Philosophy:Coherent category

Short description: Category in mathematical category theory

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 $\mathrm{S}\mathrm{u}\mathrm{b}(X)$ has finte unions and each $f^{*}:\mathrm{S}\mathrm{u}\mathrm{b}(B)\to \mathrm{S}\mathrm{u}\mathrm{b}(A)$ perserves them. (Makkai Reyes) called logical categories, and according to Makkai & Reyes (1977), the coherent category was introduced by Joyal and Gonzalo E. Reyes.

Coherent category

Axiom

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

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

Coherent functor

A functor $f:{𝒞}\to {𝒟}$ 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 $X\to 0$ is an isomorphism.
• For every object $X$ of a coherent category ${𝒞}$, the poset of subobjects $\mathrm{S}\mathrm{u}\mathrm{b}(X)$ is distributive lattice.
• If ${𝒞}$ is coherent, every functor category ${{𝒞}}^{{𝒟}}$ is again coherent.

Heyting category

A Heyting category is a coherent category ${𝒞}$ in which $f^{*}:\mathrm{S}\mathrm{u}\mathrm{b}(B)\to \mathrm{S}\mathrm{u}\mathrm{b}(A)$ has a right adjoint ${\mathrm{\forall }}_f:\mathrm{S}\mathrm{u}\mathrm{b}(A)\to \mathrm{S}\mathrm{u}\mathrm{b}(B)$. 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 ${\mathrm{\forall }}_f$.

Joyal's completeness theorem

Let ${𝒜}$ be a coherent category and $\mathrm{M}\mathrm{o}\mathrm{d}({𝒜})$ is the category of coherent functors from ${𝒜}$ to $\mathrm{S}\mathrm{e}\mathrm{t}\mathrm{s}$. Then the evaluation functor

$e_{{𝒜}}:{𝒜}\stackrel{ev}{⟶}{\mathrm{S}\mathrm{e}\mathrm{t}\mathrm{s}}^{\mathrm{M}\mathrm{o}\mathrm{d}({𝒜})}$

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

If ${𝒜}$ is a (small) Heyting category, then $e_{{𝒜}}$ 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 $\mathrm{S}\mathrm{u}\mathrm{b}(X)$ is small) and $\mathrm{S}\mathrm{u}\mathrm{b}(X)$ 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). 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. https://books.google.com/books?id=DxpHDwAAQBAJ&pg=PT39. 
• 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. https://doi.org/10.1093/oso/9780198534259.003.0001. 
• Reyes, Marie La Palme; Reyes, Gonzalo E.; Zolfaghari, Houman (2004) (in en). Generic Figures and Their Glueings: A Constructive Approach to Functor Categories. Polimetrica s.a.s.. ISBN 978-88-7699-004-5. OCLC 62251418. https://books.google.com/books?id=Z5GcOhfwoD0C&dq=joyal&pg=PA182. 
• 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. 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. 611. doi:10.1007/BFb0066201. ISBN 978-3-540-08439-6. 

External links

• Lurie, Jacob. "Lecture 4: Coherent Categories". https://www.math.ias.edu/~lurie/278xnotes/Lecture4-CoherentCategories.pdf. 
• Lurie, Jacob. "Lecture 7: Pretopoi". https://www.math.ias.edu/~lurie/278xnotes/Lecture7-Pretopoi.pdf. 
• Lurie, Jacob. "Lecture 13: Elimination of Imaginaries". https://www.math.ias.edu/~lurie/278xnotes/Lecture13-Imaginaries.pdf. 
• Hazewinkel, Michiel, ed. (2001), "Categorical logic", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Mimram, Samuel. "Notes on toposes". https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/docs/mimram_topos.pdf.