## An Adjoint Characterization of the Category of Sets (1994)

Venue: | Proc. Amer. Math. Soc |

Citations: | 5 - 2 self |

### BibTeX

@ARTICLE{Rosebrugh94anadjoint,

author = {Robert Rosebrugh and R. J. Wood},

title = {An Adjoint Characterization of the Category of Sets},

journal = {Proc. Amer. Math. Soc},

year = {1994},

volume = {122},

pages = {409--413}

}

### OpenURL

### Abstract

If a category B with Yoneda embedding Y : B \Gamma! CAT(B op ; set) has an adjoint string, U a V a W a X a Y; then B is equivalent to set. The authors gratefully acknowledge financial support from NSERC Canada. Diagrams typeset using M. Barr's diagram macros. 1 Introduction The statement of the Abstract was implicitly conjectured in [9]. Here we establish the conjecture. We will see that it suffices to assume that B has an adjoint string V a W a X a Y with V pullback preserving. A word on foundations and our notation is necessary. We write set for the category of small sets and assume that there is a Grothendieck topos, SET, of sets which contains the set of arrows of set as an object. The 2-category of category objects in SET, which we write CAT, is cartesian closed and set is an object of CAT. Thus, for C a category in CAT, CAT(C op ; set) is also an object of CAT and we abbreviate it by MC; (it was written PC in [8].) Substitution gives a 2-functor M : CAT coop \Gamma...

### Citations

32 |
structures on 2-categories
- Street, Walters, et al.
- 1978
(Show Context)
Citation Context ... in SET, which we write CAT, is cartesian closed and set is an object of CAT. Thus, for C a category in CAT, CAT(C op ; set) is also an object of CAT and we abbreviate it by MC; (it was written PC in =-=[8]-=-.) Substitution gives a 2-functor M : CAT coop \Gamma! CAT where CAT coop is the dual which reverses both arrows of CAT (functors) and 2-cells (natural transformations.) A category B in CAT is said to... |

16 | Constructive complete distributivity
- Fawcett, Wood
- 1990
(Show Context)
Citation Context ...inclusion.) This functor has a left adjoint, namely supremum, W ; precisely when B is (co)complete. It is helpful to think of X above as a generalization of W : Continuing the analogy, we recall from =-=[1]-=- that W has a left adjoint precisely when B is (constructively) completely distributive. With this in mind we say that a total category is totally distributive when it has an adjoint string, W a X a Y... |

7 | Constructive complete distributivity IV
- Rosebrugh, Wood
- 1994
(Show Context)
Citation Context ... is the Cauchy completion of A. (Since this part of the Lemma is not central to our present concerns we leave the proof of this claim as an exercise for the reader. In the ord case it is discussed in =-=[5]-=-.) It is easy to see that I is dense and Kan. (if) Given B and I as above, consider the composite B Y \Gamma! MB MI \Gamma! ME = B(I; 1 B ): Since Y and MI have left adjoints, namely X and 9I respecti... |

4 |
Notions of topos
- Street
- 1981
(Show Context)
Citation Context ... the construction of H in Lemma 2 that H preserves pullbacks so E is "lex total", meaning that the defining left adjoint for totality is left exact. (It necessarily preserves the terminal ob=-=ject.) By [6]-=-, E is a Grothendieck topos (for since E is small the size requirement in [6] is trivially satisfied). But since, by Lemma 2, E is also an ordered set it must therefore be 1: Indeed, we have true = fa... |

3 | Some remarks on total categories - Wood - 1982 |

2 |
Legitimite des categories de prefaisceaux, Diagrammes 1
- Foltz
- 1979
(Show Context)
Citation Context ...9I (\Gamma)) �� = MB(Y I; 9I (\Gamma)) �� = (MI \Delta 9I )(\Gamma) �� = \Gamma: Thus B(I; 1) : B \Gamma! ME is an equivalence. Since both E and now ME are locally small it follows from [7=-=] (see also [2]-=-) that E is small as required. If C and D are total then a functor F : C \Gamma! D preserves all colimits if and only if it has a right adjoint. If, moreover, F is Kan then preservation of all colimit... |

2 |
Idempotents in bicategories
- Par'e, Rosebrugh, et al.
- 1989
(Show Context)
Citation Context ...Y: It is convenient to define T = V Y : B \Gamma! B: Then MY \Delta 9W \Delta oe �� = MY \Delta MV \Delta oe �� = M(V Y ) \Delta oe �� = MT \Delta oe which shows that MT coinverts oe: By L=-=emma 4.3 of [4]-=-, T inverts oe. Lemma 2 A category B is equivalent to one of the form MA with A a small, complete ordered set if and only if B is totally distributive with V a W: Proof. (only if) A small, complete or... |

1 |
Unpublished manuscript
- Street
- 1979
(Show Context)
Citation Context ... �� = MB(W I; 9I (\Gamma)) �� = MB(Y I; 9I (\Gamma)) �� = (MI \Delta 9I )(\Gamma) �� = \Gamma: Thus B(I; 1) : B \Gamma! ME is an equivalence. Since both E and now ME are locally small =-=it follows from [7]-=- (see also [2]) that E is small as required. If C and D are total then a functor F : C \Gamma! D preserves all colimits if and only if it has a right adjoint. If, moreover, F is Kan then preservation ... |