Pseudoalgebras and Non-canonical Isomorphisms

Nunes, Fernando Lucatelli

doi:10.1007/s10485-018-9541-3

Cited by 4 publications

(3 citation statements)

References 9 publications

(26 reference statements)

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“…In Section 5, we show that analogous results also hold for the free finite coproduct completion pseudomonad, leading to similar novel characterisations of (finitely) extensive categories. Further, we discuss possible avenues for future work, descent theoretical considerations of our findings, and we note a result on non-canonical isomorphisms, as a direct consequence of the work of [30].…”

Section: Outlinementioning

confidence: 87%

“…In analogy with [34, Subsection 5.2], we may use the results of [30] to prove that a category C is a (Fam • L Φ )-pseudoalgebra if it has coproducts, Φ-limits, and there exists a(ny) invertible natural isomorphism is an oplax T -morphism by doctrinal adjunction [19,29]. The (codual version of the) techniques of non-canonical isomorphisms from [30] can be applied just as well to this setting.…”

Section: Non-canonical Isomorphismsmentioning

confidence: 99%

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2020

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We consider the canonical pseudodistributive law between various free limit completion pseudomonads and the free coproduct completion pseudomonad. When the class of limits includes pullbacks, we show that this consideration leads to notions of extensive categories. More precisely, we show that extensive categories with pullbacks and infinitary lextensive categories are the pseudoalgebras for the pseudomonads resulting from the pseudodistributive laws. Moreover, we introduce the notion of doubly-infinitary lextensive category, and we establish that the freely generated such categories are cartesian closed. From this result, we further deduce that, in freely generated infinitary lextensive categories, the objects with a finite number of connected components are exponentiable. We conclude our work with remarks on descent theoretical aspects of this work, along with results concerning non-canonical isomorphisms.

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Section: Outlinementioning

confidence: 87%

Section: Non-canonical Isomorphismsmentioning

confidence: 99%

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2020

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“…Definition 19 (Strictly indexed finite biproducts). A category with finite products and coproducts is semi-additive if the binary coproduct functor is naturally isomorphic to the binary product functor; see, for instance, Lack (2012), Lucatelli Nunes (2019. In this case, the product/coproduct is called biproduct, and the biproduct structure is denoted by (×, 1) or (+, 0).…”

Section: Products In Total Categoriesmentioning

confidence: 99%

CHAD for expressive total languages

Nunes

Vákár

2023

Math. Struct. Comp. Sci.

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We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár (2021). ESOP, 607–634; Vákár and Smeding (2022). ACM Transactions on Programming Languages and Systems44 (3) 20:1–20:49.) to total functional programming languages with expressive type systems featuring the combination of • tuple types; • sum types; • inductive types; • coinductive types; • function types. We achieve this by analyzing the categorical semantics of such types in $\Sigma$ -types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward- and reverse-mode automatic differentiation (AD) on total functional programming languages with expressive type systems.

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Semantic Factorization and Descent

Nunes

2022

Appl Categor Struct

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Let $${\mathbb {A}}$$ A be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the two-dimensional cokernel diagram of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category $${\mathbb {A}}$$ A .

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Pseudoalgebras and Non-canonical Isomorphisms

Cited by 4 publications

References 9 publications

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CHAD for expressive total languages

Semantic Factorization and Descent

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