The Hitchhiker's Guide to Morphisms

Posted by Jonathan Immanuel Brachthäuser on February 21, 2016 · 10 mins read

Since I always keep mixing up the names of morphisms, I finally created a cheat-sheet that now in its full glory decorates my office. The important part of the poster is, that it puts morphisms that deconstruct data (left column) with those that construct codata (right column) in contrast. So on one glance you can see that zygomorphisms are obviously dual to g-apomorphisms – of course, that’s an easy one.

Download the full poster as a PDF.

The remainder of this post is just repeating the contents of the poster for easy web-reference. I am happy for every comment that links to either easy accessible material on the topic, or good examples of how to use the various morphisms.

Least fixed point (Data)

(* → *) → *
μf = f (μf) = f (… (f 0)) = Free f 0

Instances of μf are “f-data-structures” or short “f-structures”.

Free Monads

(* → *) → * → *
Free f a = a + f (Free f a)

A finite f-structure, that can contain as. Is a functor and a monad. Monadic-bind corresponds to substitution: Substitutes as by terms that can contain bs.

Destruction Morphisms


cata ∷ ∀ a. (f a → a) → μf → a

Also known as “fold”. Deconstructs a f-structure level-by-level and applies the algebra [13, 5, 14, 6].


para ∷ ∀ a. (f (μf , a) → a) → μf → a

A.k.a. “the Tupling-Trick”. Like cata, but allows access to the full subtree during teardown. Is a special case of zygo, with the helper being the initial-algebra [16].


zygo ∷ ∀ a b. (f (a , b) → a) →
              (f b → b)       → μf → a

Allows depending on a helper algebra for deconstructing a f-structure. A generalisation of para.


histo ∷ ∀ a. (f (Cofree f a) → a) → μf → a

Deconstructs the f-structure with the help of all previous computation for the substructures (the trace). Difference to para: The subcomputation is already available and needs not to be recomputed.


prepro ∷ ∀ a. (f a → a) → (f  ⇝ f) → μf → a

Applies the natural transformation at every level, before destructing with the algebra. Can be seen as a one-level rewrite. This extension can be combined with other destruction morphisms [4].

Greatest fixed point (Codata)

(* → *) → *
νf = f (νf) = f (f (…)) = Cofree f 1

Instances of νf are “f-codata-structures” or short “f-structures”.

Cofree Comonads

(* → *) → * → *
Cofree f a = a , f (Cofree f a)

A possibly infinite f-structure, full of as. Is a functor and a comonad. Comonadic-extend corresponds to computing a new f-structure full of bs. At every level the a and the full trace are available for computing the b.

Construction Morphisms


ana ∷ ∀ a. (a → f a) → a → νf

Also known as “unfold”. Constructs a f-structure level-by-level, starting with a seed and repeatedly applying the coalgebra [13, 5].


apo ∷ ∀ a.  (a → f (a + νf)) → a → νf

A.k.a. “the Co-Tupling-Trick”™. Like ana, but also allows to return an entire substructure instead of one level only. Is a special case of g-apo, with the helper being the final-coalgebra [17, 16].


gapo ∷ ∀ a b. (a → f (a + b)) →
              (b → f b)       → a → νf

Allows depending on a helper coalgebra for constructing a f-structure. A generalisation of apo.


futu ∷ ∀ a. (a → f (Free f a)) → a → νf

Constructs a f-structure stepwise, but the coalgebra can return multiple layers of a-valued substructures at once. Difference to apo: the subtrees can again contain as [16].


postpro ∷ ∀ a. (a → f a) → (f   ⇝ f) → a → νf

Applies the natural transformation at every level, after construction with the coalgebra. Can be seen as a one-level rewrite. This extension can be combined with other construction morphisms.

Combined Morphisms

ana then cata = hylomorphism

hylo ∷ ∀ a b. (a → f a) → (f b → b) → a → b

Omits creating the intermediate structure and immediately applies the algebra to the results of the coalgebra† [13, 2, 5, 14].

ana then histo = dynamorphism

dyna ∷ ∀ a b. (a → f a) →
              (f (Cofree f b) → b) → a → b

Constructs a structure and immediately destructs it while keeping intermediate results†. Can be used to implement dynamic-programming algorithms [9, 10].

futu then histo = chronomorphism

chrono ∷ ∀ a b. (a → (Free f a)) →
                (f (Cofree f b) → b) → a → b

Can at the same time “look back” at previous results and “jump into the future” by returning seeds that are multiple levels deep† [11].

cata then conv then ana = metamorphism

meta ∷ ∀ a b. (f a → a) → (a → b) → (b → g b) →
              μf → νg

Constructs a g-structure from a f-structure while changing the internal representation in-between [7].

Other Morphisms

 Most of the above morphisms can be modified to accept generalized algebras (with w being a comonad)

GAlgebra f w a = f (w a) → a

or generalised coalgebras (with m being a monad), respectively:

GCoalgebra f m a = a → f (m a)

Also a multitude of other morphisms exist [12, 3, 1] and the combination of morphisms and distributive laws

Distr f g = ∀ a. f (g a) → g (f a)

has been studied [8, 15].

† Can also be enhanced by a representation change (natural transformation f ⇝ g), before deconstructing with a corresponding g-algebra


[1] Adámek, Jiří, Stefan Milius, and Jiří Velebil. “Elgot algebras.” Electronic Notes in
 Theoretical Computer Science, 2006.

[2] Augusteijn, Lex. “Sorting morphisms.” Advanced Functional Programming.
 Springer Berlin Heidelberg, 1998.

[3] Erwig, Martin. Random access to abstract data types. Springer Berlin
 Heidelberg, 2000.

[4] Fokkinga, Maarten M. “Law and order in algorithmics.” PhD Thesis, 1992.

[5] Gibbons, Jeremy. “Origami programming.”, 2003.

[6] Gibbons, Jeremy. “Design patterns as higher-order datatype-generic programs.”
 Proceedings of the Workshop on Generic programming. ACM, 2006.

[7] Gibbons, Jeremy. “Metamorphisms: Streaming representation-changers.”
 Science of Computer Programming, 2007.

[8] Hinze, Ralf, et al. “Sorting with bialgebras and distributive laws.” Proceedings of the Workshop on Generic programming. ACM, 2012.

[9] Hinze, Ralf, and Nicolas Wu. “Histo-and dynamorphisms revisited.” Proceedings of
 the Workshop on Generic programming. ACM, 2013.

[10] Kabanov, Jevgeni, and Varmo Vene. “Recursion schemes for dynamic programming.” Mathematics of Program Construction. Springer Berlin Heidelberg, 2006.

[11] Kmett, Edward. “Time for Chronomorphisms.”, 2008.

[12] Kmett, Edward. “Recursion Schemes: A Field Guide (Redux).”, 2009.

[13] Meijer, Erik, Maarten Fokkinga, and Ross Paterson. “Functional programming with
 bananas, lenses, envelopes and barbed wire.” Functional Programming Languages
 and Computer Architecture. Springer Berlin Heidelberg, 1991.

[14] Oliveira, Bruno, and Jeremy Gibbons. “Scala for generic programmers.”
 Proceedings of the Workshop on Generic programming. ACM, 2008.

[15] Turi, Daniele, and Gordon Plotkin. “Towards a mathematical operational
 semantics.” Logic in Computer Science. IEEE, 1997.

[16] Uustalu, Tarmo, and Varmo Vene. “Primitive (co) recursion and course-of-value
 (co) iteration, categorically.” Informatica, 1999.

[17] Vene, Varmo, and Tarmo Uustalu. “Functional programming with apomorphisms
 (corecursion).” Proceedings of the Estonian Academy of Sciences: Physics, Mathematics. Vol. 47. No. 3. 1998.