Internal sums for synthetic fibered $(\infty,1)$-categories

Weinberger, Jonathan

doi:10.48550/arxiv.2205.00386

Cited by 1 publication

(1 citation statement)

References 36 publications

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“…In this paper, we will axiomatize the 1-simplex ∆[1] as a linear order with distinct top and bottom and use the cohesive modalities to define the Čech nerve of a map and the realization or colimit of a simplicial type. We believe our approach here would pair nicely with other approaches to simplicial types for the purposes of synthetic (∞, 1)-category theory such as [42,14,49,48], where the sk 0 modality would take the core of a Rezk type. 1 Outline of the present paper.…”

Section: Mengen Kardinalen Points Codiscrete Discretementioning

confidence: 86%

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: 86%