# coproducts

Published articles for coproducts.

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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?

## Universal Properties

DevFeed: [Universal Properties](<https://devfeed.tech/articles/universal-properties-40319.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/05/24/universal-properties/>)

Published: 2013-05-24T14:53:25Z

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>), [function](<https://devfeed.tech/topics/function.md>), [object](<https://devfeed.tech/topics/object.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>), [examples](<https://devfeed.tech/tags/examples.md>), [function](<https://devfeed.tech/tags/function.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [ml](<https://devfeed.tech/tags/ml.md>), [morphisms](<https://devfeed.tech/tags/morphisms.md>), [object](<https://devfeed.tech/tags/object.md>), [product](<https://devfeed.tech/tags/product.md>), [programming](<https://devfeed.tech/tags/programming.md>), [quotients](<https://devfeed.tech/tags/quotients.md>), [types](<https://devfeed.tech/tags/types.md>), [universal-properties](<https://devfeed.tech/tags/universal-properties.md>)

### AI overview

This tutorial introduces universal properties in category theory, defining initial, final, and zero objects through unique morphisms. It illustrates the concepts with examples from mathematics and Set, and discusses constructing programs related to these properties.

### Source excerpt

Previously in this series we've seen the definition of a category and a bunch of examples, basic properties of morphisms, and a first look at how to represent categories as types in ML. In this post we'll expand these ideas and introduce the notion of a universal property. We'll see examples from mathematics and write some programs which simultaneously prove certain objects have universal properties and construct the morphisms involved.