## Values and Inclusivity in the ACT Community

27 May, 2020

In the tenth and final talk of this spring’s ACT@UCR seminar, Nina Otter will lead a discussion about diversity in the applied category theory community. This is a change from her originally scheduled talk, due to the killing of George Floyd and ensuing events.

The discussion will take place at the originally scheduled time: on Wednesday June 3rd at 5 pm UTC, which is 10 am in California, or 1 pm on the east coast of the United States, or 6 pm in the UK. It will be held online via Zoom, here:

https://ucr.zoom.us/j/607160601

Afterwards we will talk at the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/June.203rd.3A.20Nina.20Otter

You can see here slides here.

• Nina Otter, Values and Inclusivity in the Applied Category Theory Community

Abstract. Saddened by the current events, we are taking this opportunity to pause and reflect on what we can do to change the status quo and try to bring about real and long-lasting change. Thus, we are holding a discussion aimed at finding concrete solutions to make the Applied Category Theory community more inclusive, and also to reflect about the values that our community would like to stand for and endorse, in particular, in terms of which sources of funding go against our values. While this discussion is specific to the applied category theory community, we believe that many of the topics will be of interest also to people in other fields, and thus we welcome anybody with an interest to attend. The discussion will consist of two parts: we will have first several people give short talks to discuss common issues that we need to address, as well as present specific plans for initiatives that we could take. We believe that the current pandemic, and the fact that all activities are now taking place remotely, gives us the opportunity to involve people who would otherwise find it difficult to travel, because of disabilities, financial reasons or care-taking responsibilities. Thus, now we have the opportunity to come up with new types of mentoring, collaborations, and many other initiatives that might have been difficult to envision until just a couple of months ago. The second part of the discussion will take place on the category theory community server, and its purpose is to allow for a broader participation in the discussion, and ideally during this part we will be able to flesh out in detail the specific initiatives that have been proposed in the talks.

## The Legendre Transform: a Category Theoretic Perspective

26 May, 2020

In the ninth talk of the ACT@UCR seminar, Simon Willerton told us about a categorical approach to the Legendre transform, and its connection to tropical algebra.

He gave his talk on Wednesday May 27th. Afterwards we discussed it on the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/May.2027th.3A.20Simon.20Willerton

You can view or join the conversation there if you sign in.

You can see his slides here, or download a video here, or watch the video here:

• Simon Willerton, The Legendre–Fenchel transform from a category theoretic perspective.

Abstract. The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this talk I’ll show how it arises in the context of category theory using categories enriched over the extended real numbers $\overline{ \mathbb{R}}:=[-\infty,+\infty].$ It turns out that it arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (e.g. in Galois theory or algebraic geometry) arise from a relation between sets.

I won’t assume knowledge of the Legendre-Fenchel transform.

The talk is based on this paper:

• Simon Willerton, The Legendre-Fenchel transform from a category theoretic perspective.

Also see his blog article:

• Simon Willerton, The nucleus of a profunctor: some categorified linear algebra, The n-Category Café.

## A Complete Axiomatisation of Partial Differentiation

18 May, 2020

In the eighth talk of the ACT@UCR seminar, Gordon Plotkin told us about partial differentiation, viewed as a logical theory.

He gave his talk on Wednesday May 20th. Afterwards we discussed it on the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/May.2020th.3A.20Gordon.20Plotkin

You can view or join the conversation there if you sign in.

You can see his slides here, or download a video of his talk here, or watch his video here:

• Gordon Plotkin, A complete axiomatisation of partial differentiation.

Abstract. We formalise the well-known rules of partial differentiation in a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to polynomial interpretations. The proof makes use of Severi’s theorem that all multivariate Hermite problems are solvable. We also hope to present a number of related results, such as decidability and Hilbert–Post completeness.

## Formal Concepts vs Eigenvectors of Density Operators

7 May, 2020

In the seventh talk of the ACT@UCR seminar, Tai-Danae Bradley told us about applications of categorical quantum mechanics to formal concept analysis.

She gave her talk on Wednesday May 13th. Afterwards we discussed her talk at the Category Theory Community Server. You can see those discussions here if you become a member:

You can see her slides here, or download a video here, or watch the video here:

• Tai-Danae Bradley: Formal concepts vs. eigenvectors of density operators.

Abstract. In this talk, I’ll show how any probability distribution on a product of finite sets gives rise to a pair of linear maps called density operators, whose eigenvectors capture “concepts” inherent in the original probability distribution. In some cases, the eigenvectors coincide with a simple construction from lattice theory known as a formal concept. In general, the operators recover marginal probabilities on their diagonals, and the information stored in their eigenvectors is akin to conditional probability. This is useful in an applied setting, where the eigenvectors and eigenvalues can be glued together to reconstruct joint probabilities. This naturally leads to a tensor network model of the original distribution. I’ll explain these ideas from the ground up, starting with an introduction to formal concepts. Time permitting, I’ll also share how the same ideas lead to a simple framework for modeling hierarchy in natural language. As an aside, it’s known that formal concepts arise as an enriched version of a generalization of the Isbell completion of a category. Oftentimes, the construction is motivated by drawing an analogy with elementary linear algebra. I like to think of this talk as an application of the linear algebraic side of that analogy.

Her talk is based on her thesis:

• Tai-Danae Bradley, At the Interface of Algebra and Statistics.

## Separation Logic Through a New Lens

5 May, 2020

In the sixth talk of the ACT@UCR seminar, Sarah Rovner-Frydman told us about a new approach to separation logic, a way to reason about programs.

She gave her talk on May 6th, 2020. Afterwards we discussed it on the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/May.204th.3A.20Sarah.20Rovner-Frydman/near/196269053

You can view or join the conversation there if you sign in.

You can see her slides here, or download a video of her talk here, or watch the video here:

• Sarah Rovner-Frydman: Separation logic through a new lens.

Abstract. Separation logic aims to reason compositionally about the behavior of programs that manipulate shared resources. When working with separation logic, it is often necessary to manipulate information about program state in patterns of deconstruction and reconstruction. This achieves a kind of “focusing” effect which is somewhat reminiscent of using optics in a functional programming language. We make this analogy precise by showing that several interrelated techniques in the literature for managing these patterns of manipulation can be derived as instances of the general definition of profunctor optics. In doing so, we specialize the machinery of profunctor optics from categories to posets and to sets. This simplification makes most of this machinery look more familiar, and it reveals that much of it was already hiding in separation logic in plain sight.

## A Localic Approach to Dependency, Conflict, and Concurrency

28 April, 2020

In the fifth talk of the ACT@UCR seminar, Gershom Bazerman told how to use locales to study the semantics of dependency, conflict, and concurrency.

Afterwards we discussed his talk at the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/April.2029th.3A.20Gershom.20Bazerman

You can view or join the conversation there if you sign in.

You can see his slides here, or download a video here, or watch the video here:

• Gershom Bazerman, A localic approach to the semantics of dependency, conflict, and concurrency.

Abstract. Petri nets have been of interest to applied category theory for some time. Back in the 1980s, one approach to their semantics was given by algebraic gadgets called “event structures.” We use classical techniques from order theory to study event structures without conflict restrictions (which we term “dependency structures with choice”) by their associated “traces”, which let us establish a one-to-one correspondence between DSCs and a certain class of locales. These locales have an internal logic of reachability, which can be equipped with “versioning” modalities that let us abstract away certain unnecessary detail from an underlying DSC. With this in hand we can give a general notion of what it means to “solve a dependency problem” and combinatorial results bounding the complexity of this. Time permitting, I will sketch work-in-progress which hopes to equip these locales with a notion of conflict, letting us capture the full semantics of general event structures in the form of homological data, thus providing one avenue to the topological semantics of concurrent systems. This is joint work with Raymond Puzio.

## The Monoidal Grothendieck Construction

24 April, 2020

My student Joe Moeller gave a talk at the MIT Categories Seminar today! People discussed his talk at the Category Theory Community Server, and if you join that you can see the discussion here:

https://categorytheory.zulipchat.com/#narrow/stream/229457-MIT-Categories.20Seminar/topic/April.2023.20-.20Joe.20Moeller’s.20talk

You can see his slides here, and watch a video of his talk here:

The monoidal Grothendieck construction

Abstract. The Grothendieck construction gives an equivalence between fibrations and indexed categories. We will begin with a review of the classical story. We will then lift this correspondence to two monoidal variants, a global version and a fibre-wise version. Under certain conditions these are equivalent, so one can transfer fibre-wise monoidal structures to the total category. We will give some examples demonstrating the utility of this construction in applied category theory and categorical algebra.

The talk is based on this paper:

• Joe Moeller and Christina Vasilakopoulou, Monoidal Grothendieck construction.

This, in turn, had its roots in our work on network models, a setup for the compositional design of networked systems:

• John Baez, John Foley, Joe Moeller and Blake Pollard, Network models.