User dany majard - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:56:36Z http://mathoverflow.net/feeds/user/5375 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81982/what-is-the-intuition-of-connections-for-cubical-sets/82094#82094 Answer by Dany Majard for What is the intuition of connections for cubical sets? Dany Majard 2011-11-28T16:25:08Z 2011-11-29T21:45:51Z <p>As far as higher cubical categories are concerned, a connection will allow you to literally rotate a face, i.e. turn a face of one type into a face of another type, in an invertible way. In short it materializes an equivalence between the different types of faces into special degenerate cubes.</p> <p>The 2d case for example is fairly simple as one can either turn horizontal arrows into vertical arrows or vice versa.</p> <p>One advantage of a connection is therefore that it allows one to speak of commutative n-cubes in an n-tuple category with connection. To do so, you can take an n-cube, apply connections until you only have non trivial faces of one type. Then check whether the obtained cube is an identity or not. It turns out that it does not depend on the way you chose to apply the connection, if your cube gives an identity cube with one face rearrangement, it will with another. It is, to my understanding the essence of Brown and AlAlg's equivalence between cubical categories with connections and globular categories.</p> <p>So for cubical categories it is very restrictive, which is also why they are so friendly. But I am not sure about the impact on cubical sets. You surely will find good material in Tim and Ronnie's references.</p> http://mathoverflow.net/questions/44707/connections-in-double-categories Connections in double categories Dany Majard 2010-11-03T17:49:06Z 2011-06-30T09:22:12Z <p>There exist a structure on double categories due to R.Brown called a connection. The connection embodies in squares an isomorphism between the category of its vertical arrows and the category of its horizontal arrows and it allows to change boundaries of a square from one type to the other. It generates a double subcategory that is very interesting to me. I was wondering if anyone had thought of and found a way to describe this subcategory without the mention of "commutative squares" or the construction of the connection itself. I have the feeling that it is given by an adjunction but it seems to elude me.</p> <p>Please comment if you want more details.</p> http://mathoverflow.net/questions/41057/categories-first-or-categories-last-in-basic-algebra/42581#42581 Answer by Dany Majard for Categories First Or Categories Last In Basic Algebra? Dany Majard 2010-10-18T04:46:39Z 2010-10-18T04:46:39Z <p>Everyone will agree with me that there are many levels of abstraction category theory can be introduced at. It makes no sense to start undergraduate math courses with a formal approach to category theory, I don't think anyone would argue the opposite. It makes very little sense either to postpone it to higher algebra classes of late undergrad at best or, as happens in many places, in graduate studies. </p> <p>Category theory is <strong>above all</strong> a formalism, a way to frame our understanding. It has been a more and more prevalent facet of my thinking that a good notation does half the work of solving a problem, just as formulating a question properly does. Why then not start hinting towards such formulation early ? While teaching low level courses, I always have, or make a point to ensure that most of my class knows what a function is. While doing so I draw little blobs representing sets and big arrows representing a function. Then as I talk I keep presenting functions as a processes, or relations. Together with a fun example (I usually use a "friends and beer" variation) it helps them structure the knowledge they are presented with. It makes it easier to have them understand that one cannot just "add" functions by writing a plus sign in between since functions are (visually) not the same entity as numbers. It is I believe our duty to frame things as early as possible in a way that <strong>structures knowledge</strong> in the student's mind. To make another reference to food, it is better to have widely spread malleable foundations of rudimental cooking than of an elite of highly qualified cooks (of course it's best to have both).</p> <p>Moreover I would like to point out that this formalism is <strong>urgently</strong> needed in other areas of science. As a physicist by training I cannot overstate the importance of category theory in areas of science other than mathematics. And even after a MS in theoretical and mathematical physics, "functors" and "categories" were frightening words that were reserved to Jedi Masters. I am but saddened by that state of things. About everything in physics deals with processes and change and yet there seems to be very little push to spread the categorical lingua. Relativity <strong>screams</strong> category theory (equivalent views of the world in different frames yet non identical), the standard model's soul is categorical (groups, tensor structures of representations, etc...). Why should we wait so long to plant these seeds ? Why not let them germ throughout the student's curricula.</p> <p>In conclusion while it is dysfunctional to force feed students categories (why teach an intensive Japanese course to someone that just wants to make suchi ?), it is criminal to keep it, to its core, our little secret. I believe we need to join forces to move very basic categorical formalism to bigger circles, sans tambours ni trompettes (without fanfare), and without bells and whistles.</p> http://mathoverflow.net/questions/42332/on-a-special-kind-of-graph-connectig-n-point-to-n-points On a special kind of graph connectig n point to n points. Dany Majard 2010-10-15T21:42:25Z 2010-10-16T02:38:09Z <p>I don't know anything about graph theory and I was wondering about something :</p> <p>If you draw two parallel rows of n points in $\mathbb R^4$ and link each point with all the points in the opposite row except the one right in front of it, and then allow the points to move where ever, what do you get ?</p> <p>n=2 gives two lines</p> <p>n=3 gives a hexagon</p> <p>n=4 gives a cube</p> <p>n=5 ...</p> <p>Is there a general object class apart the one described just the way I did ?</p> http://mathoverflow.net/questions/63440/groupoid-actions-on-spaces Comment by Dany Majard Dany Majard 2011-04-29T23:15:43Z 2011-04-29T23:15:43Z @Michael : I believe that the space it would act on is the disjoint union of all the spaces in the image of your functor. In the definition on the action of a groupoid G on a set S, there is a surjective map $p:S\to G_0$ where $G_0$ is the set of objects in your groupoid. Then a morphism $g\in G$ acts on the fiber by p over its taget and sends it to the fiber over its source. This can be viewed as a functor $G\to Set$ that sends objects to their fiber by p. http://mathoverflow.net/questions/44707/connections-in-double-categories/51896#51896 Comment by Dany Majard Dany Majard 2011-01-13T18:31:34Z 2011-01-13T18:31:34Z The big double category in question is what is called the quintet category, it is constituted of all possible squares, not only the commutative ones). http://mathoverflow.net/questions/44707/connections-in-double-categories/44714#44714 Comment by Dany Majard Dany Majard 2010-11-09T20:02:33Z 2010-11-09T20:02:33Z Yes Buschi, I know this paper but it doesn't answer my question sadly... http://mathoverflow.net/questions/42332/on-a-special-kind-of-graph-connectig-n-point-to-n-points/42335#42335 Comment by Dany Majard Dany Majard 2010-10-15T22:52:44Z 2010-10-15T22:52:44Z So you mean that the adjacent condition on the product is an logical &quot;and&quot; on the product of the adjacent condition on the individual graphs. That makes sense. http://mathoverflow.net/questions/42332/on-a-special-kind-of-graph-connectig-n-point-to-n-points Comment by Dany Majard Dany Majard 2010-10-15T22:42:40Z 2010-10-15T22:42:40Z Will: Indeed I just wanted to get rid of knot possibilities. JBL: I initially called the question Graphs as Skeltons but thought it might be too vague a title.