The purpose of this series of lectures is to being to explain what anima are and what are they good for. This can only be a partial explanation for two reasons. First, we do not yet understand the true nature of anima. In particular, we do not know a list of basic assumptions that they must safisfy or even a language in which to express these basic assumptions. (Lurie has constructed a theory of anima within set theory, and while this gives a workable and powerful theory, it does not answer these questions.) Second, the full-fleged theory of infinity-categories is too much to cover in these lectures. So we will settle for the theory of 1-categories, which is good enough to amply display the difference with set theory and illustrate the features of the new theory.
The main distinction with set theory is that equality is not a meaningful notion in the new setting. We cannot say that two objects x and y are equal. Instead, we must explicitly say how to compare x and y. This is a big difference! For to say that x = y is a property, whereas to provide a comparison f : y → x is structure. We already know this phenomenon well from many parts of mathematics. To wit, in linear algebra, it is not meaningful to ask if two vector spaces V and W are equal. Instead we should produce a linear map f : W → V and show that it is an isomorphism. The map f : W → V tells us how to translate between V and W and not only that a translation is possible. Obviously, knowing how to translate is much more useful than knowing that a translation is possible. It is this kind of information that anima encode.
Time and place: Tuesday 2:45-4:15 on NUCT.
Lecture notes: Five lectures on category theory