On semiflexible, flexible and pie algebras

📝 Abstract

We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones. We see that in many cases, the pie algebras are precisely those “free at the level of objects” in a suitable sense; so that, for instance, a strict monoidal category is pie just when its underlying monoid of objects is free. Pie algebras are contrasted with flexible and semiflexible algebras via a series of characterisations of each class; particular attention is paid to the case of pie, flexible and semiflexible weights, these being characterised in terms of the behaviour of the corresponding weighted limit functors.

💡 Analysis

We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones. We see that in many cases, the pie algebras are precisely those “free at the level of objects” in a suitable sense; so that, for instance, a strict monoidal category is pie just when its underlying monoid of objects is free. Pie algebras are contrasted with flexible and semiflexible algebras via a series of characterisations of each class; particular attention is paid to the case of pie, flexible and semiflexible weights, these being characterised in terms of the behaviour of the corresponding weighted limit functors.

📄 Content

arXiv:1112.1448v1 [math.CT] 6 Dec 2011 ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS JOHN BOURKE AND RICHARD GARNER Abstract. We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones. We see that in many cases, the pie algebras are precisely those “free at the level of objects” in a suitable sense; so that, for instance, a strict monoidal category is pie just when its underlying monoid of objects is free. Pie algebras are contrasted with flexible and semiflexible algebras via a series of characterisations of each class; particular attention is paid to the case of pie, flexible and semiflexible weights, these being characterised in terms of the behaviour of the corresponding weighted limit functors.

1. Introduction One category-theoretic approach to universal algebra is based on the theory of monads. Single-sorted (possibly infinitary) algebraic theories correspond with monads on the category of sets, and models of a theory with algebras for the associ- ated monad; this justifies our regarding monads on other categories as generalised algebraic theories, and many basic aspects of classical universal algebra may be re- constructed in this broader context. A further generalisation is obtained on passing from the study of monads on categories to that of 2-monads on 2-categories, which yields a kind of “two-dimensional universal algebra”. The simplest case studies 2-monads on Cat, which encode many familiar structures that may be borne by a category: thus there are 2-monads whose algebras are monoidal categories, or cat- egories with finite products, or distributive categories, or cocomplete categories, and so on. In the passage from 1- to 2-dimensional monad theory, a number of new phe- nomena come into being. As well as strict algebras for a 2-monad, we also have pseudo and lax ones, for which the algebra axioms have been weakened to hold only up to coherent 2-cells, invertible in the former case but not in the latter. Likewise, between the algebras for a 2-monad we have not only the strict morphisms, but also pseudo and lax ones, which preserve the algebra structure in correspondingly weakened manners. The interplay between strict, pseudo and lax in 2-dimensional monad theory provides an abstract setting for the study of coherence problems of the kind exemplified by Mac Lane’s famous result [26] that every monoidal cat- egory is monoidally equivalent to a strict one. The study of coherence from the standpoint of 2-monad theory was championed by Max Kelly, who initiated the Date: November 7, 2018. 2000 Mathematics Subject Classification. Primary: 18D05, 18C15. The first author acknowledges the support of the Eduard ˇCech Center for Algebra and Ge- ometry, grant number LC505. The second author acknowledges the support of an Australian Research Council Discovery Project, grant number DP110102360. 1 2 JOHN BOURKE AND RICHARD GARNER programme in [8, 9] and with his collaborators, brought it to a particularly fine expression in [3]. A subtle and crucial notion in Kelly’s framework is that of flexibility; this first arose in [9], where was introduced the notion of a flexible 2-monad. One of the most important properties of flexible 2-monads, as described in [14, Theorem 3.3], is that strict algebra structure may be transported along equivalences: which is to say that if A ≃B, and A bears strict algebra structure for a flexible 2-monad, then so does B. This is the case, for example, with the 2-monad on Cat whose algebras are monoidal categories, but not for the 2-monad whose algebras are strict monoidal categories; and indeed, the former 2-monad is flexible, whilst the latter is not. This particular good behaviour of flexible 2-monads is a consequence of a more fundamental one: that every pseudoalgebra for a flexible 2-monad is isomorphic to a strict one. Intuitively, we think that flexibility of a 2-monad expresses a certain “looseness” in the structure it imposes on its algebras, and this intuition has been expressed for 2-monads on Cat in the following way: that such a 2- monad is flexible if its algebras can be presented as categories equipped with basic operations and with natural transformations between derived operations, satisfying equations between derived natural transformations but with no equations being imposed between derived operations themselves. The scope of the concept of flexibility was vastly expanded in [3], where was introduced the notion of a flexible algebra for a 2-monad with rank on a complete and cocomplete 2-category. With suitable cardinality constraints, 2-monads may themselves be viewed as algebras for such a 2-monad, so that flexible 2-monads are an instance of the more general notion. The flexible algebras exhibit the same kind of “looseness” as we saw above, this now being manifested in the result that each pseudomorphism out of a flexible algebra is isomorphic to a strict one; which is in

This content is AI-processed based on ArXiv data.