Tool and Object

Krömer, Ralf

doi:10.1007/978-3-7643-7524-9

Cited by 57 publications

(4 citation statements)

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“…Thus, if they are not included in Feferman's (R3) then that requirement does not accurately capture the desire for unlimited category theory to fit with the rest of mathematics. 16 For readers interested in a full understanding of the history I recommend Krömer (2007) or Marquis (2009).…”

Section: To Graphmentioning

confidence: 99%

The Prospects of Unlimited Category Theory: Doing What Remains to Be Done

Ernst¹

2015

The Review of Symbolic Logic

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Section: To Graphmentioning

confidence: 99%

The Prospects of Unlimited Category Theory: Doing What Remains to Be Done

Ernst¹

2015

The Review of Symbolic Logic

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“…Thus, in their book, Eilenberg & Steenrod gives axiom of homology theories and these axioms stipulates that a homology theory is a functor from a category of topological spaces to an algebraic category, e.g. groups, abelian groups, modules, vector spaces, (24) See [58] and [64] for more on these aspects of the history of category theory. (25) From the point of view of what used to be called "combinatorial topology", it took quite some time to show that combinatorial methods could be applied to an arbitrary topological space.…”

Section: The Abstract Framework: Categorical Contextsmentioning

confidence: 99%

Mathematical Models of Abstract Systems: Knowing abstract geometric forms

Marquis¹

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

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“…I found some brief excursions into mathematical logic unavoidable, but I have tried to explain all logical notions as they occur and relegate the more complicated logical discussion to footnotes. Related papers include [Fef69, Appendix II] (by G. Kreisel) and [Bla84]; a historical survey can be found in [Krö07]. Since my goal is expository, I have tried to restrict citations to those most useful for the student, when such exist.…”

Section: Introductionmentioning

confidence: 99%

Set theory for category theory

Shulman

2008

Preprint

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Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number of such "set-theoretic foundations for category theory," and describe their implications for the everyday use of category theory. We assume the reader has some basic knowledge of category theory, but little or no prior experience with formal logic or set theory.Theorem 2.1. If a category A has products indexed by the collection Arr(A) of arrows in A, then A is a preorder. In particular, any small complete category is a preorder, and no large category that is not a preorder can admit products indexed by proper classes.Proof. Suppose that we had two different arrows f, g : a → b, and form the product Arr(A) b. Then f and g give us 2 |Arr(A)| different arrows a → Arr(A) b; but there are only |Arr(A)| total arrows in A, a contradiction. 1

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Tool and Object

Cited by 57 publications

References 0 publications

The Prospects of Unlimited Category Theory: Doing What Remains to Be Done

The Prospects of Unlimited Category Theory: Doing What Remains to Be Done

Mathematical Models of Abstract Systems: Knowing abstract geometric forms

Set theory for category theory

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