## Strict 2-toposes (2006)

### BibTeX

@MISC{Weber06strict2-toposes,

author = {Mark Weber},

title = {Strict 2-toposes},

year = {2006}

}

### OpenURL

### Abstract

Abstract. A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category [Str74b] and provides a self-contained development of the necessary background material on yoneda structures.

### Citations

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Citation Context ...iminaries 2.1. Finitely complete 2-categories. Recall that a 2-category K is finitely complete when it admits all limits weighted by 2-functors I : J→CAT, in the sense of CAT-enriched category theory =-=[Kel82]-=-, such that the set of 2-cells of J and the sets of arrows of the I(j) for j ∈ J, are all finite. This means that for such I and T : J→K there is an object lim(I, T) of K and isomorphisms of categorie... |

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Citation Context ...pullbacks, pseudo pullbacks and lax pullbacks in Cat(E). One can either do this directly, or by interpretting the explicit description of these constructions in CAT in the internal language of E (see =-=[Joh02]-=-). A nice consequence of this is that for any pullback preserving functor E→E ′ , the 2-functor Cat(E)→Cat(E ′ ) it induces preserves pullbacks, pseudo pullbacks and lax pullbacks. Lax pullbacks satis... |

143 | Higher-dimensional algebra and topological quantum field theory
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Citation Context ...s facilitated. This will then be used in future work on reconciling the notions of higher dimensional categories in [BD98] and [Bat98b], and in an operadic exploration of the stabilisation hypothesis =-=[BD95]-=-. In the 1970’s internal colimits were understood from a 2-categorical perspective in the work of Ross Street and Bob Walters [SW78]. In this paper the concept of a yoneda structure on a 2-category wa... |

55 |
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Citation Context ...lters [SW78]. In this paper the concept of a yoneda structure on a 2-category was discovered; inspired largely by the work of Bill Lawvere on the foundational importance of the category of categories =-=[Law70]-=-. Logical motivations notwithstanding, the perspective of this paper is that the point of having a yoneda structure on a 2-category K, is that one can then say what it means for an object of K to be c... |

49 |
Sheaves in geometry and logic
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Citation Context ...0Ω0 in E such that for all E ∈ E the function E(E, Ω0) → Sub0(E) given by pulling back τ0, is a bijection. Since E is locally small Sub0 actually lands in Set. Moreover it is easily verified (or see =-=[MM91]-=- page 33 for example) that Ω•0 is a terminal object, so we will denote it as 1. Subobjects form not just a set but a poset, and as we shall recall below in example(4.3), τ0 is the object part of an in... |

44 | Higher-dimensional algebra VI: Lie 2-algebras, to appear in Theory and Applications of Categories. Also available as math.QA/0307263
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Citation Context ...ample in the work of Michael Batanin [Bat98b] [Bat98a] [Bat02] [Bat03] internal category theory is the backdrop for the theory of higher operads. In the work of John Baez and his collaborators [BL04] =-=[BC04]-=- [BS05] we see internal categories as fundamental in the process of categorifying differential geometry and gauge theory with a view to applications in physics. This paper is about doing category theo... |

41 | Higher Gauge Theory
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Citation Context ...n the work of Michael Batanin [Bat98b] [Bat98a] [Bat02] [Bat03] internal category theory is the backdrop for the theory of higher operads. In the work of John Baez and his collaborators [BL04] [BC04] =-=[BS05]-=- we see internal categories as fundamental in the process of categorifying differential geometry and gauge theory with a view to applications in physics. This paper is about doing category theory inte... |

38 | Fibred categories and the foundations of naive category theory - Bénabou - 1985 |

30 |
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Citation Context ...sheaves on A. The main results of this paper describe when the yoneda structures that arise from 2-toposes have presheaves which are cocomplete. Our basic references for background on 2-categories is =-=[KS74]-=- and [Str80b]. Another important background article is [Str74b] although efforts are made in this paper to keep the exposition relatively self-contained. The 2-categorical background pertinent to this... |

29 | Homotopical algebraic geometry I: topos theory, preprint available at http://arxiv.org/PS cache/math/pdf/0207/0207028.pdf - Toen, Vezzosi |

28 |
Fibred and cofibred categories
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Citation Context ...an work 2-categorically and follow [Str74b], or one can regard K as a bicategory following [Str80b]. For the case K = CAT the 2-categorical definition of fibration coincides with that of Grothendieck =-=[Gra66]-=- whereas the bicategorical definition of fibration is more general. In this paper we shall consider only the stronger notion. Let f : A→B be a functor. A morphism α : a1→a2 in A is f-cartesian when fo... |

22 |
Higher-dimensional algebra V: 2-groups, Th
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Citation Context ... For example in the work of Michael Batanin [Bat98b] [Bat98a] [Bat02] [Bat03] internal category theory is the backdrop for the theory of higher operads. In the work of John Baez and his collaborators =-=[BL04]-=- [BC04] [BS05] we see internal categories as fundamental in the process of categorifying differential geometry and gauge theory with a view to applications in physics. This paper is about doing catego... |

13 |
Operads within monoidal pseudo algebras, preprint available at www.mathstat.uottawa.ca/ mwebe937/research/Preprints.html
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(Show Context)
Citation Context ...theory internally with a particular focus on the theory of colimits. It was motivated by the need to manipulate internal colimits more easily in order to push forward the theory of higher operads. In =-=[Web]-=- the results and notions of the present paper are used to bring all the operad theory of [Bat98b] to the level of generality of [Web05] so that the theory of higher symmetric operads is facilitated. T... |

12 |
Categories ´ Fibrees ´ et Descente
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Citation Context ...ten A/A. Some other notation: we shall denote a terminal object of K by 1 and for A ∈ K the unique map A→1 by tA. 2.2. Fibrations. The concept of a fibration between categories is due to Grothendieck =-=[Gro70]-=-. Fibrations can be defined internal to any finitely complete 2-category K. In fact there are two approaches: one can work 2-categorically and follow [Str74b], or one can regard K as a bicategory foll... |

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Citation Context ...s due to Ehresmann [Ehr58]. This notion has experienced somewhat of a resurgence in recent times because of developments in higher category theory. For example in the work of Michael Batanin [Bat98b] =-=[Bat98a]-=- [Bat02] [Bat03] internal category theory is the backdrop for the theory of higher operads. In the work of John Baez and his collaborators [BL04] [BC04] [BS05] we see internal categories as fundamenta... |

10 |
Gattungen von lokalen Strukturen
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Citation Context ...y complete 2category [Str74b] and provides a self-contained development of the necessary background material on yoneda structures. 1. Introduction The idea of an internal category is due to Ehresmann =-=[Ehr58]-=-. This notion has experienced somewhat of a resurgence in recent times because of developments in higher category theory. For example in the work of Michael Batanin [Bat98b] [Bat98a] [Bat02] [Bat03] i... |

8 | Some properties of Fib as a fibred 2-category
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Citation Context ...n CAT/C. In particular notice that the condition that f be a fibration over C is in general weaker than the condition that f be a fibration, although they are equivalent when f lives in Fib(C) [B´85] =-=[Her99]-=- as we shall now recall. For such an f let φ : b→fa. To obtain a cartesian lift for φ one first takes a cartesian lift φ1 : a1→a of β(φ). Since f preserves cartesian arrows φ factors uniquely as b φ2 ... |

8 |
Elementary Cosmoi I
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Citation Context ...adigmatic example described above is an important observation of Bill Lawvere. A 2-categorical expression of this property is provided in the work of Ross Street by the notion of a fibrational cosmos =-=[Str74a]-=- [Str80a]. The 2-toposes of this work are simply cosmoses whose underlying 2-category is cartesian closed and comes equipped with a duality involution1 . This notion is isolated here because it is eas... |

5 |
Connected limits, familial representability and
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Citation Context ...at the objects of K/A are arrows f : X→A, and a morphism f1→f2 is an arrow g of K such that f2g = f1. However a 2-cell γ : g1⇒g2 of K/A is a 2-cell γ : g1⇒g2 of K such that f2γ is an identity. Recall =-=[CJ95]-=- that the 2-functor K/A→K whose object map takes the domain of an arrow into A, creates any connected limits that exist in K. When K is finitely complete and f : A→B in K, the processes of taking the ... |

4 |
of internal categories
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(Show Context)
Citation Context ... example described above is an important observation of Bill Lawvere. A 2-categorical expression of this property is provided in the work of Ross Street by the notion of a fibrational cosmos [Str74a] =-=[Str80a]-=-. The 2-toposes of this work are simply cosmoses whose underlying 2-category is cartesian closed and comes equipped with a duality involution1 . This notion is isolated here because it is easier to ex... |

3 |
globular categories as a natural environment for the theory of weak n-categories
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(Show Context)
Citation Context ...ategory is due to Ehresmann [Ehr58]. This notion has experienced somewhat of a resurgence in recent times because of developments in higher category theory. For example in the work of Michael Batanin =-=[Bat98b]-=- [Bat98a] [Bat02] [Bat03] internal category theory is the backdrop for the theory of higher operads. In the work of John Baez and his collaborators [BL04] [BC04] [BS05] we see internal categories as f... |

3 |
algebra III: n-categories and the algebra of opetopes, Adv
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- 1998
(Show Context)
Citation Context ... the level of generality of [Web05] so that the theory of higher symmetric operads is facilitated. This will then be used in future work on reconciling the notions of higher dimensional categories in =-=[BD98]-=- and [Bat98b], and in an operadic exploration of the stabilisation hypothesis [BD95]. In the 1970’s internal colimits were understood from a 2-categorical perspective in the work of Ross Street and Bo... |

2 |
On the size of categories, Theory and applications of categories 1
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Citation Context ...s will be small in our sense: A and Â are admissible. The converse, that A and Â admissible implies that A is equivalent to a category which has a small set of arrows, is a result of Freyd and Street =-=[FS95]-=-. Remark 6.3. The results of [FS95] apply also when “small” is taken to mean finite, or more generally “of cardinality less than λ” where λ is a regular cardinal. Thus throughout this work, one could ... |

2 |
and yoneda’s lemma in a 2-category
- Fibrations
- 1974
(Show Context)
Citation Context ...t to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2category =-=[Str74b]-=- and provides a self-contained development of the necessary background material on yoneda structures. 1. Introduction The idea of an internal category is due to Ehresmann [Ehr58]. This notion has expe... |

2 |
indexed by category-valued 2-functors
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(Show Context)
Citation Context ...0 : 1→Ω0. A Lawvere-Tierney topology on E amounts to an idempotent monad j on Ω in Cat(E), such that j preserves ∧. As a monad in a finitely complete 2-category we can take its Eilenberg-Moore object =-=[Str76]-=-, part of which is the forgetful arrow u j : Ω j →Ω. Since j is idempotent, u j is fully faithful and by proposition(4.4) Ω j • �� 1 τj Ω j u j � Ω τ τj is a classifying discrete opfibration, and it i... |

2 |
within monoidal pseudo algebras, Applied Categorical Structures 13
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- 2005
(Show Context)
Citation Context ...e easily in order to push forward the theory of higher operads. In [Web] the results and notions of the present paper are used to bring all the operad theory of [Bat98b] to the level of generality of =-=[Web05]-=- so that the theory of higher symmetric operads is facilitated. This will then be used in future work on reconciling the notions of higher dimensional categories in [BD98] and [Bat98b], and in an oper... |