# Diagonal morphism

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For the particular instance of the notion in algebraic geometry, see [diagonal embedding](/source/Diagonal_embedding).

In [category theory](/source/Category_theory), a branch of [mathematics](/source/Mathematics), for every [object](/source/Object_(category_theory)) A in every [category](/source/Category_(mathematics)) \mathcal{C} where the [product](/source/Product_(category_theory)) A\times A exists, there exists the **diagonal morphism**[1][2][3][4][5][6]

- \delta_A : A \rightarrow A \times A

satisfying

- \pi_k \circ \delta_A = \operatorname{id}_A for k \in \{ 1,2 \},

where \pi_k is the [canonical projection morphism](/source/Canonical_projection_morphism) to the k-th component. The existence of this [morphism](/source/Morphism) is a consequence of the [universal property](/source/Universal_property) that [characterizes](/source/Characterization_(mathematics)) the product ([up to](/source/Up_to) [isomorphism](/source/Isomorphism)). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The [image](/source/Image_(category_theory)) of a diagonal morphism in the [category of sets](/source/Category_of_sets), as a [subset](/source/Subset) of the [Cartesian product](/source/Cartesian_product), is a [relation](/source/Relation_(mathematics)) on the [domain](/source/Domain_of_a_function), namely [equality](/source/Equality_(mathematics)).

For [concrete categories](/source/Concrete_categories), the diagonal morphism can be simply described by its action on elements x of the object A. Namely, \delta_A(x) = \langle x,x \rangle, the [ordered pair](/source/Ordered_pair) formed from x. The reason for the name is that the [image](/source/Image_(mathematics)) of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism \mathbb{R} \rightarrow \mathbb{R}^2 on the [real line](/source/Real_line) is given by the line that is the [graph](/source/Graph_of_a_function) of the equation y=x. The diagonal morphism into the [infinite product](/source/Infinite_product) X^\infty may provide an [injection](/source/Injective_function) into the [space of sequences](/source/Space_of_sequences) valued in X; each element maps to the constant [sequence](/source/Sequence) at that element. However, most notions of sequence spaces have [convergence](/source/Convergent_series) restrictions that the image of the diagonal map will fail to satisfy.

The dual notion of a diagonal morphism is a **codiagonal morphism**. For every object B in a category \mathcal{C} where the [coproducts](/source/Coproducts) B \sqcup B exists, the codiagonal[3][2][7][5][6] is the canonical morphism

- \delta_B \colon B \sqcup B \stackrel{[Id,Id]} \to B

satisfying

- \delta_B \circ \tau_l = \operatorname{id}_B for l \in \{ 1,2 \}.

where \tau_l is the injection morphism to the l-th component.

## See also

- [Diagonal functor](/source/Diagonal_functor)
- [Diagonal embedding](/source/Diagonal_embedding)
- [Category Theory/(Co-)cones and (co-)limits](https://en.wikibooks.org/wiki/Category_Theory%2F(Co-)cones_and_(co-)limits)

## References

1. (Carter et al. 2008)

1. (Faith 1973)

1. (Popescu & Popescu 1979, Exercise 7.2.)

1. (Diagonal in nlab)

1. (Laurent 2013)

1. (Masakatsu 1972, Definition 4.)

1. (codiagonal in nlab)

## Bibliography

- Awodey, s. (1996). "Structure in Mathematics and Logic: A Categorical Perspective". *Philosophia Mathematica*. **4** (3): 209–237. [doi:10.1093/philmat/4.3.209](https://doi.org/10.1093/philmat/4.3.209)
- Baez, John C. (2004). "Quantum Quandaries: A Category-Theoretic Perspective". *The Structural Foundations of Quantum Gravity*. pp. 240–265. [arXiv:quant-ph/0404040](https://arxiv.org/abs/quant-ph/0404040). [Bibcode:2004quant.ph..4040B](https://ui.adsabs.harvard.edu/abs/2004quant.ph..4040B). [doi:10.1093/acprof:oso/9780199269693.003.0008](https://doi.org/10.1093/acprof:oso/9780199269693.003.0008). ISBN 978-0-19-926969-3.
- Carter, J. Scott; Crans, Alissa; Elhamdadi, Mohamed; Saito, Masahico (2008). ["Cohomology of Categorical Self-Distributivity"](http://tcms.org.ge/Journals/JHRS/xvolumes/2008/n1a2/v3n1a2.pdf). *Journal of Homotopy and Related Structures*. **3** (1): 13–63. [arXiv:math/0607417](https://arxiv.org/abs/math/0607417). [Bibcode:2006math......7417C](https://ui.adsabs.harvard.edu/abs/2006math......7417C)
- Faith, Carl (1973). [[Google Books](https://books.google.com/books?id=vsfyCAAAQBAJ) "Product and Coproduct"]. *Algebra*. pp. 83–109. [doi:10.1007/978-3-642-80634-6_4](https://doi.org/10.1007/978-3-642-80634-6_4). ISBN 978-3-642-80636-0.
- Kashiwara, Msakia & Schapira, Pierre (2006). [[Google Books](https://books.google.com/books?id=K-SjOw_2gXwC) "Limits"]. *Categories and Sheaves*. Vol. 332. Grundlehren der mathematischen Wissenschaften. pp. 35–69. [doi:10.1007/3-540-27950-4_3](https://doi.org/10.1007/3-540-27950-4_3). ISBN 978-3-540-27949-5.
- Mitchell, Barry (1965). [[Google Books](https://books.google.com/books?id=hgJ3pTQSAd0C) *Theory of Categories*]. Academic Press. ISBN 978-0-12-499250-4.
- Masakatsu, Uzawa (1972). ["Some categorical properties of complex spaces Part II"](https://opac.ll.chiba-u.jp/da/curator/900025755/KJ00004239811.pdf). *Bulletin of the Faculty of Education, Chiba University*. **21**: 83-93. [ISSN 0577-6856](https://www.worldcat.org/issn/0577-6856)
- Popescu, Nicolae & Popescu, Liliana (1979). [[Google Books](https://books.google.com/books?id=YnHwCAAAQBAJ) "Categories and functors"]. *Theory of categories*. pp. 1–148. [doi:10.1007/978-94-009-9550-5_1](https://doi.org/10.1007/978-94-009-9550-5_1). ISBN 978-94-009-9552-9.
- Pupier, R. (1964). ["Petit guide des catégories"](http://eudml.org/doc/273359) (in French). *Publications du Département de Mathématiques (Lyon)*. **1** (1): 1–18.

## External links

- Aubert, Clément (2019). ["Categories for Me, and You?"](https://hal.science/hal-02308858). [arXiv:1910.05172](https://arxiv.org/abs/1910.05172)
- Herscovich, Estanislao (2020). ["Lectures on basic homological algebra"](https://www-fourier.ujf-grenoble.fr/~eherscov/Master2020/Basic-homology.pdf)
- Laurent, Olivier (2013). ["Categories for Me \[note\]"](https://perso.ens-lyon.fr/olivier.laurent/categories.pdf). *perso.ens-lyon.fr*
- ["codiagonal"](https://ncatlab.org/nlab/show/codiagonal). *ncatlab.org*
- ["diagonal morphism"](https://ncatlab.org/nlab/show/diagonal+morphism). *ncatlab.org*

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Adapted from the Wikipedia article [Diagonal morphism](https://en.wikipedia.org/wiki/Diagonal_morphism) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Diagonal_morphism?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.