# Seven dogmas of category theory

Joseph Goguen gave seven dogmas in his paper A Categorical Manifesto.

1. To each species of mathematical structure, there corresponds a category whose objects have that structure, and whose morphisms preserve it.
2. To any natural construction on structures of one species, yielding structures of another species, there corresponds a functor from the category of the first species to the category of the second.
3. To each natural translation from a construction F : A -> B to a construction G: A -> B there corresponds a natural transformation F => G.
4. A diagram D in a category C can be seen as a system of constraints, and then a limit of D represents all possible solutions of the system.
5. To any canonical construction from one species of structure to another corresponds an adjuction between the corresponding categories.
6. Given a species of structure, say widgets, then the result of interconnecting a system of widgets to form a super-widget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected.
7. Given a species of structure C, then a species of structure obtained by “decorating” or “enriching” that of C corresponds to a comma category under C (or under a functor from C).

Although category theory is all about general patterns, it can be hard to learn what the general patterns of category theory are. The list above is the best high-level description of category theory I’ve seen.

## 5 thoughts on “Seven dogmas of category theory”

2. David W. Locke

3. Thanks, this is a nice summary.

There is probably not much in “pure” category theory. This is why math216.wordpress.com just introduces the language and then goes on to talk about something of actual interest.

Another fun example is look on arXiv for fast khovanov homology computations or Lee’s spectral sequence paper (2002).

Without CT you cannot say “ABC embeds functorially into XYZ”, which is a kind of statement you want to be able to make and clearly communicates an idea that is hard to otherwise state. That doesn’t mean that functors themselves (or NT’s) are of interest.

4. Anatoly

Just start to learn the subject and this is very insightful points! Thank you !

5. Vigalchin

Tom Leinster wrote a book, “Basic Category Theory”, that published by Cambridge press. He has very kindly put a downloadable copy at https://arxiv.org/abs/1612.09375. Fantastic book.

Also Emily Riehl also published an excellent book “Categories in Context”. Here is a pre pub online copy: http://www.math.jhu.edu/~eriehl/context.pdf

Both are professors

Vasily