Strictification

In mathematics, specifically in 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.

• Every monoidal category is monoidally equivalent to a strict monoidal category.^[1] This is (essentially) the Mac Lane coherence theorem.

• Coherence condition

• Schauenburg, Peter (2001). "Turning monoidal categories into strict ones". The New York Journal of Mathematics [Electronic Only]. 7: 257–265. ISSN 1076-9803.
• Joyal, A.; Street, R. (1993). "Braided Tensor Categories". Advances in Mathematics. 102 (1): 20–78. doi:10.1006/aima.1993.1055.
• Lack, Stephen (2002). "Codescent objects and coherence". Journal of Pure and Applied Algebra. 175 (1–3): 223–241. doi:10.1016/S0022-4049(02)00136-6.
• Mac Lane, Saunders (1978). "Symmetry and Braidings in Monoidal Categories". Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5. pp. 251–266. doi:10.1007/978-1-4757-4721-8_12. ISBN 978-1-4419-3123-8. §3. Strict Monoidal Categories
• Shulman, Michael A. (2012). "Not every pseudoalgebra is equivalent to a strict one". Advances in Mathematics. 229 (3): 2024–2041. arXiv:1005.1520. doi:10.1016/j.aim.2011.01.010.

• Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. "18.769, Spring 2009, Graduate Topics in Lie Theory: Tensor Categories §.Lecture 2". MIT Open Course Ware.