Modalities and Parametric Adjoints

Gratzer, Daniel; Cavallo, Evan; Kavvos, G. A.; Guatto, Adrien; Birkedal, Lars

doi:10.1145/3514241

Cited by 5 publications

(11 citation statements)

References 33 publications

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“…The presence of multiple notions of crispness forces a more complex context structure than spatial type theory's separation of the context into a crisp zone and cohesive zone. Similar to many other modal type theories [30,22,21,9,38], we annotate each variable with modal information, here, the focuses for which that variable is crisp. The typing rules for the modalities of each focus then work essentially independently.…”

Section: Mengen Kardinalen Points Codiscrete Discretementioning

confidence: 99%

“…Ours is far from the only extension of type theory with multiple modalities, but as we discuss in more detail later, no existing theory has the combination of features that we are looking for: dependent types (ruling out [30]) that may depend on modal variables (ruling out [9]), multiple commuting comodalities (ruling out [47,11,38]) each with a with right-adjoint modality (ruling out [33]) and no further left-adjoints (ruling out [22,21] and [16, §14]).…”

Section: Mengen Kardinalen Points Codiscrete Discretementioning

confidence: 99%

“…Later work [21] describes a multimodal type theory where each mode morphism becomes a (more convenient) negative type former. The semantic requirements are even stronger: the functor corresponding to the modality must be a dependent right-adjoint [11], whose left adjoint is itself a parametric right adjoint.…”

Section: Ctx-emptymentioning

confidence: 99%

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Commuting Cohesions

Myers¹,

Riley²

2023

Preprint

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Shulman's spatial type theory internalizes the modalities of Lawvere's axiomatic cohesion in a homotopy type theory, enabling many of the constructions from Schreiber's modal approach to differential cohomology to be carried out synthetically. In spatial type theory, every type carries a spatial cohesion among its points and every function is continuous with respect to this. But in mathematical practice, objects may be spatial in more than one way at the same time; a simplicial space has both topological and simplicial structures. Moreover, many of the constructions of Schreiber's differential cohomology and Schreiber and Sati's account of proper equivariant orbifold cohomology require the interplay of multiple sorts of spatiality -differential, equivariant, and simplicial.In this paper, we put forward a type theory with "commuting focuses" which allows for types to carry multiple kinds of spatial structure. The theory is a relatively painless extension of spatial type theory, and enables us to give a synthetic account of simplicial, differential, equivariant, and other cohesions carried by the same types. We demonstrate the theory by showing that the homotopy type of any differential stack may be computed from a discrete simplicial set derived from the Čech nerve of any good cover. We also give other examples of multiple cohesions, such as differential equivariant types and supergeometric types, laying the groundwork for a synthetic account of Schreiber and Sati's proper orbifold cohomology.

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Section: Mengen Kardinalen Points Codiscrete Discretementioning

confidence: 99%

Section: Mengen Kardinalen Points Codiscrete Discretementioning

confidence: 99%

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Commuting Cohesions

Myers¹,

Riley²

2023

Preprint

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“…In particular, generalizing the elimination rule proved to be problematic. Later work would show that these elimination rule's good behavior relied on additional structure [Gra+22]. In the case of a single modality, this additional structure was often present on the syntax as an admissiblity, but for multiple modalities it was necessary to manually postulate.…”

Section: Related Workmentioning

confidence: 99%

Under Lock and Key: A Proof System for a Multimodal Logic

A.¹,

Gratzer²

2023

Bull. symb. log

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“…The more recent frameworks MTT [12] and FitchTT [11] resolve these problems: they are dependently typed, with a well-behaved definitional equality, and only ever extend the context; all indications suggest their implementability [10,40]. However, their naïve semantics requires the functors interpreting the modal operators to have additional left adjoints ("context locks"), which are not visible internally in the syntax.…”

Section: Introductionmentioning

confidence: 99%

Semantics of multimodal adjoint type theory

Shulman

2023

Electronic Notes in Theoretical Informatics and Computer Science

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We show that contrary to appearances, Multimodal Type Theory (MTT) over a 2-category M can be interpreted in any M-shaped diagram of categories having, and functors preserving, M-sized limits, without the need for extra left adjoints. This is achieved by a construction called "co-dextrification" that co-freely adds left adjoints to any such diagram, which can then be used to interpret the "context lock" functors of MTT. Furthermore, if any of the functors in the diagram have right adjoints, these can also be internalized in type theory as negative modalities in the style of FitchTT. We introduce the name Multimodal Adjoint Type Theory (MATT) for the resulting combined general modal type theory. In particular, we can interpret MATT in any finite diagram of toposes and geometric morphisms, with positive modalities for inverse image functors and negative modalities for direct image functors.

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Modalities and Parametric Adjoints

Cited by 5 publications

References 33 publications

Commuting Cohesions

Commuting Cohesions

Under Lock and Key: A Proof System for a Multimodal Logic

Semantics of multimodal adjoint type theory

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