Citation Context

...sed by taking for C some subcategory of the category of (small) categories and instead of Set some other big category K (e.g. K = Ab and C = Cat). ♦ Example 1.3 Let E be an elementary topos (see e.g. =-=[Joh]-=-). Then E( , Ω) : E op → Ha is a contravariant functor from E to the category Ha of Heyting algebras and their morphisms. ♦ 3Example 1.4 Let C be the category CRng of commutative rings with 1. Then w...

Citation Context

...ause gl(∆) has internal sums whereas Fam(E) doesn’t! Consider also the following somewhat weaker counterexample. Let A be a partial combinatory algebra, RT[A] the realizability topos over A (see e.g. =-=[vOo]-=-) and Γ ⊣ ∇ : Set → RT[A] the geometric morphism where Γ = RT[A](1, −) is the global elements functor. Then gl(∇) = ∇ ∗ PRT[A] is a fibration with stable and disjoint internal sums over Set although f...

Citation Context

...for his lectures and many personal tutorials where he explained to me various aspects of his work on fibred categories. I also want to thank J.-R. Roisin for making me available his handwritten notes =-=[Ben2]-=- of Des Catégories Fibrées, a course by Jean Bénabou given at the University of Louvain-la-Neuve back in 1980. The current notes are based essentially on [Ben2] and a few other insights of J. Bénabou ...

Citation Context

... in 1980. The current notes are based essentially on [Ben2] and a few other insights of J. Bénabou that I learnt from him personally. The last four sections are based on results of J.-L. Moens’ Thése =-=[Moe]-=- from 1982 which itself was strongly influenced by [Ben2]. 1Contents 1 Motivation and Examples 3 2 Basic Definitions 6 3 Split Fibrations and Fibred Yoneda Lemma 8 4 Closure Properties of Fibrations ...

Citation Context

...gories with finite limits the fibration gl(F ) = F ∗ PC has internal sums and small global sections iff F preserves pullbacks and has a right adjoint. 13 13 This was already observed by J. Bénabou in =-=[Ben1]-=-. 69Thus, for categories B with finite limits we get a 1–1–correspondence (up to equivalence) between geometric morphisms to B (i.e. adjunctions F ⊣ U : C → B where C has finite limits and F preserve...