Theories of analytic monads

Szawiel, Stanis law; Zawadowski, Marek

doi:10.1017/s0960129513000868

Cited by 16 publications

(23 citation statements)

References 22 publications

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“…As explained in [15,29] such polynomial monads may be identified with ordinary (coloured symmetric) operads whose symmetric group actions are Σ-free. Thus via [33] these operads may also be regarded as polynomial monads now over Cat.…”

Section: F(g) : F(r) → F(t ) Is An Exact Symmetric Monoidal Functormentioning

confidence: 99%

Algebraic Kan extensions along morphisms of internal algebra classifiers

Weber

2016

Tbilisi Math. J.

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An "algebraic left Kan extension" is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of conformal field theories. In the most interesting examples, the functor along which we left Kan extend goes between categories that enjoy universal properties which express the meaning of the calculation we are trying to understand. These universal properties say that the categories in question are universal examples of some categorical structure possessing some kind of internal structure, and so fall within the theory of "internal algebra classifiers" described in earlier work of the author. In this article conditions of a monad-theoretic nature are identified which give rise to morphisms between such universal objects, which satisfy the key condition of Guitart-exactness, which guarantees the algebraicness of left Kan extending along them. The resulting setting explains the algebraicness of the left Kan extensions arising in operad theory, for instance from the theory of "Feynman categories" of Kaufmann and Ward, generalisations thereof, and also includes the situations considered by Batanin and Berger in their work on the homotopy theory of algebras of polynomial monads.

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Section: F(g) : F(r) → F(t ) Is An Exact Symmetric Monoidal Functormentioning

confidence: 99%

Algebraic Kan extensions along morphisms of internal algebra classifiers

Weber

2016

Tbilisi Math. J.

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“…The analytic functors were introduced in [9], the semi-analytic monads were introduced in [18] under the name of collection monads. The categories An, San, AnMnd and SanMnd were studied extensively in [21,22].…”

Section: Coreflective Subcategories Of the Categories Of Monadsmentioning

confidence: 99%

“…For the explicite formulas for the coefficient functors of the composed functors in categories An and San the reader can consult [21,22]. Thus we have described a diagram of (strictly monoidal) categories andIf a monoidal category has a terminal object 1, then there is a unique structure of a monoid on 1 and this monoid is the terminal monoid.…”

Section: Coreflective Subcategories Of the Categories Of Monadsmentioning

confidence: 99%

Generalized Płonka Sums and Products

Zawadowski

2013

Appl Categor Struct

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We give an abstract categorical treatment of Płonka sums and products using lax and oplax morphisms of monads. Płonka sums with sup-semilattices as arities were originally defined as operations on algebras of regular theories. It turns out that even more general operations are available on the categories of algebras of semi-analytic monads. Their arities are the categories of the regular polynomials over any sup-semilattice, i.e. any algebra for the terminal semi-analytic monad. We also show that similar operations can be defined on the category of algebras of any analytic monad. This time we can allow the arities to be the categories of linear polynomials over any commutative monoid, i.e. any algebra for the terminal analytic monad. There are also dual operations of Płonka products. They can be defined on Kleisli categories of commutative monads.

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“…These equivalences induce a correspondence between various subcategories. In [14] we have given an intrinsic characterizations of equational theories and Lawvere theories that correspond to the analytic and polynomial monads on Set. This was achieved via correspondence with symmetric and rigid operads.…”

Section: Introductionmentioning

confidence: 99%

“…In [14] symmetric operads proved to be very useful in describing correspondences of this kind. Full and regular operads (cf.…”

Section: Introductionmentioning

confidence: 99%