// Workers AI · dad joke modeWhat did the coherent category say? It was well-categorized.

Not to be confused with coherent topos.

In category theory in mathematics, a coherent category is a regular category in which the poset of subobjects $\mathrm {Sub} (X)$ has finte unions and each $f^{*}:\mathrm {Sub} (B)\rightarrow \mathrm {Sub} (A)$ perserves them. Makkai & Reyes (1977) called logical categories, and according to Makkai & Reyes (1977), the coherent category was introduced by Joyal and Gonzalo E. Reyes.

Coherent category

Axiom

Let ${\mathcal {C}}$ be a category. We will say that ${\mathcal {C}}$ is coherent category if it satisfies the following axioms:

• The category ${\mathcal {C}}$ admits finite limits.
• Every morphism $f:X\rightarrow Z$ in ${\mathcal {C}}$ admits a factorization $X\mathrel {\stackrel {g}{\longrightarrow }} Y\mathrel {\stackrel {h}{\longrightarrow }} Z$ where g is an effective epimorphism and h is a monomorphism.
• For every object $X\in {\mathcal {C}}$, the poset $\mathrm {Sub} (X)$ have "finite" unions which are stable under pullback, then $\mathrm {Sub} (X)$ is an upper semilattice.
• The collection of effective epimorphisms in ${\mathcal {C}}$ is stable under pullback.
• For every morphism $f:X\rightarrow Y$ in ${\mathcal {C}}$, the map $f^{-1}:\mathrm {Sub} (Y)\rightarrow \mathrm {Sub} (X)$ is a homomorphism of upper semilattices.

Coherent functor

A functor $f:{\mathcal {C}}\rightarrow {\mathcal {D}}$ between coherent categories is called coherent functor if it is a regular functor which preserves finite unions.

Example

• Every coherent category admits an initial object $0$ which is strict, that is every morphism $X\rightarrow 0$ is an isomorphism.
• For every object $X$ of a coherent category ${\mathcal {C}}$, the poset of subobjects $\mathrm {Sub} (X)$ is distributive lattice.
• If ${\mathcal {C}}$ is coherent, every functor category ${\mathcal {C}}^{\mathcal {D}}$ is again coherent.

Heyting category

A Heyting category is a coherent category ${\mathcal {C}}$ in which $f^{*}:\mathrm {Sub} (B)\rightarrow \mathrm {Sub} (A)$ has a right adjoint $\forall _{f}:\mathrm {Sub} (A)\rightarrow \mathrm {Sub} (B)$. The binary operation on subobjects thus defined is stable under pullback.

Heyting functor

A Heyting functor between Heyting category is a coherent functor which commutes up to isomorphism with right adjoints $\forall _{f}$.

Joyal's completeness theorem

Let ${\mathcal {A}}$ be a coherent category and $\mathrm {Mod} ({\mathcal {A}})$ is the category of coherent functors from ${\mathcal {A}}$ to $\mathrm {Sets}$. Then the evaluation functor

$e_{\mathcal {A}}:{\mathcal {A}}\mathrel {\stackrel {ev}{\longrightarrow }} \mathrm {Sets} ^{\mathrm {Mod} ({\mathcal {A}})}$

is conservative and preserves all finite limits, stable finite sups, stable images and stable $\forall _{f}({\mathcal {A}})$ existing in ${\mathcal {A}}$.

If ${\mathcal {A}}$ is a (small) Heyting category, then $e_{\mathcal {A}}$ is a conservative Heyting functor.

Geometric category (a.k.a. Infinitary coherent category)

A geometric category is a regular category which is well-powered (every $\mathrm {Sub} (X)$ is small) and $\mathrm {Sub} (X)$ have all unions which are stable under pullback. A geometric category is Heyting category by the adjoint functor theorem for posets. Also, every Grothendieck topos (in the sense of Giraud's axioms) is a geometric category.

References

• Borceux, Francis; Campanini, Federico; Gran, Marino (2022). "The stable category of preorders in a pretopos I: General theory". Journal of Pure and Applied Algebra. 226 (9) 106997. arXiv:2201.05992. doi:10.1016/j.jpaa.2021.106997.
• Cigoli, Alan S.; Gray, James R. A.; Linden, Tim Van der (2015). "Algebraically coherent categories". Theory and Applications of Categories. 30: 1864–1906. doi:10.70930/tac/l546hee4.
• Caramello, Olivia (19 January 2018). Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'. Oxford University Press. ISBN 978-0-19-107675-6.
• Johnstone, Peter T. (2002). "Regular and Cartesia's Closed Categories (A 1.4)". Sketches of an Elephant a Topos Theory Cornpendiurn. pp. 3–67. doi:10.1093/oso/9780198534259.003.0001. ISBN 978-0-19-853425-9.
• Reyes, Marie La Palme; Reyes, Gonzalo E.; Zolfaghari, Houman (2004). Generic Figures and Their Glueings: A Constructive Approach to Functor Categories. Polimetrica s.a.s. ISBN 978-88-7699-004-5. OCLC 62251418.
• Marquis, Jean-Pierre; Reyes, Gonzalo E. (2012). "The History of Categorical Logic: 1963–1977". Sets and Extensions in the Twentieth Century. Handbook of the History of Logic. Vol. 6. pp. 689–800. doi:10.1016/B978-0-444-51621-3.50010-4. ISBN 978-0-444-51621-3.
• Makkai, Michael (1 July 1995). "On Gabbay's Proof of the Craig Interpolation Theorem for Intuitionistic Predicate Logic". Notre Dame Journal of Formal Logic. 36 (3). doi:10.1305/ndjfl/1040149353.
• Reyes, Gonzalo E. (2 October 2019). "Topos Theory in Montréal in the 1970s: My Personal Involvement". History and Philosophy of Logic. 40 (4): 389–402. doi:10.1080/01445340.2018.1554471.
• Marra, Vincenzo; Reggio, Luca (2020). "A characterisation of the category of compact Hausdorff spaces". Theory and Applications of Categories. 35: 1871–1906. doi:10.70930/tac/e0lnn7uy.
• Makkai, Michael; Reyes, Gonzalo E. (1977). First Order Categorical Logic. Lecture Notes in Mathematics. Vol. 611. doi:10.1007/BFb0066201. ISBN 978-3-540-08439-6.

External links

• Lurie, Jacob. "Lecture 4: Coherent Categories" (PDF).
• Lurie, Jacob. "Lecture 7: Pretopoi" (PDF).
• Lurie, Jacob. "Lecture 13: Elimination of Imaginaries" (PDF).
• Lambek, J. (2001) [1994], "Categorical logic", Encyclopedia of Mathematics, EMS Press
• Mimram, Samuel. "Notes on toposes" (PDF).