Monads in double categories

Introduction The development of the formal theory of monads, begun in [23] and continued in [15], shows that much of the theory of monads [1] can be generalized from the setting of the 2-category Cat of small categories, functors and natural transformations to that of a general 2-category. The generalization, which involves defining the 2-category Mnd(K) of monads, monad maps and monad 2-cells in a 2-category K , is useful for studying homogeneously a variety of important mathematical structures. For example, as explained in [17], categories, operads, multicategories and T -multicategories can all be seen as monads in appropriate bicategories. However, the most natural notions of a morphism between these mathematical structures do not appear as instances of the notion of a monadmap. For example, it is well-known that, while categories can be viewed as monads in the bicategory of spans [2], functors are not monad maps therein. To address this issue, we define the double category Mnd(C) of monads, horizontal monad maps, vertical monad maps and monad squares in a double category C. Monads and horizontal monad maps in C are exactly monads and monad maps in the horizontal 2-category of C, while the definitions of vertical monad maps and monad squares in C involve vertical arrows of C that are not necessarily identities. This combination of horizontal and vertical arrows of C in the definition of Mnd(C) allows us to describe mathematical structures and morphisms between them as monads and vertical monad maps in appropriate double categories. For example, small categories and functors can be viewed as monads and vertical monad maps in the double category of spans. For a double categoryC, we define also the double category End(C) of endomorphisms, horizontal endomorphismmaps, vertical endomorphism maps and endomorphism squares. The double categories Mnd(C) and End(C) are related by a forgetful double functor U : Mnd(C) → End(C), mapping a monad to its underlying endomorphism. By definition, a double categoryC is said to admit the construction of free monads if U has a vertical left adjoint. In view of our applications, we consider the construction of free monads in double categories that satisfy the additional assumption of being framed ∗ Corresponding author at: Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, via Archirafi 34, 90123 Palermo, Italy. E-mail addresses: [email protected] (T.M. Fiore), [email protected] (N. Gambino), [email protected] (J. Kock). 0022-4049/$ – see front matter© 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2010.08.003 2 T.M. Fiore et al. / Journal of Pure and Applied Algebra ( ) – bicategories, in the sense of [21]. Our main result shows that a framed bicategory satisfying some mild assumptions admits the construction of free monads if its horizontal 2-category does. Here, the notion of a 2-category admitting the construction of free monads is obtained by generalizing the characterization of the free monads in the 2-category Cat obtained in [22, Section 6.1]. We apply the general theory to the study of two free constructions. First, we consider the construction of the free category on a graph (relatively to a category with finite limits), which plays an important role in Joyal’s abstract treatment of Gödel’s incompleteness theorems [18]. We show that if E is a pretopos with parametrized list objects, then the double category of spans in E admits the construction of free monads. Secondly, we consider the construction of the free monad on a polynomial endofunctor (relatively to a locally cartesian closed category, which is always assumed here to have a terminal object), which contributes to the category-theoretic analysis of Martin-Löf’s types of well-founded trees, begun in [19] and continued in [7,8]. We show that if E is a locally cartesian closed category with finite disjoint coproducts and W-types, then the double category of polynomials in E admits the construction of free monads. Both of these results are obtained by application of our main result, which is possible since the double categories of interest are framed bicategories. Examples of categories E satisfying the hypotheses above abound: for example, every elementary topos with a natural numbers object is both a pretopos with parametrized list objects and a locally cartesian closed category with finite disjoint coproducts and W-types [19]. Thus, our theory applies in particular to the category Set of sets and functions and to categories of sheaves. The double categories of spans and of polynomials are defined such that if we consider the vertical part of the freemonad double adjunction, we recover exactly the adjunction between graphs and categories [16, Section II.7] and the adjunction between polynomial endofunctors and polynomial monads [8, Section 4.6]. Hence, we both strengthen these adjointness results and put them in a general context. Indeed, one of the original motivations for the research presented here was to make precise the analogy between the two constructions. In both cases, the application of our main theorem simplifies a problem regarding double categories by reducing it to a question on 2-categories. Note, however, that the combination of horizontal and vertical arrows is exploited essentially to recover the existing results, since the free monad construction acts on endomorphisms (which are defined using horizontal arrows) but its universal property is expressed with respect to vertical endomorphism maps. Some double-categorical aspects of monads have also been investigated within the theory of fc-multicategories in [17, Chapter 5] and within the theory of framed bicategories in [21, Section 11]. However, the notion of a horizontal monad map considered there generalizes the ring-theoretic notion of a bimodule, whereas our horizontal monad maps are essentially the monad maps of Street [23]. Plan of the paper. Section 1 discusses monads in a 2-category, recalling some basic notions from [23] and giving a characterization of the freemonads in a 2-category. Section 2 introduces the double categoryMnd(C) ofmonads in a double category C and illustrates its definition with examples. Section 3 establishes some special properties of Mnd(C) under the assumption that C is a framed bicategory. In particular, we state our main result, Theorem 3.7, and apply it to our examples. Finally, Section 4 contains the proof of Theorem 3.7. 1. Monads in a 2-category Preliminaries. We recall some definitions concerning endomorphisms, monads and their algebras in a 2-category. Let K be a 2-category. An endomorphism in K is a pair (X, P) consisting of an object X and a 1-cell P : X → X . An endomorphism map (F , φ) : (X, P) → (Y ,Q ) consists of a 1-cell F : X → Y and a 2-cell φ : QF → FP , which is not required to satisfy any condition. An endomorphism 2-cell α : (F , φ) → (F , φ) is a 2-cell α : F → F ′ making the following diagram commute: QF φ / Qα FP αP QF ′ φ / F P. We write End(K) or EndK for the 2-category of endomorphisms, endomorphism maps and endomorphism 2-cells in K . There is a 2-functor Inc : K → End(K) which sends an object X ∈ K to the identity endomorphism (X, 1X ) on X . Let us now consider a fixed endomorphism (Y ,Q ) in K . For X ∈ K , the category Q -algX of X-indexed Q -algebras, in the sense of Lambek, is defined by letting Q -algX =def EndK((X, 1X ), (Y ,Q )). Explicitly, an X-indexed Q -algebra consists of a 1-cell F : X → Y , called the underlying 1-cell of the algebra, and a 2-cell f : QF → F , called the structure map of the algebra. Note that the structure map is not required to satisfy any conditions. These definitions extend to a 2-functor Q -alg(−) : K → Cat. We write U(−) : Q -alg(−) → K(−, Y ) for the 2-natural transformations whose components are the forgetful functors UX : Q -algX → K(X, Y ) mapping an X-indexed Q -algebra to its underlying 1-cell. T.M. Fiore et al. / Journal of Pure and Applied Algebra ( ) – 3 We write Mnd(K) or MndK for the 2-category of monads, monad maps and monad 2-cells in K , as defined in [23]. As usual, we refer to a monad by mentioning only its underlying endomorphism, leaving implicit its multiplication and unit. With a minor abuse of notation, we write Inc : K → Mnd(K) for the 2-functor mapping an object X to the monad (X, 1X ). If (Y ,Q ) is a monad, for every X ∈ K we may consider not only the category Q -algX of Lambek algebras for its underlying endomorphism, but also the category Q -AlgX of X-indexed Eilenberg–Moore Q -algebras, which is defined by letting Q -AlgX =def MndK((X, 1X ), (Y ,Q )). Note that we write Q -algX for the category of algebras for the endomorphism and Q -AlgX for the category of Eilenberg– Moore algebras for the monad. Explicitly, an X-indexed Eilenberg–Moore Q -algebra consists of a 1-cell F : X → Y and a 2-cell f : QF → F satisfying the axioms QQF Qf / μQ F QF f QF f / F , F ηQ F / 1F QF f F . Again, these definitions extend to a 2-functor Q -Alg(−) : K → Cat and there is a 2-natural transformation U(−) : Q -Alg(−) → K(−, Y ), with components given by the evident forgetful functors. Since (Y ,Q ) is assumed to be a monad, for every X ∈ K the forgetful functor UX : Q -AlgX → K(X, Y ) has a left adjoint, defined by composition with Q : Y → Y . A characterization of free monads. We generalize the characterization of the free monad on an endomorphism given by Staton in [22, Theorem 6.1.5] from the 2-category Cat to an arbitrary 2-category K . The generalization is essentially straightforward, but we indicate the main steps of the proof. See [1, Section 9.4] for background material on free monads and [13] for a general account of several examples of the free monad construction. Theorem 1.1. Let (Y ,Q ) be an endomorphism in a 2-category K . For a monad (Y ,Q ) and a 2-cell ιQ : Q → Q , the following conditions are equivalent. (i) The endomorphism map (1Y , ιQ ) : (Y ,Q ) → (Y ,Q ) is universal, in the sense that for every monad (X, P), composition with (1Y , ιQ ) induces an isomorphism fitting in the diagram MndK((X, P), (Y ,Q )) ∼= / ( R R R R R R R R R R R R R EndK((X, P), (Y ,Q )) vmmm mmm mmm mmm m