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This video defines the category of programs and discusses how it is related to computation. It introduces the concept of a monad and explains how it can be used to compose functions. Finally, it discusses how the category of programs is related to the category of sets.

**00:00:00**This talk will introduce readers to the concepts of functions and composition, and motivate the definition of a category. It will then explain how mathematicians think about computations and monads, and compare the two. Finally, the talk will describe how a LaVere theory is just a category, and how it can be used to model computations.**00:05:00**This talk discusses the importance of computation and its effects, focusing on the concept of a program. A monad is introduced as a structure that allows programs to be turned into arrows in a category. The levere theory is visible inside this category, and its arrows are programs. A finite number of elements is necessary for a category to be a category, and a category is important for understanding composition of functions.**00:10:00**Category theory is a branch of mathematics that helps to study the properties of objects and arrows that do not depend on what they are. Category theory is useful for modeling complicated mathematical objects.**00:15:00**This video provides a categorical view of computational effects, describing how objects (types) and arrows (programs) are related in a way that is not important for most programming purposes. Later in the video, the author mentions monads and states that they are technically functors, but that they are not needed for today's discussion. He then goes on to describe a function with side-effects and state that it is partially defined relative to a fixed set F.**00:20:00**In this video, the speaker discusses the concept of a program, which is a function from one set ( A ) to another set ( G ). He explains that a tea program is a function from A to G of B, which is different than a function from A to T of B. He also introduces notation for programs, using a squiggly arrow to indicate that they represent a different kind of arrow. Finally, he discusses how programs relate to computations, and how they can be used to represent arrows from one set to another.**00:25:00**The author suggests that computer programs should be viewed as a category and that they should be composable. He defines a T program as a function from a set of input values to a set of output values, and states that a program is really an arrow from A to B. He provides a metal state theorem which states that a notion of computation defines a monad, and that programs defined relative to it define the arrows in a category.**00:30:00**In this video, the presenter defines the category of computational effects, which includes monads and identity arrows. He explains that the category is defined in a particular way so that monads are automatically defined. The presenter then discusses the category of sets and how it is similar to the category of programs, which includes identity arrows. He concludes by outlining the category of categories and how it is similar to the category of sets.**00:35:00**The author is trying to define a category whose arrows are programs, and he explains that programs are defined relative to key and composition of programs relative to T. He also provides an example of an F function, which is a function from input values to prime factor lists.**00:40:00**The video discusses the concept of a monad, which is a structure that allows for the composition of functions from one category to another. The examples used to illustrate this concept include the concept of a category for computation with partial addition and the natural numbers.**00:45:00**The video discusses the concept of a partial function, and how it can be thought of as an arrow from one input to another with a defined output, but with some inputs that have undefined results. It also discusses the concept of a category program, which is a way of specifying a partial function as a real function with a subset of input values. The composition operation is then defined, and it is shown that it is the same as the operation on real functions.**00:50:00**In the video, Professor Van Rossum explains that a category of partially defined computations (T programs) is a levere theory or a subcategory of a low-V R theory. He goes on to say that an arrow in this category of list programs from 1 to 6 is a list of 6 elements, just as an arrow in the category of programs from 1 to n is a list of n elements. Finally, he argues that whenever an arrow in this category has a composition relation between its operations, that relation is between two operations in the category of lists programs.**00:55:00**In this video, the author discusses the category of programs and its algebraic theory. He defines a model for this theory, and explains that each operation in the model corresponds to a function between two inputs and one output. The composite of these functions is a function. The author then discusses the relationship between monads and the category of programs, and provides references for further study.

This video explains the concept of computational effects, and how they can be expressed in terms of categorical terms. It explains how monads, which are a form of computational effect, can be recovered from a category of operations and equations. This provides a way of understanding monads that is more general than the standard approach.

**01:00:00**In this talk, the presenter explains how computational effects can be categorically viewed, and goes on to show how LaVere theories are equivalent to monads. The advantages of using LaVere theories over monads include their ability to be equivalently viewed as endofunctors of different categories, and their ability to preserve finiteness.**01:05:00**This talk is about how to think about monads in terms of computational effects, and how they can be seen as universal algebra concepts. One example given is the levira theory, which is acategory that is not always obvious how to transport them. However, the models of Ollivier Theory can be defined in any category, and this is an advantage. The levira theory is often freely generated by a very small collection of operations and equations between them, which makes it a good perspective to look at. There is one monad that does not fit into this framework, continuations, because it is involving a power set.**01:10:00**This video explains the concept of computational effects, and how they can be expressed in terms of categorical terms. It explains how monads, which are a form of computational effect, can be recovered from a category of operations and equations.

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