Quasitoposes as elementary quotient completions
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2021
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913536073007104 |
|---|---|
| author | Maietti, Maria Emilia Pasquali, Fabio Rosolini, Giuseppe |
| author_facet | Maietti, Maria Emilia Pasquali, Fabio Rosolini, Giuseppe |
| contents | The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category with finite products and weak equalisers. In this paper we characterise when an elementary quotient completion is a quasi-topos. We obtain as a corollary a complete characterisation of when an elementary quotient completions is an elementary topos. As a byproduct we determine also when the elementary quotient completion of a tripos is equivalent to the doctrine obtained via the tripos-to-topos construction. Our results are reminiscent of other works regarding exact completions and put those under a common scheme: in particular, Carboni and Vitale's characterisation of exact completions in terms of their projective objects, Carboni and Rosolini's characterisation of locally cartesian closed exact completions, also in the revision by Emmenegger, and Menni's characterisation of the exact completions which are elementary toposes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_15299 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quasitoposes as elementary quotient completions Maietti, Maria Emilia Pasquali, Fabio Rosolini, Giuseppe Logic Category Theory The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category with finite products and weak equalisers. In this paper we characterise when an elementary quotient completion is a quasi-topos. We obtain as a corollary a complete characterisation of when an elementary quotient completions is an elementary topos. As a byproduct we determine also when the elementary quotient completion of a tripos is equivalent to the doctrine obtained via the tripos-to-topos construction. Our results are reminiscent of other works regarding exact completions and put those under a common scheme: in particular, Carboni and Vitale's characterisation of exact completions in terms of their projective objects, Carboni and Rosolini's characterisation of locally cartesian closed exact completions, also in the revision by Emmenegger, and Menni's characterisation of the exact completions which are elementary toposes. |
| title | Quasitoposes as elementary quotient completions |
| topic | Logic Category Theory |
| url | https://arxiv.org/abs/2111.15299 |