## The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads (2007)

Citations: | 19 - 1 self |

### BibTeX

@MISC{Hyland07thecategory,

author = {Martin Hyland and John Power},

title = { The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads},

year = {2007}

}

### OpenURL

### Abstract

Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future.

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Citation Context ...nition of Lawvere theory, every monad arises thus, uniquely up to coherent isomorphism. Monads immediately became the more common category theoretic formulation of universal algebra, see for instance =-=[31]-=-. In retrospect, that surprises the current authors: the notion of Lawvere theory arose directly from universal algebra while that of monad did not; Lawvere theories relate more closely to universal a... |

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Citation Context ...ly check that the monad TL is finitary for every Lawvere theory L. When the base category is Set, finitariness characterises the image of the construction, but that was an observation of a later time =-=[19]-=-. For a converse, first observe that for any monad T on Set, the Kleisli category Kl(T) has all small coproducts and the canonical functor I : Set −→ Kl(T) preserves them: for the canonical functor I ... |

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Citation Context ...s of one general construction. He had in mind models for exceptions, side-effects, interactive input/output and non-determinism, as well as partiality and continuations. Probabalistic non-determinism =-=[18]-=- was soon to be added to the list. Monads have come to be used as a tool in computer science in contexts other than those originally proposed (data bases [5], pure function languages [48]), but it see... |

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Citation Context ...heoretic formulation of universal algebra, which was in terms of monads, has a more complicated history. Monads typically arise from adjoint pairs of functors; and in such a case, the Eilenberg-Moore =-=[8]-=- and Kleisli [22] categories of algebras for the monad provide adjoint pairs which one can regard as approximations to the original adjoint pair. This notion of monad (or triple or standard constructi... |

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Citation Context ...ard as approximations to the original adjoint pair. This notion of monad (or triple or standard construction) arose in algebraic topology for reasons distinct from universal algebra, see for instance =-=[10]-=-. In [8] Eilenberg and Moore noted that in case T is the free group monad, their category of T-algebras is the category of groups. Then Linton [29] made the general connection between monads and Lawve... |

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Citation Context ...lopment of thinking and a narrower one typically concerned with the logic of properties. In the epoch we consider, the former was undergoing substantial development largely stimulated by Lawvere (see =-=[24,25,26,27]-=-). But it was only in the 1970s that a sophisticated categorical logic in the narrower sense emerged; and ironically Lawvere theories fit naturally within this narrow reading of logic. Lawvere theorie... |

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Citation Context ...lgebra. The earlier was by Bill Lawvere in his doctoral thesis in 1963 [23]. Nowadays, his central construct is usually called a Lawvere theory, more prosaically a single-sorted finite product theory =-=[2,3]-=-. It is a more flexible version of the universal algebraist’s notion 1 This work is supported by EPSRC grant GR/586372/01: A Theory of Effects for Programming Languages. 2 Email: M.Hyland@dpmms.cam.ac... |

38 | algebras and cohomology
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Citation Context ...as taken forward by Beck, Barr, and others whose expertise lay primarily in algebraic topology, where the notion of monad had arisen; that led to important mathematical developments, see for instance =-=[1,4]-=- and the account in [2], but the focus of these results is quite different from that of the Lawvere theory notion. 6 Computational Effects We now jump from the mid 1960s to the late 1980s. In the mean... |

36 |
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Citation Context ... [11] mildly generalise Horn clauses; and more generally still, one has regular theories, coherent theories, and geometric theories, all fitting into the topos theory perspective on categorical logic =-=[32,2]-=-. In the context of that work, monads are isolated. Though they are part of logic in the broader sense, in contrast to Lawvere theories, they do not immediately fit into this logical hierarchy. But th... |

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Citation Context ...ry was first formulated by Eilenberg and Kelly in 1966 in [7], but it took time for it to enter the mainstream, and a definitive account of finiteness in the enriched setting only appeared in 1982 in =-=[20]-=-. The reason enriched categories help is because, given a notion of finiteness, the generalisation of Lawvere theory to that setting is easy and covers a wide class of the examples of structures that ... |

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(Show Context)
Citation Context ...thematics: theory has had to be developed adequately to ground the practice, and the computationally significant examples have had to be worked through. This has resulted in a series of recent papers =-=[37,38,39,40,41,42]-=-. But one should recognise that issues of presentation are also mathematically significant. Famously for instance the theory of groups has a presentation in terms of division, but that presentation ge... |

32 | Combining effects: Sum and tensor
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(Show Context)
Citation Context ...is part of Lawvere’s adjunction between semantics and algebraic structure [23]. The ideas are extended and refined in the treatment of algebraic operations in [38]. A major thrust of recent work (see =-=[15]-=- and [13]) has been to understand computationally natural ways to combine computational effects in terms of operations on Lawvere theories. We briefly review here two natural ways to combine Lawvere t... |

27 |
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Citation Context ...tion of universal algebra, which was in terms of monads, has a more complicated history. Monads typically arise from adjoint pairs of functors; and in such a case, the Eilenberg-Moore [8] and Kleisli =-=[22]-=- categories of algebras for the monad provide adjoint pairs which one can regard as approximations to the original adjoint pair. This notion of monad (or triple or standard construction) arose in alge... |

24 |
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Citation Context ...L ′ is trivial. (There are analogous issues with Triv0.) One can also consider the tensor product L ⊗ L ′ of Lawvere theories L and L ′ which can be explained as follows, (but see also discussions in =-=[9,45]-=-). The category ℵ0 not only has finite coproducts, but also has finite products, which we denote by n × n ′ . The object n × n ′ may also be seen as the coproduct of n copies of n ′ . So, given an arb... |

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15 |
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Citation Context ...eral remark we observe that properties of and structure on the category of Lawvere theories remains an interesting area of study in its own right. A survey together with a list of problems appears in =-=[25]-=-, and the commentary to the TAC reprint updates the material. 4 Monads Soon after Lawvere gave his characterisation of the clone of an algebraic theory, Linton showed that every Lawvere theory yields ... |

14 |
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(Show Context)
Citation Context ...rise from computationally natural operations, such as raise for exceptions, lookup and update for side-effects, read and write for interactive input/output, nondeterministic ∨ for nondeterminism, and =-=[0,1]-=--many binary operations +r for probabilistic nondeterminism, subject to computationally natural equations. So there are evident scientific issues relating to the computational justification of (presen... |

12 | Countable Lawvere Theories and Computational Effects
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Citation Context ...iderable effort to resolve the situation [47]. In contrast with the situation for distributive laws, the various kinds of composite of Lawvere theories, such as sum, tensor, and a distributive tensor =-=[17]-=-, always exist. Those we know include the combinations of computational effects of primary interest; and it seems likely that further combinations of computational effects which may well arise in part... |

11 |
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Citation Context ...arbitrary size, with a generalised theory no longer a small category or fully determined by one. The construction of a monad from a Lawvere theory then generalises to an equivalence of categories. In =-=[30]-=-, Linton gives generalisations of Lawvere’s treatments of semantics and algebraic structure. It is implied that the case treated by Lawvere should be seen as a special case of the more general theory.... |

9 |
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Citation Context ...s of sheaves, of differentiable manifolds, of schemes and so on. A special case of interest is that of models of LG in the category Group ≃ Mod(LG,Set) of small groups. By the Eckmann-Hilton argument =-=[6]-=-, which was published in 1962, these are abelian groups; and this explains why the higher homotopy groups are abelian. These examples give some indication how the notion of Lawvere theory brought prec... |

9 | Combining Algebraic Effects with Continuations, Theoretical Computer Science 375
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Citation Context ...f Lawvere’s adjunction between semantics and algebraic structure [23]. The ideas are extended and refined in the treatment of algebraic operations in [38]. A major thrust of recent work (see [15] and =-=[13]-=-) has been to understand computationally natural ways to combine computational effects in terms of operations on Lawvere theories. We briefly review here two natural ways to combine Lawvere theories, ... |

7 |
Adjointness in Foundations, Dialectica 23
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(Show Context)
Citation Context ...lopment of thinking and a narrower one typically concerned with the logic of properties. In the epoch we consider, the former was undergoing substantial development largely stimulated by Lawvere (see =-=[24,25,26,27]-=-). But it was only in the 1970s that a sophisticated categorical logic in the narrower sense emerged; and ironically Lawvere theories fit naturally within this narrow reading of logic. Lawvere theorie... |

3 | Pseudo-closed 2-categories and pseudocommutativities - Hyland, Power |

2 |
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(Show Context)
Citation Context ...lopment of thinking and a narrower one typically concerned with the logic of properties. In the epoch we consider, the former was undergoing substantial development largely stimulated by Lawvere (see =-=[24,25,26,27]-=-). But it was only in the 1970s that a sophisticated categorical logic in the narrower sense emerged; and ironically Lawvere theories fit naturally within this narrow reading of logic. Lawvere theorie... |

2 | Some aspects of equational theories - Linton - 1966 |

2 |
Lawvere theories over a general base
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Citation Context ... an account of Lawvere theories relative to an arbitrary base category subject to axiomatically defined conditions. But that is awkward: it was only finally resolved in 2005, the details appearing in =-=[36]-=-. How history might have developed had finiteness and enrichment been resolved by 1966 and had the motivating experience of more general categorical logic been available then is imponderable. Lawvere ... |

1 |
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(Show Context)
Citation Context ...We shall give some details of enriched Lawvere theories shortly. Third, developments in categorical logic which favour the Lawvere theory perspective were yet to come. As discussed in the Appendix to =-=[28]-=- there are two extant senses to logic, a broader one concerning the development of thinking and a narrower one typically concerned with the logic of properties. In the epoch we consider, the former wa... |