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generate yaml files for all properties of categories
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‎databases/catdat/data_yaml/category-properties/CIP.yaml‎

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id: CIP
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relation: satisfies
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description: |-
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A category satisfies <i>CIP</i> ("coproducts inject into products") if it has zero morphisms, products, coproducts, and for every family of objects $(X_i)_{i \in I}$ the canonical morphism
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$$\textstyle \alpha : \coprod_i X_i \to \prod_{i \in I} X_i$$
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defined by $p_j \circ \alpha \circ \iota_i = \delta_{i,j}$ is a monomorphism. This is no standard terminology. This property has been added to clarify relationships between other properties, in particular those concerning the commutation between limits and colimits.
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dual_property_id: CSP
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invariant_under_equivalences: true
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related_properties:
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- coproducts
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- counital
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- filtered-colimit-stable monomorphisms
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- products
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- zero morphisms

‎databases/catdat/data_yaml/category-properties/CSP.yaml‎

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id: CSP
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relation: satisfies
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description: |-
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A category satisfies <i>CSP</i> ("coproducts surject onto products") if it has zero morphisms, products, coproducts, and for every family of objects $(X_i)_{i \in I}$ the canonical morphism
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$$\textstyle \alpha : \coprod_i X_i \to \prod_{i \in I} X_i$$
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defined by $p_j \circ \alpha \circ \iota_i = \delta_{i,j}$ is an epimorphism. This is no standard terminology. This property has been added to clarify relationships between other properties, in particular those concerning the commutation between limits and colimits.
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dual_property_id: CIP
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invariant_under_equivalences: true
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related_properties:
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- cofiltered-limit-stable epimorphisms
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- coproducts
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- products
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- unital
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- zero morphisms

‎databases/catdat/data_yaml/category-properties/Cauchy complete.yaml‎

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id: Cauchy complete
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relation: is
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description: "A category is <i>Cauchy complete</i> if every idempotent splits. That is, every endomorphism $e : X \\to X$ with $e^2 = e$ may be written as $e = i \\circ p$ for some morphisms $p : X \\to Y$ and $i : Y \\to X$ with $p \\circ i = \\id_Y$. Equivalently, the pair $e,\\id_X : X \\rightrightarrows X$ has an equalizer (or an coequalizer)."
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nlab_link: https://ncatlab.org/nlab/show/Cauchy+complete+category
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dual_property_id: Cauchy complete
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invariant_under_equivalences: true
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related_properties:
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- finitely complete

‎databases/catdat/data_yaml/category-properties/Grothendieck abelian.yaml‎

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id: Grothendieck abelian
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relation: is
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description: A category is <i>Grothendieck abelian</i> if it is abelian, has coproducts (and is therefore cocomplete), a generator, and filtered colimits commute with finite limits. These categories play an important role in homological algebra.
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nlab_link: https://ncatlab.org/nlab/show/Grothendieck+category
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invariant_under_equivalences: true
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related_properties:
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- abelian
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- cocomplete
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- exact filtered colimits
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- generator

‎databases/catdat/data_yaml/category-properties/Grothendieck topos.yaml‎

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id: Grothendieck topos
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relation: is a
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description: A <i>Grothendieck topos</i> is a category that is equivalent to the category of sheaves on a site.
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nlab_link: https://ncatlab.org/nlab/show/Grothendieck+topos
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invariant_under_equivalences: true
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related_properties:
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- elementary topos

‎databases/catdat/data_yaml/category-properties/Malcev.yaml‎

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id: Malcev
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relation: is
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description: A category is <i>Malcev</i> when it has finite limits and every internal reflexive relation is an internal equivalence relation. That is, if $R \subseteq X^2$ is a subobject with $\Delta_X \subseteq R$, then $R$ is symmetric and transitive.
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nlab_link: https://ncatlab.org/nlab/show/Malcev+category
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dual_property_id: co-Malcev
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invariant_under_equivalences: true
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related_properties:
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- finitely complete

‎databases/catdat/data_yaml/category-properties/abelian.yaml‎

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id: abelian
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relation: is
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description: A category is <i>abelian</i> if it is additive, every morphism has a kernel and a cokernel, and every monomorphism and epimorphism is normal. Equivalently, it is additive, has equalizers and coequalizers, and it is mono-regular and epi-regular. As opposed to other types of categories (such as monoidal categories), being abelian turns out to be a mere property.
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nlab_link: https://ncatlab.org/nlab/show/abelian+category
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dual_property_id: abelian
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invariant_under_equivalences: true
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related_properties:
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- additive
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- cokernels
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- conormal
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- kernels
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- normal

‎databases/catdat/data_yaml/category-properties/accessible.yaml‎

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id: accessible
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relation: is
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description: Let $\kappa$ be a regular cardinal. A category is <i>$\kappa$-accessible</i> if it has $\kappa$-filtered colimits and there is a (small) set $G$ of $\kappa$-presentable objects such that every object is a $\kappa$-filtered colimit of objects in $G$. A category is <i>accessible</i> if it is $\kappa$-accessible for some regular cardinal $\kappa$.
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nlab_link: https://ncatlab.org/nlab/show/accessible+category
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dual_property_id: coaccessible
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invariant_under_equivalences: true
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related_properties:
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- finitely accessible
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- locally multi-presentable
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- locally poly-presentable
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- locally presentable
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- ℵ₁-accessible

‎databases/catdat/data_yaml/category-properties/additive.yaml‎

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id: additive
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relation: is
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description: A category is <i>additive</i> if it is preadditive and has finite products (equivalently, finite coproducts). Note that in the context of finite products, the preadditive structure <a href="/lemma/preadditive_structure_unique">is unique</a>.
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nlab_link: https://ncatlab.org/nlab/show/additive+category
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dual_property_id: additive
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invariant_under_equivalences: true
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related_properties:
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- biproducts
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- finite coproducts
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- finite products
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- preadditive

‎databases/catdat/data_yaml/category-properties/aleph1-accessible.yaml‎

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id: ℵ₁-accessible
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relation: is
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description: This is the special case of the notion of $\kappa$-accessible categories, where $\kappa = \aleph_1$ is the first uncountable cardinal.
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nlab_link: https://ncatlab.org/nlab/show/accessible+category
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invariant_under_equivalences: true
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related_properties:
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- accessible
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- finitely accessible
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- locally ℵ₁-presentable

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