What is an Operad? Part 1

Why are operads useful?

The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar

Why are operads useful?

The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar

Opera Neon. This browser is built to act.

Opera Neon is an agentic browser that understands your intent, assists with tasks, and takes actions. It helps you move faster and get things done.

Operads, Type Level Nats, and Tic-Tac-Toe

This summer I spent some time talking with Edward Kmett about lots of things. (Which really means that he was talking and I was trying to keep up.) One of the topics was operads. The ideas behind o…

Opera unveils an AI agent that runs natively within the browser | TechCrunch

Browser company Opera has unveiled a new AI agent called Browser Operator that can complete tasks for you on different websites. In a demo video, the

A Very Early History of Algebraic Data Types

Been quiet around here! I’ve been putting almost all of my writing time into Logic for Programmers and my whole brain is book-shaped. Trust me, you do not want to read my 2000-word rant on Sphinx post-build LaTeX customization. But I spent the past week in a historical rabbit hole and had to share what I found. It started with Algebraic [Data] Types are not Scary, Actually. The post covers AlgDTs1 in more detail, but a quick overview is:

Meet Opera’s AI Browser Operator

We're introducing an AI agent into the browser making us the first major browser with AI-based agentic browsing.

The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits

Wiring diagrams, as seen in digital circuits, can be nested hierarchically and thus have an aspect of self-similarity. We show that wiring diagrams form the morphisms of an operad $\mcT$, capturing this self-similarity. We discuss the algebra $\Rel$ of mathematical relations on $\mcT$, and in so doing use wiring diagrams as a graphical language with which to structure queries on relational databases. We give the example of circuit diagrams as a special case. We move on to show how plug-and-play devices and also recursion can be formulated in the operadic framework as well. Throughout we include many examples and figures.

ravarcheon

mostly music, art, some math, a bit of code, webstuff, girlstuff, etc

Opera Mini is a game change : Cloud-Based Browsers and Its important to know them.

Index Introduction

Generalized Lens Categories via functors $\mathcal{C}^{\rm op}\to\mathsf{Cat}$

Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category $\mathsf{Lens}_F$ for any category $\mathcal{C}$ and functor $F\colon \mathcal{C}^{\rm op}\to\mathsf{Cat}$, using a variant of the Grothendieck construction. All of the mathematics in this note is straightforward; the purpose is simply to see lenses in a broader context where some closely-related examples, such as ringed spaces and open continuous dynamical systems, can be included.

Algebraic data integration

In this paper, we develop an algebraic approach to data integration by combining techniques from functional programming, category theory, and database theory. In our formalism, database schemas and instances are algebraic (multi-sorted equational) theories of a certain form. Schemas denote categories, and instances denote their initial (term) algebras. The instances on a schema S form a category, S–Inst, and a morphism of schemas F : S → T induces three adjoint data migration functors: ΣF : S–Inst → T–Inst, defined by substitution along F, which has a right adjoint ΔF : T–Inst → S–Inst, which in turn has a right adjoint ΠF : S–Inst → T–Inst. We present a query language based on for/where/return syntax where each query denotes a sequence of data migration functors; a pushout-based design pattern for performing data integration using our formalism; and describe the implementation of our formalism in a tool we call AQL (Algebraic Query Language).

Dynamic task delegation for hierarchical agents

This is the fourth installment in a series of papers offering models of hierarchical structure for dynamical systems, using the language of polynomial functors. The operad underlying the symmetric monoidal category $(\mathbf{Poly}, \otimes, \mathcal{y})$ can be viewed as defining the behavior of hierarchical delegation. In particular, a morphism $\mathbf{Poly}(p_1 \otimes \cdots \otimes p_m, q)$ turns the outputs of subordinates with interfaces $p_i$ into the output of an agent with interface $q$ and turns a task given to the agent into a task for each of the subordinates. In this article, we extend the framework so that subordinates may be invoked asynchronously depending on the outcomes of other subordinates. We prove that the free (co)monad (co)monad extends to a (co)monad on $\mathbf{Org}$. From the perspective of programs/pattern, this extension implies the existence of a $\mathbf{Cat}$-enriched operad $\mathbf{Org}_\mathfrak{m}$, and from the perspective of behavior/matter, it implies the existence of a $\mathbf{Cat}$-enriched operad $\mathbf{Org}^\mathfrak{c}$. Second, we crispen the relationship between the programmatic and behavioral perspectives via a functor $[-, t] \colon \mathbf{Org}_{\mathfrak{m}}^\textrm{op} \to \mathbf{Org}^\mathfrak{c}$ for any polynomial monad $t$.

Operads for compositional reasoning in LLMs

Question decomposition, i.e. breaking a complex query into simpler sub-queries whose answers are composed to produce a final answer, is a widely used strategy for improving LLM reasoning, yet it currently lacks a rigorous mathematical foundation. In this paper, we propose operads, mathematical structures that model many-in, one-out operations and compositions thereof, as a natural framework for describing question decomposition. We define the questions operad $Q$, in which operations correspond to question templates and composition corresponds to substitution of sub-answers, and show how QA models can be interpreted as algebras over $Q$. Beyond reframing existing practice, this operadic perspective points toward new methods, in particular a notion of operadic consistency, which measures whether a QA model's answers agree across the partial collapses of a question decomposition tree. Empirical evaluation of operadic consistency is reported in our companion paper (Bottman, Liu, and Richardson, 2026), which finds it strongly correlated with accuracy across twelve LLMs and four multi-hop QA datasets and outperforming standard temperature-based self-consistency baselines. We argue that operads are the natural mathematical home for question decomposition, and that invariants such as operadic consistency open new directions for analyzing and improving the reliability of multi-step reasoning.