In mathematics, 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 ...
, a strictification refers to statements of the form “every weak structure of some sort is equivalent to a stricter one.” Such a result was first proven for monoidal categories by
Mac Lane, and it is often possible to derive strictifications from coherence results and vice versa.
Monoidal category
*Every
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
is monoidally equivalent to a strict monoidal category.
This is (essentially) the
Mac Lane coherence theorem In category theory, a branch of mathematics, Mac Lane's coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. But regarding a result about certain commutative diagrams, Kelly is states as follows: "no longer be
s ...
.
See also
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Coherence condition
In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".
The adjectives such as "pseudo-" and "lax-" ...
Notes
Reference
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* §3. Strict Monoidal Categories
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External links
*{{cite web , last1=Etingof , first1=Pavel , last2=Gelaki , first2=Shlomo , last3=Nikshych , first3=Dmitri , last4=Ostrik , first4=Victor , title=18.769, Spring 2009, Graduate Topics in Lie Theory: Tensor Categories §.Lecture 2 , url=https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/resources/mit18_769s09_lec02/ , website=MIT Open Course Ware
Category theory