Categories for the Working Mathematician

Lane, Saunders Mac

doi:10.1007/978-1-4757-4721-8

Cited by 520 publications

(475 citation statements)

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“…Fun.J; D/. These functors can be described explicitly in terms of the over-categories f # j and under-categories j # f (Mac Lane [15,II.6]…”

Section: 5mentioning

confidence: 99%

Representations of spaces

Chachólski¹,

Scherer

2008

Algebr. Geom. Topol.

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We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Spaces. 55U35, 18G55; 18G10

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“…Fun.J; D/. These functors can be described explicitly in terms of the over-categories f # j and under-categories j # f (Mac Lane [15,II.6]…”

Section: 5mentioning

confidence: 99%

Representations of spaces

Chachólski¹,

Scherer

2008

Algebr. Geom. Topol.

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“…If T is an R-ring spectrum, we may consider algebras over the monad [23,VI] T ∧ R − in D R . These are objects M of D R with an action map T ∧ R M → M in D R , satisfying the associativity and unit axioms.…”

Section: Notation and Terminologymentioning

confidence: 99%

I–adic towers in topology

Wüthrich

2005

Algebr. Geom. Topol.

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A large variety of cohomology theories is derived from complex cobordism M U * (−) by localizing with respect to certain elements or by killing regular sequences in M U * . We study the relationship between certain pairs of such theories which differ by a regular sequence, by constructing topological analogues of algebraic I -adic towers. These give rise to Higher Bockstein spectral sequences, which turn out to be Adams spectral sequences in an appropriate sense. Particular attention is paid to the case of completed Johnson-Wilson theory E(n) and Morava K -theory K(n) for a given prime p.

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“…The trace has an equivalent definition, based on the standard construction of the subdivision category [27] which allows any coend to be expressed as a colimit. For a category Q its subdivision category Q is defined as follows.…”

Section: Definition 18 Formentioning

confidence: 99%

“…For general reasons π * 1 has a right adjoint (constructed as a right Kan extension-see [27,19]). Thus π * 1 and, similarly, π * 2 are left adjoints and so preserve (surjective) open maps.…”

Section: Definition 26mentioning

confidence: 99%

Relational Semantics of Non-Deterministic Dataflow

Hildebrandt¹,

Panangaden²,

Winskel³

1997

BRICS

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We recast dataflow in a modern categorical light using profunctors as a generalization of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higher-order dataflow as a biproduct, essentially by following the geometry of interaction programme.

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Categories for the Working Mathematician

Cited by 520 publications

References 0 publications

Representations of spaces

Representations of spaces

I–adic towers in topology

Relational Semantics of Non-Deterministic Dataflow

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