Extranatural Transformation
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically in
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, an extranatural transformation
Eilenberg Eilenberg is a surname. Notable people with the surname include: * Samuel Eilenberg (1913–1998), Polish mathematician * Richard Eilenberg (1848–1927), German composer Named after Samuel * Eilenberg–MacLane space * Eilenberg–Moore algebra ...
and
Kelly Kelly may refer to: Art and entertainment * ''Kelly'' (Kelly Price album), 2011 * ''Kelly'' (Andrea Faustini album) * ''Kelly'' (musical), by Mark Charlap, 1965 * "Kelly" (song), by Kelly Rowland, 2018 * ''Kelly'' (film), Canada, 1981 * ...
, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966) is a generalization of the notion of
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
.


Definition

Let F:A\times B^\mathrm\times B\rightarrow D and G:A\times C^\mathrm\times C\rightarrow D be two
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s of categories. A family \eta (a,b,c):F(a,b,b)\rightarrow G(a,c,c) is said to be natural in ''a'' and extranatural in ''b'' and ''c'' if the following holds: *\eta(-,b,c) is a natural transformation (in the usual sense). * (extranaturality in ''b'') \forall (g:b\rightarrow b^\prime)\in \mathrm\, B, \forall a\in A, \forall c\in C the following diagram commutes :: \begin F(a,b',b) & \xrightarrow & F(a,b',b') \\ _\downarrow\qquad & & _\downarrow\qquad \\ F(a,b,b) & \xrightarrow & G(a,c,c) \end * (extranaturality in ''c'') \forall (h:c\rightarrow c^\prime)\in \mathrm\, C, \forall a\in A, \forall b\in B the following diagram commutes :: \begin F(a,b,b) & \xrightarrow & G(a,c',c') \\ _\downarrow\qquad & & _\downarrow\qquad \\ G(a,c,c) & \xrightarrow & G(a,c,c') \end


Properties

Extranatural transformations can be used to define wedges and thereby
ends End, END, Ending, or ENDS may refer to: End Mathematics *End (category theory) *End (topology) *End (graph theory) *End (graph_theory)#Cayley_graphs, End (group theory) (a subcase of the previous) *End (endomorphism) Sports and games *End (gridir ...
Fosco Loregian, ''This is the (co)end, my only (co)friend'', arXiv preprin

/ref> (dually co-wedges and co-ends), by setting F (dually G) constant. Extranatural transformations can be defined in terms of
dinatural transformation In category theory, a branch of mathematics, a dinatural transformation \alpha between two functors :S,T : C^\times C\to D, written :\alpha : S\ddot\to T, is a function that to every object c of C associates an arrow :\alpha_c : S(c,c)\to T(c,c ...
s, of which they are a special case.


See also

*
Dinatural transformation In category theory, a branch of mathematics, a dinatural transformation \alpha between two functors :S,T : C^\times C\to D, written :\alpha : S\ddot\to T, is a function that to every object c of C associates an arrow :\alpha_c : S(c,c)\to T(c,c ...


References


External links

* {{nlab, id=extranatural+transformation Higher category theory