Category theory seminar abstracts
Abstracts and references saved with the November and October 2024 log entries. Each abstract is attributed to its speaker.
November 2024
“Finitely complete co-pretoposes are ideally exact.” James R. A. Gray
abstract. Semi-abelian categories were introduced in (JMT) to provide a categorical context in which to give a general treatment of the properties of the categories of groups, Lie algebras, and other related algebraic structures. Certain categories which have been of interest in classical algebra are not semi-abelian; these include rings with identity, Boolean algebras, and Heyting algebras (among others). Ideally exact categories were introduced in (J) as a non-pointed counterpart to semi-abelian categories and include the above-mentioned categories as examples.
A topos provides a categorical context in which one can “do mathematics.” Examples of toposes include the category of sets or, more generally, functor categories of sets, as well as the category of sheaves on a site. It turns out that every cotopos is ideally exact (see (B) for exactness and protomodularity). The category of compact Hausdorff spaces is not a topos but satisfies two important properties of a topos, namely, it is extensive and exact. It was shown in (BC) that the opposite of the category of pointed objects of the category of compact Hausdorff spaces is semi-abelian.
In this talk, we will briefly outline the known proof showing that co-toposes are ideally exact and give an outline of the proof that finitely complete co-pretoposes are ideally exact.
(B) D. Bourn, Protomodular aspect of the dual of a topos, Adv. Math. 187 (2004), no. 1, 240-255
(BC) F. Borceux, and M. M. Clementino, On toposes, algebraic theories, semi-abelian categories and compact Hausdorff spaces
(JMT) G. Janelidze, L. M´arki, and W. Tholen, Semi-abelian categories, Category theory 1999 (Coimbra), J. Pure Appl. Algebra 168 (2002), no. 2-3, 367-386
(J) G. Janelidze, Ideally exact categories, Theory and Applications of Categories, Vol. 41, No. 11, 2024, pp. 414–425.
Category Theory Seminar Talk, November 6
Logical notions
The November entry also recorded this passage from Alfred Tarski:
consider the class of all one-one transformations of the space, or universe of discourse, or “world”, onto itself. What will be the science which deals with the notions invariant under this widest class of transformations? Here we will have very few notions, all of a very general character. I suggest that they are the logical notions, that we call a notion “logical” if it is invariant under all possible one-one transformations of the world onto itself (“What are logical notions?”, Alfred Tarski, 1986, p. 149).
October 2024
“On extensivity of morphisms.” Michael Hoefnagel
abstract. Extensivity of a category may be described as a property of coproducts in the category, namely, that they are disjoint and universal. An alternative viewpoint is that it is a property of morphisms in a category. This talk explores this point of view through a natural notion of extensive and coextensive morphism. There is an interesting interplay between the algebraic and the categorical. One the one hand, through these notions several topics in algebra related to (unique) factorisation and refinement of direct products, such as the strict refinement or Fraser-Horn properties, take categorical form, and thereby enjoy all the benefits of categorical generalisation. On the other hand, the algebraic theory surrounding these topics inspire categorical results. One such result is that a Barr-exact category is coextensive if and only if every split monomorphism in the category coextensive.
Category Theory Seminar Talk, October 30