## Tags - Part of: [[Mathematics]], [[Category Theory]] - Related: - Includes: - Additional: ## Significance - ## Intuitive summaries - ## Definitions - ## Technical summaries - ## Main resources - <iframe src="https://en.wikipedia.org/wiki/Topos" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> [topos in nLab](https://ncatlab.org/nlab/show/topos) [Category Theory For Beginners: Topos Theory And Subobjects - YouTube](https://www.youtube.com/watch?v=o-yBDYgUqZQ&t=8034s&pp=ygUPc291dGh3ZWxsIHRvcG9z) [Toposes - "Nice Places to Do Math" - YouTube](https://www.youtube.com/watch?v=gKYpvyQPhZo&pp=ygUidG9wb3NlcyAtICJuaWNlIHBsYWNlcyB0byBkbyBtYXRoItIHCQmyCQGHKiGM7w%3D%3D) - Set theory is the traditional foundation for all mathematics, providing a common language [[00:06](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=6), [00:14](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=14)]. - However, mathematicians found that other approaches exist, each offering a new interpretation that might be better suited for specific problems [[00:30](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=30), [00:38](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=38)]. - These alternative approaches can be envisioned as distinct "universes," each with its own rules for mathematical objects, and methods for comparing objects across these universes [[00:44](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=44), [00:50](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=50)]. - A "Topos" is the term for such a mathematical universe, described as a "nice place to do mathematics" [[00:56](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=56), [01:05](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=65)]. - The video uses the algebraic theory of unital commutative rings as an example, noting that each Topos has its own interpretation of this theory, with the set theory interpretation being the standard one taught in abstract algebra [[01:10](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=70), [01:18](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=78)]. - Similarly, the theory of local unital commutative rings has an interpretation in each Topos, where a local ring is a commutative ring [[01:25](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=85), [01:32](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=92)]. - A key point is that within set theory alone, a universal construction of a local ring from a ring is impossible, as there can be multiple ways to make a ring local, often without a single canonical choice [[01:39](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=99), [01:46](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=106)]. - However, by stepping outside a given Topos, a universal local ring construction from rings becomes possible, though it resides in a different Topos [[01:53](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=113), [02:00](http://www.youtube.com/watch?v=gKYpvyQPhZo&t=120)]. ## Landscapes - ## Contents - ## Deep dives - ## Brain storming - ## Additional resources - ## Related - ## Related resources - ## Explanation by AI - ## Written by AI (may include factually incorrect information) - ## Deep dives by AI - ## Written by AI (may include factually incorrect information) - ## Other - ## Other AI -