COOKIES: By using this website you agree that we can place Google Analytics Cookies on your device for performance monitoring. |

University of Cambridge > Talks.cam > Category Theory Seminar > Generalising the functor of points approach

## Generalising the functor of points approachAdd to your list(s) Download to your calendar using vCal - Zhen Lin Low - DPMMS
- Tuesday 19 May 2015, 14:15-15:15
- MR5, Centre for Mathematical Sciences.
If you have a question about this talk, please contact Dr Ignacio Lopez Franco. The passage from commutative rings to schemes has three main steps: first, one identifies a distinguished class of ring homomorphisms corresponding to open immersions of schemes; second, one defines the notion of an open covering in terms of these distinguished homomorphisms; and finally, one embeds the opposite of the category of commutative rings in an ambient category in which one can glue (the formal duals of) commutative rings along (the formal duals of) distinguished homomorphisms. Traditionally, the ambient category is taken to be the category of locally ringed spaces, but following [Demazure and Gabriel], one could equally well work in the category of sheaves for the large Zariski site – this is the so-called ‘functor of points approach’. The three procedures described above can be generalised to other contexts. The first step essentially amounts to reconstructing the class of open embeddings from the class of closed embeddings. Once we have a suitable class of open embeddings, the class of open coverings is a subcanonical Grothendieck pretopology. We then define a notion of ‘charted space’ in the category of sheaves. This gives a uniform way of defining locally Hausdorff spaces, schemes, locally finitely presented C^\infty-schemes etc. as special sheaves on their respective categories of local models, taking as input just the class of closed embeddings. We can also get many variations on manifolds by skipping the first step and working directly with a given class of open embeddings. This talk is part of the Category Theory Seminar series. ## This talk is included in these lists:- All CMS events
- All Talks (aka the CURE list)
- CMS Events
- Category Theory Seminar
- DPMMS Lists
- DPMMS Pure Maths Seminar
- DPMMS info aggregator
- DPMMS lists
- Hanchen DaDaDash
- Interested Talks
- MR5, Centre for Mathematical Sciences
- School of Physical Sciences
- bld31
- ndb35's list
- yk373's list
Note that ex-directory lists are not shown. |
## Other listsCMS Events Von Hügel Institute events Social Anthropology Post-Doc Seminar## Other talksMEMS Particulate Sensors Cooperation, Construction, Coercion, Consent: Understanding the Role of Reimagined Urban Space within Nazi Germany and Fascist Italy Britain, Jamaica and the modern global financial order, 1800-50 The Ambonese Rumphius and his inter-island information networks How could education systems research prompt a change to how DFIS works on education |