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memo a

Hartshorne, Robin (1970). Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics. Vol. 156. doi:10.1007/BFb0067839. ISBN 978-3-540-05184-8.

https://books.google.com/books?id=okHfUv4l4vgC&pg=PA57 ^[1]

memo b

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memo c

memo d

Let $X_{1},\dots ,X_{n}\in C$ . Let $P_{1},P_{2}$ be arbitrary two parenthesized products of $X_{1},\dots ,X_{n}$ (in this order) with arbitrary insertions of unit objects $1$ . Let $f,g:P_{1}\rightarrow P_{2}$ be two isomorphisms, obtained by composing associativity and unit isomorphisms and their inverses possibly tensored with identity morphisms. Then $f=g$ .

memo g

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strictification

Schauenburg, Peter (2001). "Turning monoidal categories into strict ones". The New York Journal of Mathematics [Electronic Only]. 7: 257–265. ISSN 1076-9803.

^[6]

Mac Lane, Saunders (1998). Categories for the working mathematician. New York: Springer. ISBN 0-387-98403-8. OCLC 37928530.

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,

memo i

In monoidal category $C$ , the following two conditions are called coherence conditions:

Let a bifunctor $\otimes \colon \mathbf {C} \times \mathbf {C} \to \mathbf {C}$ called the tensor product, a natural isomorphism $\alpha _{A,B,C}$ , called the associator:

\alpha _{A,B,C}\colon (A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)

Also, let $I$ an identity object and $I$ has a left identity, a natural isomorphism $\lambda _{A}$ called the left unitor:

\lambda _{A}:I\otimes A\rightarrow A

as well as, let

I

has a right identity, a natural isomorphism

\rho _{A}

called the right unitor:

\rho _{A}:A\otimes I\rightarrow A

.

"pentagon identity". ncatlab.org.

zigang identity

"triangle identity". ncatlab.org.

Wikipedia talk:WikiProject Mathematics/Archive/2024/Oct#(Mac Lane's) coherence theorem vs. Coherence condition

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memo j

^ Costa, Laura; Miró-Roig, Rosa María; Pons-Llopis, Joan (2021). Ulrich Bundles. doi:10.1515/9783110647686. ISBN 9783110647686.
^ Mitchell, Barry (1965). Theory of Categories. Academic Press. ISBN 978-0-12-499250-4.
^ Faith, Carl (1973). "Product and Coproduct". Algebra. pp. 83–109. doi:10.1007/978-3-642-80634-6_4. ISBN 978-3-642-80636-0.
^ Leinster, Tom (2014). Basic Category Theory. arXiv:1612.09375. doi:10.1017/CBO9781107360068. ISBN 978-1-107-04424-1.
^ Category Theory. Oxford University Press. 17 June 2010. ISBN 978-0-19-958736-0.
^ MacLane, Saunders; Paré, Robert (1985). "Coherence for bicategories and indexed categories". Journal of Pure and Applied Algebra. 37: 59–80. doi:10.1016/0022-4049(85)90087-8.
^ Ben-Moshe, Shay (2024). "Naturality of the ∞-categorical enriched Yoneda embedding". Journal of Pure and Applied Algebra. 228 (6). arXiv:2301.00601. doi:10.1016/j.jpaa.2024.107625.

[1] Costa, Laura; Miró-Roig, Rosa María; Pons-Llopis, Joan (2021). Ulrich Bundles. doi:10.1515/9783110647686. ISBN 9783110647686.

[2] Mitchell, Barry (1965). Theory of Categories. Academic Press. ISBN 978-0-12-499250-4.

[3] Faith, Carl (1973). "Product and Coproduct". Algebra. pp. 83–109. doi:10.1007/978-3-642-80634-6_4. ISBN 978-3-642-80636-0.

[4] Leinster, Tom (2014). Basic Category Theory. arXiv:1612.09375. doi:10.1017/CBO9781107360068. ISBN 978-1-107-04424-1.

[5] Category Theory. Oxford University Press. 17 June 2010. ISBN 978-0-19-958736-0.

[6] MacLane, Saunders; Paré, Robert (1985). "Coherence for bicategories and indexed categories". Journal of Pure and Applied Algebra. 37: 59–80. doi:10.1016/0022-4049(85)90087-8.

[7] Ben-Moshe, Shay (2024). "Naturality of the ∞-categorical enriched Yoneda embedding". Journal of Pure and Applied Algebra. 228 (6). arXiv:2301.00601. doi:10.1016/j.jpaa.2024.107625.

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