [37], without establishing the name "lax descent objects", it is shown that given a 2-monad T , for each pair y, z of strict T -algebras, there is a diagram of categories for which its lax descent category (object) is the category of lax morphisms between y and z. We establish a generalization of this result for lax algebras: Proposition 4.5.
In order to establish such result, our approach in defining the lax descent objects is different from [37], commencing with the definition of our "domain 2-category", denoted by ∆ ℓ .

The 2-category 9
∆ ℓ is, herein, the locally preordered 2-category freely generated by 9 ℓ . The full sub-2-category of 9 ∆ ℓ with objects 1, 2, 3 is denoted by ∆ ℓ and the full inclusion by t : ∆ ℓ Ñ 9 ∆ ℓ . We consider also the computad 9 which is defined as the computad 9 ℓ with one extra 2-cell d 0 d ñ d 1 d. We denote by 9 ∆ the locally preordered 2-category freely generated by 9 . Of course, there is also a full inclusion j : ∆ ℓ Ñ 9 ∆.

Proposition. Let A be a 2-category.
There is a bijection between the 2-functors ∆ ℓ Ñ A and the maps of computads ℓ Ñ UA. In other words, ∆ ℓ is the 2-category freely generated by the computad ℓ .
Also, there is a bijection between 2-functors D : 9 ∆ ℓ Ñ A and the maps of computads D : ℓ Ñ UA which satisfy the following equations: -Associativity:
Let A be a 2-category and D : ∆ ℓ Ñ A be a pseudofunctor. If the weighted bilimit ! 9 ∆p0, j´q, D ) bi exists, we say that Analogously, if such D is a 2-functor and the (strict) weighted 2-limit exists, we call it the strict descent object of D. Finally, the (strict) weighted 2-limit is called the strict lax descent object of D, if it exists.
The dual notions of lax descent object and descent object are called the codescent object and the lax codescent object. If A : ∆ op ℓ Ñ A is a 2-functor, the codescent object of A is, if it exists, 9 ∆p0, j´q˚b i A and the lax codescent object of A is 9 ∆ ℓ p0, t´q˚b i A if it exists.
Also, the weighted colimits 9 ∆p0, j´q˚A, 9 ∆ ℓ p0, t´q˚A are called, respectively, the strict codescent object and the strict lax codescent object of A.
Thereby, we can describe the strict lax descent object of D : ∆ ℓ Ñ Cat explicitly as follows: 1. Objects are 2-natural transformations f : 9 ∆ ℓ p0, t´q ÝÑ D. We have a bijective correspondence between such 2-natural transformations and pairs pf, @ f D q in which f is an object of D1 and @ f D : Dpd 1 qf Ñ Dpd 0 qf is a morphism in D2 satisfying the following equations: -Associativity: -Identity:`D ps 0 qp @ f D q˘`Dpn 1 q f˘"`D pn 0 q fȊ f f : 9 ∆ ℓ p0,´q ÝÑ D is a 2-natural transformation, we get such pair by the correspondence f Þ Ñ pf 1 pdq, f 2 pϑqq.

The morphisms are modifications. In other words, a morphism
Furthermore, there is a full inclusion such that the objects of are precisely the pairs pf, @ f D q (described above) with one further property: @ f D is actually an isomorphism in D2.

Pseudomonads and lax algebras
Pseudomonads in 2-Cat are defined in [27,28]. The definition agrees with the theory of pseudomonads for Gray-categories [29,21,30,31] and with the definition of doctrines of [39].
For each pseudomonad T on a 2-category B, there is an associated (right biadjoint) forgetful 2-functor Ps-T -Alg Ñ B, in which Ps-T -Alg is the 2-category of pseudoalgebras. In this section, we give the definitions of the 2-category of lax algebras Lax-T -Alg ℓ and its associated forgetful 2-functor Lax-T -Alg ℓ Ñ B, which are slight generalizations of the definitions given in [36,22].
Recall that a pseudomonad T on a 2-category B consists of a sextuple pT , m, η, µ, ι, τ q, in which T : B Ñ B is a pseudofunctor, m : T 2 ÝÑ T , η : Id B ÝÑ T are pseudonatural transformations and τ : Id T ùñ pmqpT ηq, ι : pmqpηT q ùñ Id T , µ : m pT mq ñ m pmT q are invertible modifications satisfying the following coherence equations: -Associativity: in which x T ι :" pt T q´1 pT ιq`t pmqpηT q˘x T µ :"`t pmqpmT q˘´1 pT µq`t pmqpT mq˘.
Recall that the Cat-enriched notion of monad is a pseudomonad T " pT , m, η, µ, ι, τ q such that the invertible modifications µ, ι, τ are identities and m, η are 2-natural transformations. In this case, we say that T " pT , m, ηq is a 2-monad, omitting the identities.

Definition. [Lax algebras] Let
T " pT , m, η, µ, ι, τ q be a pseudomonad on B. We define the 2-category Lax-T -Alg ℓ as follows: 1. Objects: lax T -algebras are defined by z " pZ, alg z , z, z 0 q in which alg z : T Z Ñ Z is a morphism of B and z : alg z T palg z q ñ alg z m Z , z 0 : Id Z ñ alg z η Z are 2-cells of B satisfying the coherence axioms: The compositions are defined in the obvious way and these definitions make Lax-T -Alg ℓ a 2-category. The full sub-2-category of the pseudoalgebras of Lax-T -Alg ℓ is denoted by Ps-T -Alg ℓ . Also, the locally full sub-2-category consisting of lax algebras and pseudomorphisms between them is denoted by Lax-T -Alg. Finally, the full sub-2-category of the pseudoalgebras of Lax-T -Alg is denoted by Ps-T -Alg. In short, we have locally full inclusions: Remark. If T " pT , m, ηq is a 2-monad, we denote by T -Alg ℓ the full sub-2-category of strict algebras of Lax-T -Alg ℓ . That is to say, the objects of T -Alg ℓ are the lax T -algebras y " pY, alg y , y, y 0 q such that its 2-cells y, y 0 are identities. Also, we denote by T -Alg the locally full sub-2-category of T -Alg ℓ consisting of strict algebras and pseudomorphisms between them. Finally, T -Alg s is the locally full sub-2category T -Alg ℓ consisting of strict algebras and strict morphisms between them. That is to say, the 1-cells of T -Alg s are the pseudomorphisms f " pf, identity. In this case, we have locally full inclusions in which the vertical arrows are full. -The "free 2-monad" T on Cat whose pseudoalgebras are unbiased monoidal categories. This is defined by T X :" 8 ž n"0 X n , in which X n`1 :" X nˆX and X 0 :" I is the terminal category, with the obvious pseudomonad structure. In this case, the T -pseudomorphisms are the so called strong monoidal functors, while the lax T -morphisms are the lax monoidal functors [25].
-The most simple example is the pseudomonad rising from a monoidal category. A monoidal category M is just a pseudomonoid [42] of Cat and, therefore, it gives rise to a pseudomonad T : Cat Ñ Cat defined by T X " MˆX with obvious unit and multiplication (and invertible modifications) coming from the monoidal structure of M. The pseudoalgebras and lax algebras of this pseudomonad are called, respectively, the pseudoactions and lax actions of M. Lax actions of a monoidal category M are also called graded monads (see [11]).
The inclusion Set Ñ Cat is a strong monoidal functor w.r.t. the cartesian structures, since this functor preserves products. In particular, it takes monoids of Set to monoids of Cat. In short, this means that we can see a monoid M as a (discrete) strict monoidal category. Therefore, a monoid M gives rise to a 2-monad T X " MX as defined above. In this case, the 2-categories T -Alg s , Ps-T -Alg and Lax-T -Alg ℓ are, respectively, the 2-categories of (strict) actions, pseudoactions (as defined in [8]) and lax actions of this monoid M on categories. A lax action of the trivial monoid on a category is the same as a monad. 4.6. Remark. In the context of the proposition above, we can define a pseudofunctor T y : ∆ ℓˆL ax-T -Alg Ñ Cat in which T y p´, zq :" T y z , since the morphisms defined above are actually pseudonatural in z w.r.t. T -pseudomorphisms and T -transformations.
Assume that the triangles below are commutative, R is a right biadjoint pseudofunctor and the arrows without labels are the forgetful 2-functors of Remark 4.4. By Corollary 2.4, it follows from Proposition 4.5 (and last remark) that, whenever A has lax codescent objects, ℓ˝J is right biadjoint to a pseudofunctor G. Also, for each lax algebra y, there is a diagram A y such that Gy » 9 ∆ ℓ p0, t´q˚b i A y defines the left biadjoint to ℓ˝J. Moreover, J is right biadjoint as well if A has codescent objects of these diagrams A y . Next section, we give precisely the diagrams A y and prove a strict version of our theorem as a consequence of Lemma 2.2.

Lifting of biadjoints to lax algebras
In this section, we give our results on lifting right biadjoints to the 2-category of lax algebras of a given pseudomonad. As explained above, we already have such results by Corollary 2.4 and Proposition 4.5. But, in this section, we present an explicit calculation of the diagrams A y whose lax codescent objects are needed in the construction of our left biadjoint.

Definition.
Let pE % R, ρ, ε, v, wq be a biadjunction and T " pT , m, η, µ, ι, τ q a pseudomonad on B such that is commutative, in which U is the forgetful 2-functor defined in Remark 4.4. In this setting, for each lax T -algebra y " pY, alg y , y, y 0 q, we define the 2-functor A y : in which A y pσ 21 q :" e´1 palg y qpm U y q¨E pyq¨e palg y qpT palg y qq A y pn 0 q :" e´1 palg y qpη U y q¨E py 0 q¨e U y A y pn 1 q :"ˆ´id ε EU y¯˚ˆe´1 palg J EU y T pρ U y qqpη U y q¨E pid alg J EU y˚η´1 ρ U y q¨EpJEU y 0˚i d ρ U y q˙˙¨v U y A y pσ 20 q :"ˆε´1 y qq˙Ȧ y pσ 00 q :"ˆid ε EU y˚ˆe´1 palg J EU y T pρ U y qqpm U y q¨E pid alg J EU y˚m´1 ρ U y q¨EpJEU y˚id [Biadjoint Triangle Theorem] Let pE % R, ρ, ε, v, wq be a biadjunction, T " pT , m, η, µ, ι, τ q a pseudomonad on B and ℓ : Lax-T -Alg Ñ Lax-T -Alg ℓ the inclusion. Assume that A is commutative. The pseudofunctor ℓ˝J is right biadjoint if and only if A has the lax codescent object of the diagram A y : ∆ op ℓ Ñ A for every lax T -algebra y. In this case, the left biadjoint G is defined by Gy " 9 ∆ ℓ p0, t´q˚b i A y Furthermore, J is right biadjoint if and only if A has the codescent object of the diagram A y : ∆ op ℓ Ñ A for every lax T -algebra y. In this case, the left biadjoint G 1 is defined by G 1 y " 9 ∆p0, j´q˚b i A y :" χ pT 2 U y,Aq : BpT 2 Uy, RAq Ñ ApET 2 Uy, Aq in which χ : Bp´, R´q » ApE´,´q is the pseudonatural equivalence corresponding to the biadjunction pE % R, ρ, ε, v, wq (see Remark 1.4). Also, pψ y s 0 q f :" id ε A˚e pf qpη U y q pψ y d 1 q f :" id ε A˚e pf qpalg y q pψ y B 1 q f :" id ε A˚e pf qpm U y q pψ y B 2 q f :" id ε A˚e pf qpT palg y qq pψ y d 0 q f :"´id ε A˚´E pid alg J A˚T pw A q˚id T pf q q¨Epid alg J A˚t pRpε A qqpρ RA q˚i d T pf q q¯¯ï Lax-T -Alg are commutative triangles, in which Lax-T -Alg s Ñ Lax-T -Alg is the locally full inclusion of the 2-category of lax algebras and strict T -morphisms into the 2-category of lax algebras and T -pseudomorphisms. The pseudofunctor ℓ˝J is right biadjoint if and only if A has the strict lax codescent object of the diagram A y : ∆ op ℓ Ñ A for every lax T -algebra y. In this case, the left 2-adjoint G is defined by Gy " 9 ∆ ℓ p0, t´q˚A y Furthermore, J is right 2-adjoint if and only if A has the strict codescent object of the diagram A y : ∆ op ℓ Ñ A for every lax T -algebra y. Proof. We have, in particular, the setting of Theorem 5.2. Therefore, we can define ψ as it is done in the last proof. However, in our setting, we get a 2-natural transformation which is an objectwise isomorphism. Therefore ψ is a 2-natural isomorphism.
By Lemma 2.2, Proposition 4.5 and Remark 4.6, this completes our proof.

Coherence
As mentioned in the introduction, the 2-monadic approach to coherence consists of studying the inclusions induced by a 2-monad T of Remark 4.2 to get general coherence results [5,22,33]. Given a 2-monad pT , m, ηq on a 2-category B, the inclusions of Remark 4.2 and the forgetful functors of Remark 4.4 give in particular the commutative diagram below, in which Ps-T -Alg Ñ B is right biadjoint and T -Alg s Ñ B is right 2-adjoint.
In this section, we are mainly concerned with the triangles involving the 2-category of lax algebras. We refer to [27] for the remaining triangles involving the 2-category of pseudoalgebras. The inclusion T -Alg s Ñ Lax-T -Alg ℓ is also studied in [22]. Therein, it is proved that it has a left 2-adjoint whenever the 2-category T -Alg s has the lax codescent objects of some diagrams called therein lax coherence data. This is of course the immediate consequence of Theorem 5.3 applied to the large triangle above.
Actually, we can study other inclusions of Remark 4.2 with the techniques of this paper. For instance, by Theorem 5.3 and Corollary 2.5, the inclusion of T -Alg into any 2category of T -algebras and lax T -morphisms of Remark 4.2 has a left biadjoint provided that T -Alg has lax codescent objects. Also, the inclusion of this 2-category into any 2-category of T -algebras and T -pseudomorphisms (i.e. vertical arrows with domain in T -Alg of Remark 4.2) has a left biadjoint provided that T -Alg has codescent objects.
In the more general context of pseudomonads, we can apply Theorem 5.2 and Theorem 5.3 to understand precisely when the inclusions Ps-T -Alg Ñ Lax-T -Alg ℓ and Ps-T -Alg Ñ Lax-T -Alg have left biadjoints. In particular, we have: 6.1. Theorem. Let T " pT , m, η, µ, ι, τ q be a pseudomonad on a 2-category B. If Ps-T -Alg has lax codescent objects, then the inclusion Ps-T -Alg Ñ Lax-T -Alg has a left biadjoint. Furthermore, if Ps-T -Alg has codescent objects, Ps-T -Alg Ñ Lax-T -Alg has a left biadjoint.
In particular, if T " pT , m, η, µ, ι, τ q is a pseudomonad that preserves lax codescent objects, then Ps-T -Alg has lax codescent objects and, therefore, satisfies the hypothesis of the first part of the result above. Similarly, if T preserves codescent objects, it satisfies the hypothesis of the second part.