diff --git a/src/Cat/Bi/Instances/Relations.lagda.md b/src/Cat/Bi/Instances/Relations.lagda.md index 28cc82560..f2bfe0bb9 100644 --- a/src/Cat/Bi/Instances/Relations.lagda.md +++ b/src/Cat/Bi/Instances/Relations.lagda.md @@ -48,6 +48,7 @@ monic.], but in finite-products categories, the definition takes an especially pleasant form: A relation $\phi : a \rel b$ is a [subobject] of the [Cartesian product] $a \times b$. +[bicategory]: Cat.Bi.Base.html [regular categories]: Cat.Regular.html [pullback-stable]: Cat.Diagram.Pullback.html#stability [image factorisations]: Cat.Diagram.Image.html diff --git a/src/Cat/Diagram/Colimit/Base.lagda.md b/src/Cat/Diagram/Colimit/Base.lagda.md index 641f1dc1c..25cf69aca 100644 --- a/src/Cat/Diagram/Colimit/Base.lagda.md +++ b/src/Cat/Diagram/Colimit/Base.lagda.md @@ -663,6 +663,8 @@ coproducts indexed by its class of morphisms, then it is automatically [thin]. Since excluded middle is independent of type theory, we can not prove that any non-thin categories have arbitrary colimits. +[thin]: Order.Base.html + Instead, categories are cocomplete _with respect to_ a pair of universes: A category is **$(o, \ell)$-cocomplete** if it has colimits for any diagram indexed by a precategory with objects in $\ty\ o$ and diff --git a/src/Cat/Diagram/Colimit/Cocone.lagda.md b/src/Cat/Diagram/Colimit/Cocone.lagda.md index fb539f42e..0bbbef47d 100644 --- a/src/Cat/Diagram/Colimit/Cocone.lagda.md +++ b/src/Cat/Diagram/Colimit/Cocone.lagda.md @@ -150,7 +150,7 @@ We can then rephrase the universality from the definition of left Kan extension by asking that a particular cocone be [initial] in the category we have just constructed. -[initial object]: Cat.Diagram.Initial.html +[initial]: Cat.Diagram.Initial.html ```agda is-initial-cocone→is-colimit diff --git a/src/Cat/Displayed/Cartesian.lagda.md b/src/Cat/Displayed/Cartesian.lagda.md index eeb409f75..870ea8050 100644 --- a/src/Cat/Displayed/Cartesian.lagda.md +++ b/src/Cat/Displayed/Cartesian.lagda.md @@ -340,6 +340,7 @@ invertible→cartesian ``` If $f$ is cartesian, it's also a [weak monomorphism]. + [weak monomorphism]: Cat.Displayed.Morphism.html#weak-monos ```agda diff --git a/src/Cat/Displayed/Cartesian/Right.lagda.md b/src/Cat/Displayed/Cartesian/Right.lagda.md index 12213e326..2a7fee03f 100644 --- a/src/Cat/Displayed/Cartesian/Right.lagda.md +++ b/src/Cat/Displayed/Cartesian/Right.lagda.md @@ -35,6 +35,8 @@ open Cat.Displayed.Reasoning ℰ A [cartesian fibration] $\cE$ is said to be a **right fibration** if every morphism in $\cE$ is cartesian. +[cartesian fibration]: Cat.Displayed.Cartesian.html + ```agda record Right-fibration : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality diff --git a/src/Cat/Displayed/Cocartesian/Indexing.lagda.md b/src/Cat/Displayed/Cocartesian/Indexing.lagda.md index 687d114e4..503da4571 100644 --- a/src/Cat/Displayed/Cocartesian/Indexing.lagda.md +++ b/src/Cat/Displayed/Cocartesian/Indexing.lagda.md @@ -38,7 +38,7 @@ fibrations. [Opfibrations]: Cat.Displayed.Cocartesian.html [fibrations]: Cat.Displayed.Cartesian.html -[reindex along morphisms in the base] Cat.Displayed.Cartesian.Indexing.html +[reindex along morphisms in the base]: Cat.Displayed.Cartesian.Indexing.html This difference in variance gives opfibrations a distinct character. Reindexing in fibrations can be thought of a sort of restriction map. diff --git a/src/Cat/Displayed/Cocartesian/Weak.lagda.md b/src/Cat/Displayed/Cocartesian/Weak.lagda.md index 77fbfc9e7..d749ab0bb 100644 --- a/src/Cat/Displayed/Cocartesian/Weak.lagda.md +++ b/src/Cat/Displayed/Cocartesian/Weak.lagda.md @@ -473,6 +473,9 @@ record is-weak-cocartesian-fibration : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) wher --> Weak opfibrations are dual to [weak fibrations]. + +[weak fibrations]: Cat.Displayed.Cartesian.Weak.html#is-weak-cartesian-fibration + ```agda weak-op-fibration→weak-opfibration : is-weak-cartesian-fibration (ℰ ^total-op) diff --git a/src/Cat/Displayed/Composition.lagda.md b/src/Cat/Displayed/Composition.lagda.md index 775b32844..c269f0dee 100644 --- a/src/Cat/Displayed/Composition.lagda.md +++ b/src/Cat/Displayed/Composition.lagda.md @@ -26,7 +26,7 @@ $\int \cE$, then we can construct a new category $\cE \cdot \cF$ displayed over $\cB$ that contains the data of both $\cE$ and $\cF$. -[total category] Cat.Displayed.Total.html +[total category]: Cat.Displayed.Total.html To actually construct the composite, we bundle up the data of $\cE$ and $\cF$ into pairs, so an object in $\cE \cdot \cF$ diff --git a/src/Cat/Displayed/GenericObject.lagda.md b/src/Cat/Displayed/GenericObject.lagda.md index 99556f01c..aabf52e5b 100644 --- a/src/Cat/Displayed/GenericObject.lagda.md +++ b/src/Cat/Displayed/GenericObject.lagda.md @@ -158,10 +158,10 @@ structure of $t'$. This condition is quite strong: for instance, the family fibration of a strict category $\cC$ has a skeletal generic object if and only if $\cC$ -is [skeletal] (See [here] for a proof). +is [skeletal] (See [here](Cat.Displayed.Instances.Family.html#skeletal-generic-objects) +for a proof). [skeletal]: Cat.Skeletal.html -[here]: Cat.Displayed.Instances.Family.html#skeletal-generic-objects ```agda is-skeletal-generic-object : ∀ {t} {t′ : Ob[ t ]} → Generic-object t′ → Type _ @@ -194,10 +194,10 @@ must, as well. We can also expand on what this means for the family fibration: $\rm{Fam}({\cC})$ has a gaunt generic object if and only if $\cC$ is itself -[gaunt] (See [here] for the proof). +[gaunt] (See [here](Cat.Displayed.Instances.Family.html#gaunt-generic-objects) +for the proof). [gaunt]: Cat.Gaunt.html -[here]: Cat.Displayed.Instances.Family.html#gaunt-generic-objects