# Invited Paper Session Abstracts - Category Theory for All

Saturday, August 4, 1:30 p.m. - 4:20 p.m., Plaza Ballroom D, Plaza Building

Category theory can be thought of as being "very abstract algebra". It is typically taught at graduate school or in some select cases to advanced undergraduates. In this session we will show ways in which category theory can be taught in a meaningful way to undergraduates and those without particularly aptitude or expertise in math, even high school and middle school students. In the process, we will emphasize important aspects of mathematics that are not to do with solving problems, proving theorems, or getting the right answer, including: making connections between different situations, illuminating deep structures, finding fundamental reasons for things, and improving the clarity of our thinking. The talks will be of interest for general enrichment as well as pedagogy.

Oragnizer:
Eugenia Cheng, School of the Art Institute of Chicago

#### Making Distinctions: Interpreting the Notion of Sameness

1:30 p.m. - 2:05 p.m.
Alissa Crans, Loyola Marymount University

Walgreens or CVS? Same difference if you’re just stopping on the way home for a box of tissues. But certainly not to the employee whose shift starts tonight at 5 pm! As we know, a fundamental problem in mathematics consists of determining whether two given mathematical structures are the same'. But what exactly do we mean when we say that two gadgets are the same? Often, we mean “sufficiently the same for our purposes,” and that purpose naturally differs from field to field. We will explore mathematical interpretations of being the same' by carefully examining the concept of equality and comparing it to weaker notions of sameness. No prior knowledge of category theory will be assumed.

#### Social Choice and Functoriality

2:15 p.m. - 2:50 p.m.
Sarah Yeakel, University of Maryland

A town wants to vote on the placement of a statue in a park. Is it possible to find a solution where everyone gets an equal say? The question boils down to a topological problem; what does the park look like? But whether such a vote will result in a happy town is more easily answered by looking at some group theory. The reason this comparison works is because of a piece of category theory that facilitates moving between different mathematical worlds. Functoriality is the esssentially principle of mapping from one world to another while respecting aspects of its mathematical structure. It enables us to study one world via a different one that we might understand better. In this talk we will discuss how this helps us in the question of the statue, to give a taste of how it helps in broader mathematics. No prior knowledge of group theory, topology or category theory will be assumed.

#### Unifying Different Worlds in Mathematics

3:00 p.m. - 3:35 p.m.
Angélica Osorno, Reed College

Category theory can be thought of as a language and a framework for making comparisons between different worlds. Often the comparisons consist of forgetting certain details in two worlds so that we see a sense in which the two worlds are the same deep down. In this talk we'll apply this principle to two constructions that might at first sight seem very different: the free group generated by a set, and the discrete topology on a set. We will find a property satisfied by both, that we can make precise using category theory. Using this example and many others, we will show that category theory allows us to unify concepts from different areas of mathematics and work with them as if they were the same. No prior knowledge of group theory, topology or category theory will be assumed.

#### From Arithmetic to Category Theory

3:45 p.m. - 4:20 p.m.
Emily Riehl, Johns Hopkins University

You probably know that a x (b+c) = a x b + a x c because of the distributive law of multiplication over addition. But why is it true? Typically we might prove it using repeated addition, or perhaps by a geometric method involving areas of rectangles. In this talk we'll prove it via a roundabout method that takes us on a tour through several deep ideas from category theory including categorification, the Yoneda lemma, universal properties, and adjunctions. The point here isn't to re-prove a familiar result, but to show how ideas in category theory can stem from familiar basic math and yet be generalized to encompass wildly diverse examples. No prior knowledge of category theory will be assumed.

Year:
2018