# Tagged Questions

Category theory is used to formalize mathematics and its concepts as a collection of objects and arrows (also called morphisms). Category theory can be used to formalize concepts of other high-level abstractions such as set theory, ring theory, and group theory. (By Steve Awodey)

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### Generators in category theory

An object K in a category C is called a generator if, for all pairs of morphisms f, g : A → B between arbitrary objects A and B, f = g iff ∀e : K → A. f ◦ e = g ◦ e. Source: http://events.cs....
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### How precise is the statement “STLC is the internal language of CCCs”?

I'm studying some basic category theory in the context of type theory and came across the statement "simply typed lambda calculus is the internal language of cartesian closed categories". However ...
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### Scott/Lawson topology for function space domain

Given two domains, $D_1$, $D_2$, already equipped with Scott (or Lawson) topology, the product domain $D=D_1\times D_2$ has the Tychonoff product topology, e.g., Mathematical Theory of Domains, ...
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### DFAs as Categories

I've been recently investigating metacategories (arrows and objects) alongside Automata theory and noticed that a category is a sort of parent container for DFAs, which are just a specific type of ...
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### Tool/app for learning category theory?

Being a programmer I appreciate the errors given by a compiler for a programming language and come to rely on the compiler's error as a safety net. In learning category theory I would like to have ...
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### Substitution by structural recursion

Following the article's notation, I write $\mathcal{F}$ for the category of presheaves on a (suitable) category $\mathbb{F}$, $TV$ for the presheaf of terms, $\delta$ for the context extension, and \$\...
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### What is meant by Category theory doesn't yet know how to deal with higher-order functions?

In reading Uday Reddy's answer to What is the relation between functors in SML and Category theory? Uday states Category theory doesn't yet know how to deal with higher-order functions. Some day,...
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### What is the relation between functors in SML and Category theory?

Along the same thinking as this statement by Andrej Bauer in this answer The Haskell community has developed a number of techniques inspired by category theory, of which monads are best known ...
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### How are programming languages and foundations of mathematics related?

Basically I am aware of three foundations for math Set theory Type theory Category theory So in what ways are programming languages and foundations of mathematics related? EDIT The original ...
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### Are monoids useful in optimization?

Many common operations are monoids. Haskell has leveraged this observation to make many higher-order functions more generic (Foldable being one example). There is ...