# functor

Published articles for functor.

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## Functoriality in Category Theory: Mappings That Preserve Morphisms

DevFeed: [Functoriality in Category Theory: Mappings That Preserve Morphisms](<https://devfeed.tech/articles/functoriality-40326.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/07/14/functoriality/>)

Published: 2013-07-14T10:03:29Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Category Theory](<https://devfeed.tech/topics/category-theory.md>), [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>)

Tags: [categories](<https://devfeed.tech/tags/categories.md>), [category-theory](<https://devfeed.tech/tags/category-theory.md>), [coproducts](<https://devfeed.tech/tags/coproducts.md>), [functor](<https://devfeed.tech/tags/functor.md>), [homology](<https://devfeed.tech/tags/homology.md>), [ml](<https://devfeed.tech/tags/ml.md>), [morphisms](<https://devfeed.tech/tags/morphisms.md>), [universal-properties](<https://devfeed.tech/tags/universal-properties.md>)

### AI overview

This tutorial introduces functoriality in category theory. It explains functors as mappings between categories that assign objects and morphisms while preserving identity morphisms and composition, with homology as an example of a functorial construction.

### Source excerpt

Last time we worked through some basic examples of universal properties, specifically singling out quotients, products, and coproducts. There are many many more universal properties that we will mention as we encounter them, but there is one crucial topic in category theory that we have only hinted at: functoriality. As we've repeatedly stressed, the meat of category theory is in the morphisms. One natural question one might ask is, what notion of morphism is there between categories themselves?

## Categories as Types

DevFeed: [Categories as Types](<https://devfeed.tech/articles/categories-as-types-40316.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/05/04/categories-as-types/>)

Published: 2013-05-04T19:30:35Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Programming](<https://devfeed.tech/topics/programming.md>), [Standard ML](<https://devfeed.tech/topics/standard-ml.md>), [Programming language](<https://devfeed.tech/topics/programming-language.md>)

Tags: [categories](<https://devfeed.tech/tags/categories.md>), [class](<https://devfeed.tech/tags/class.md>), [exception](<https://devfeed.tech/tags/exception.md>), [functions](<https://devfeed.tech/tags/functions.md>), [functor](<https://devfeed.tech/tags/functor.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [ml](<https://devfeed.tech/tags/ml.md>), [object](<https://devfeed.tech/tags/object.md>), [programming](<https://devfeed.tech/tags/programming.md>), [programming-language](<https://devfeed.tech/tags/programming-language.md>), [type](<https://devfeed.tech/tags/type.md>)

### AI overview

The post explains two ways to represent a category as an ML type: a tuple of functions and an ML functor. It discusses representing objects and morphisms with types, checking composability, and providing identity morphisms while noting which category-theory properties cannot be enforced directly.

### Source excerpt

In this post we'll get a quick look at two ways to define a category as a type in ML. The first way will be completely trivial: we'll just write it as a tuple of functions. The second will involve the terribly-named "functor" expression in ML, which allows one to give a bit more structure on data types. The reader unfamiliar with the ML programming language should consult our earlier primer.