https://julesh.com/forest/index/ index /forest/index/ Forest Back to home This is my forest, a loosely structured collection of notes. https://julesh.com/forest/all/ all /forest/all/ Index of all trees in reverse order of creation 2025 10 25 https://julesh.com/forest/000O/ 000O /forest/000O/ The history functor for cartesian colinks Let be a category with finite products and let be the category of cartesian colinks. We define the history functor to be the covariant functor represented by the tensor unit . Concretely, on objects is given by . factors through the forwards fibration via the functor . For simple colinks in the category of sets, the history functor and the forwards fibration coincide. The dual of the history functor is the continuation functor. 2025 10 25 https://julesh.com/forest/000N/ 000N /forest/000N/ The continuation functor for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. We define the continuation functor as the contravariant functor represesented by the tensor unit . Concretely, is given as follows: On objects, Given a cartesian colink with forwards pass and backwards pass , the function takes a morphism to the composite We use the term continuation to refer to a morphism of when regarded as an element of . The dual of the continuation functor is the history functor. 2025 10 25 https://julesh.com/forest/000L/ 000L /forest/000L/ The forwards fibration for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. Since is constructed as the total category of an indexed category it is equipped with a fibration called the forwards fibration. Concretely, is given as follows: On objects, On morphisms, takes each cartesian colink to its forwards pass Given an object of and a morphism of , the cartesian lifting is the colink with forwards pass and backwards pass the projection The forwards fibration is closely related to the history functor. They coincide for simple colinks in the category of sets. 2025 10 25 https://julesh.com/forest/000K/ 000K /forest/000K/ Tensor product of cartesian colinks Recall the construction of the category of cartesian colinks from the indexed category of simple slices, Recall that the simple slice categories have finite products given by products in , and that reindexing preserves them. Therefore the opposite categories have finite coproducts given by products in , and reindexing preserves them. This structure defines a symmetric monoidal product in , called the tensor product. In general the tensor product is neither cartesian nor cocartesian. Reference: Shulman, Framed bicategories and monoidal fibrations Concretely, on objects the tensor product is given by pairwise monoidal product: The monoidal unit is denoted , where is the terminal object of . The tensor product of cartesian colinks is fibrewise dual to the cartesian product of cartesian links. 2025 10 25 https://julesh.com/forest/000J/ 000J /forest/000J/ Construction of cartesian colinks from simple slices Let be a category with finite products, and let be the indexed category of simple slices. We define the category of cartesian colinks as the total category of the opposite indexed category: Notice that since the simple slice categories all have the same class of objects, namely the objects of , an object of is a pair of objects of . 2025 10 24 https://julesh.com/forest/000I/ 000I /forest/000I/ Simple slices as polynomial categories Let be a category with finite products and an object of . Recall that the simple slice category has finite products. We can universally characterise as the smallest category with finite products that contains and another morphism . Based on the analogous construction of polynomial rings, the simple slice is sometimes called a polynomial category. This is the origin of its notation. Reference: Lambek & Scott, Introduction to higher-order categorical logic 2025 10 24 https://julesh.com/forest/000H/ 000H /forest/000H/ Products in the simple slice category Let be a category with finite products, and let be an object of . Then the simple slice category has finite products, given by the underlying products in . To prove this, we construct the Kleisli adjunction between and . The left adjoint is given by: On objects, is taken to given by the pairing of the projection and The right adjoint is given by: is identity on objects is taken to given by precomposition with the projection The hom-set isomorphism is proven as follows: Since is a right adjoint it preserves limits. Since has the same objects as and since is identity on objects, this establishes that has finite products given by finite products in . 2025 10 24 https://julesh.com/forest/000I/ 000I /forest/000I/ Simple slices as polynomial categories Let be a category with finite products and an object of . Recall that the simple slice category has finite products. We can universally characterise as the smallest category with finite products that contains and another morphism . Based on the analogous construction of polynomial rings, the simple slice is sometimes called a polynomial category. This is the origin of its notation. Reference: Lambek & Scott, Introduction to higher-order categorical logic 2025 10 24 https://julesh.com/forest/000G/ 000G /forest/000G/ Indexed category structure of simple slices Let be a category with finite products. A morphism of induces a functor where: is identity on objects For a morphism , is given by This yields an indexed category . This structure can also be seen as arising from the comonad morphisms between the parameters comonads, since morphisms of comonads lift contravariantly to functors between Kleisli categories. Since is identity on objects, it also preserves products in simple slice categories. Thus has indexed finite products. 2025 10 24 https://julesh.com/forest/000F/ 000F /forest/000F/ The simple slice category Let be a category with finite products, and let be an object of . We write for the Kleisli category of the parameters comonad . Concretely, is given as follows: Objects of are objects of Morphisms in are morphisms in The identity morphism in is the projection in The composition of and is given by Reference: Categorical logic and type theory 2025 10 24 https://julesh.com/forest/000H/ 000H /forest/000H/ Products in the simple slice category Let be a category with finite products, and let be an object of . Then the simple slice category has finite products, given by the underlying products in . To prove this, we construct the Kleisli adjunction between and . The left adjoint is given by: On objects, is taken to given by the pairing of the projection and The right adjoint is given by: is identity on objects is taken to given by precomposition with the projection The hom-set isomorphism is proven as follows: Since is a right adjoint it preserves limits. Since has the same objects as and since is identity on objects, this establishes that has finite products given by finite products in . 2025 10 24 https://julesh.com/forest/000I/ 000I /forest/000I/ Simple slices as polynomial categories Let be a category with finite products and an object of . Recall that the simple slice category has finite products. We can universally characterise as the smallest category with finite products that contains and another morphism . Based on the analogous construction of polynomial rings, the simple slice is sometimes called a polynomial category. This is the origin of its notation. Reference: Lambek & Scott, Introduction to higher-order categorical logic 2025 10 24 https://julesh.com/forest/000G/ 000G /forest/000G/ Indexed category structure of simple slices Let be a category with finite products. A morphism of induces a functor where: is identity on objects For a morphism , is given by This yields an indexed category . This structure can also be seen as arising from the comonad morphisms between the parameters comonads, since morphisms of comonads lift contravariantly to functors between Kleisli categories. Since is identity on objects, it also preserves products in simple slice categories. Thus has indexed finite products. 2025 10 24 https://julesh.com/forest/000E/ 000E /forest/000E/ The parameters comonad Let be a category with finite products, and let be an object of . The functor given by can be given the structure of a commutative comonad, called the parameters comonad, as follows: The counit is given by the projection The comultiplication is given by the diagonal The commutative strength is the natural isomorphism Given a morphism of , the natural transformation extends to a morphism of comonads . 2025 10 24 https://julesh.com/forest/000D/ 000D /forest/000D/ Costrong profunctors Let be a monoidal category and let and be -actegories. A costrong profunctor or Tambara comodule is a functor equipped with a natural transformation called the costrength, The dual concept of costrong profunctors is strong profunctors. 2025 10 24 https://julesh.com/forest/000C/ 000C /forest/000C/ Strong profunctors Let be a monoidal category and let and be -actegories. A strong profunctor or Tambara module is a functor equipped with a natural transformation called the strength, The dual of strong profunctors is costrong profunctors. 2025 10 24 https://julesh.com/forest/000B/ 000B /forest/000B/ Simple colinks in the category of sets Let be sets. A simple colink is given by: A function called the forwards pass A function called the backwards pass Simple colinks are so-called to distinguish them from indexed colinks, which also exist in the category of sets. Given simple colinks and , there is a compsite colink whose forwards pass is the composite function and whose backwards pass is the function defined by the colink composition law In this case we can directly prove that colink composition is associative as follows: Since the category of sets is cartesian closed, the category of colinks in the category of sets can be constructed either as a category of cartesian colinks or as a category of colinks in a monoidal closed category. 2025 10 24 https://julesh.com/forest/000A/ 000A /forest/000A/ Cartesian colinks Let be a category with finite products, and let be objects of . A cartesian colink is given by: A morphism called the forwards pass A morphism called the backwards pass Given colinks and , there is a composite colink whose forwards pass is the composite morphism and whose backwards pass is the composite It is not obvious from this definition that colink composition is associative. A direct proof is possible but tedious, so we prefer to prove it using machinery of fibred categories. For a category with finite products, there is a category where: Objects of are pairs of objects of , written Morphisms of are colinks in The identity morphism has as forwards pass the identity morphism in , and as backwards pass the projection Composite morphisms are given by colink composition 2025 10 24 https://julesh.com/forest/000F/ 000F /forest/000F/ The simple slice category Let be a category with finite products, and let be an object of . We write for the Kleisli category of the parameters comonad . Concretely, is given as follows: Objects of are objects of Morphisms in are morphisms in The identity morphism in is the projection in The composition of and is given by Reference: Categorical logic and type theory 2025 10 24 https://julesh.com/forest/000H/ 000H /forest/000H/ Products in the simple slice category Let be a category with finite products, and let be an object of . Then the simple slice category has finite products, given by the underlying products in . To prove this, we construct the Kleisli adjunction between and . The left adjoint is given by: On objects, is taken to given by the pairing of the projection and The right adjoint is given by: is identity on objects is taken to given by precomposition with the projection The hom-set isomorphism is proven as follows: Since is a right adjoint it preserves limits. Since has the same objects as and since is identity on objects, this establishes that has finite products given by finite products in . 2025 10 24 https://julesh.com/forest/000I/ 000I /forest/000I/ Simple slices as polynomial categories Let be a category with finite products and an object of . Recall that the simple slice category has finite products. We can universally characterise as the smallest category with finite products that contains and another morphism . Based on the analogous construction of polynomial rings, the simple slice is sometimes called a polynomial category. This is the origin of its notation. Reference: Lambek & Scott, Introduction to higher-order categorical logic 2025 10 24 https://julesh.com/forest/000G/ 000G /forest/000G/ Indexed category structure of simple slices Let be a category with finite products. A morphism of induces a functor where: is identity on objects For a morphism , is given by This yields an indexed category . This structure can also be seen as arising from the comonad morphisms between the parameters comonads, since morphisms of comonads lift contravariantly to functors between Kleisli categories. Since is identity on objects, it also preserves products in simple slice categories. Thus has indexed finite products. 2025 10 25 https://julesh.com/forest/000J/ 000J /forest/000J/ Construction of cartesian colinks from simple slices Let be a category with finite products, and let be the indexed category of simple slices. We define the category of cartesian colinks as the total category of the opposite indexed category: Notice that since the simple slice categories all have the same class of objects, namely the objects of , an object of is a pair of objects of . 2025 10 25 https://julesh.com/forest/000L/ 000L /forest/000L/ The forwards fibration for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. Since is constructed as the total category of an indexed category it is equipped with a fibration called the forwards fibration. Concretely, is given as follows: On objects, On morphisms, takes each cartesian colink to its forwards pass Given an object of and a morphism of , the cartesian lifting is the colink with forwards pass and backwards pass the projection The forwards fibration is closely related to the history functor. They coincide for simple colinks in the category of sets. 2025 10 25 https://julesh.com/forest/000K/ 000K /forest/000K/ Tensor product of cartesian colinks Recall the construction of the category of cartesian colinks from the indexed category of simple slices, Recall that the simple slice categories have finite products given by products in , and that reindexing preserves them. Therefore the opposite categories have finite coproducts given by products in , and reindexing preserves them. This structure defines a symmetric monoidal product in , called the tensor product. In general the tensor product is neither cartesian nor cocartesian. Reference: Shulman, Framed bicategories and monoidal fibrations Concretely, on objects the tensor product is given by pairwise monoidal product: The monoidal unit is denoted , where is the terminal object of . The tensor product of cartesian colinks is fibrewise dual to the cartesian product of cartesian links. 2025 10 25 https://julesh.com/forest/000O/ 000O /forest/000O/ The history functor for cartesian colinks Let be a category with finite products and let be the category of cartesian colinks. We define the history functor to be the covariant functor represented by the tensor unit . Concretely, on objects is given by . factors through the forwards fibration via the functor . For simple colinks in the category of sets, the history functor and the forwards fibration coincide. The dual of the history functor is the continuation functor. 2025 10 25 https://julesh.com/forest/000N/ 000N /forest/000N/ The continuation functor for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. We define the continuation functor as the contravariant functor represesented by the tensor unit . Concretely, is given as follows: On objects, Given a cartesian colink with forwards pass and backwards pass , the function takes a morphism to the composite We use the term continuation to refer to a morphism of when regarded as an element of . The dual of the continuation functor is the history functor. 2025 10 24 https://julesh.com/forest/0009/ 0009 /forest/0009/ Colinks Colinks, usually called lenses, are pairs of morphisms with the backwards morphism also having an input in the forwards direction. Composition of colinks is an abstraction of the chain rule of calculus. There are many definitions of colinks that apply with different assumptions on the base category. In each of these cases, for a category with the appropriate structure we will obtain a category whose morphisms are colinks in . 2025 10 24 https://julesh.com/forest/000B/ 000B /forest/000B/ Simple colinks in the category of sets Let be sets. A simple colink is given by: A function called the forwards pass A function called the backwards pass Simple colinks are so-called to distinguish them from indexed colinks, which also exist in the category of sets. Given simple colinks and , there is a compsite colink whose forwards pass is the composite function and whose backwards pass is the function defined by the colink composition law In this case we can directly prove that colink composition is associative as follows: Since the category of sets is cartesian closed, the category of colinks in the category of sets can be constructed either as a category of cartesian colinks or as a category of colinks in a monoidal closed category. 2025 10 24 https://julesh.com/forest/000A/ 000A /forest/000A/ Cartesian colinks Let be a category with finite products, and let be objects of . A cartesian colink is given by: A morphism called the forwards pass A morphism called the backwards pass Given colinks and , there is a composite colink whose forwards pass is the composite morphism and whose backwards pass is the composite It is not obvious from this definition that colink composition is associative. A direct proof is possible but tedious, so we prefer to prove it using machinery of fibred categories. For a category with finite products, there is a category where: Objects of are pairs of objects of , written Morphisms of are colinks in The identity morphism has as forwards pass the identity morphism in , and as backwards pass the projection Composite morphisms are given by colink composition 2025 10 24 https://julesh.com/forest/000F/ 000F /forest/000F/ The simple slice category Let be a category with finite products, and let be an object of . We write for the Kleisli category of the parameters comonad . Concretely, is given as follows: Objects of are objects of Morphisms in are morphisms in The identity morphism in is the projection in The composition of and is given by Reference: Categorical logic and type theory 2025 10 24 https://julesh.com/forest/000H/ 000H /forest/000H/ Products in the simple slice category Let be a category with finite products, and let be an object of . Then the simple slice category has finite products, given by the underlying products in . To prove this, we construct the Kleisli adjunction between and . The left adjoint is given by: On objects, is taken to given by the pairing of the projection and The right adjoint is given by: is identity on objects is taken to given by precomposition with the projection The hom-set isomorphism is proven as follows: Since is a right adjoint it preserves limits. Since has the same objects as and since is identity on objects, this establishes that has finite products given by finite products in . 2025 10 24 https://julesh.com/forest/000I/ 000I /forest/000I/ Simple slices as polynomial categories Let be a category with finite products and an object of . Recall that the simple slice category has finite products. We can universally characterise as the smallest category with finite products that contains and another morphism . Based on the analogous construction of polynomial rings, the simple slice is sometimes called a polynomial category. This is the origin of its notation. Reference: Lambek & Scott, Introduction to higher-order categorical logic 2025 10 24 https://julesh.com/forest/000G/ 000G /forest/000G/ Indexed category structure of simple slices Let be a category with finite products. A morphism of induces a functor where: is identity on objects For a morphism , is given by This yields an indexed category . This structure can also be seen as arising from the comonad morphisms between the parameters comonads, since morphisms of comonads lift contravariantly to functors between Kleisli categories. Since is identity on objects, it also preserves products in simple slice categories. Thus has indexed finite products. 2025 10 25 https://julesh.com/forest/000J/ 000J /forest/000J/ Construction of cartesian colinks from simple slices Let be a category with finite products, and let be the indexed category of simple slices. We define the category of cartesian colinks as the total category of the opposite indexed category: Notice that since the simple slice categories all have the same class of objects, namely the objects of , an object of is a pair of objects of . 2025 10 25 https://julesh.com/forest/000L/ 000L /forest/000L/ The forwards fibration for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. Since is constructed as the total category of an indexed category it is equipped with a fibration called the forwards fibration. Concretely, is given as follows: On objects, On morphisms, takes each cartesian colink to its forwards pass Given an object of and a morphism of , the cartesian lifting is the colink with forwards pass and backwards pass the projection The forwards fibration is closely related to the history functor. They coincide for simple colinks in the category of sets. 2025 10 25 https://julesh.com/forest/000K/ 000K /forest/000K/ Tensor product of cartesian colinks Recall the construction of the category of cartesian colinks from the indexed category of simple slices, Recall that the simple slice categories have finite products given by products in , and that reindexing preserves them. Therefore the opposite categories have finite coproducts given by products in , and reindexing preserves them. This structure defines a symmetric monoidal product in , called the tensor product. In general the tensor product is neither cartesian nor cocartesian. Reference: Shulman, Framed bicategories and monoidal fibrations Concretely, on objects the tensor product is given by pairwise monoidal product: The monoidal unit is denoted , where is the terminal object of . The tensor product of cartesian colinks is fibrewise dual to the cartesian product of cartesian links. 2025 10 25 https://julesh.com/forest/000O/ 000O /forest/000O/ The history functor for cartesian colinks Let be a category with finite products and let be the category of cartesian colinks. We define the history functor to be the covariant functor represented by the tensor unit . Concretely, on objects is given by . factors through the forwards fibration via the functor . For simple colinks in the category of sets, the history functor and the forwards fibration coincide. The dual of the history functor is the continuation functor. 2025 10 25 https://julesh.com/forest/000N/ 000N /forest/000N/ The continuation functor for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. We define the continuation functor as the contravariant functor represesented by the tensor unit . Concretely, is given as follows: On objects, Given a cartesian colink with forwards pass and backwards pass , the function takes a morphism to the composite We use the term continuation to refer to a morphism of when regarded as an element of . The dual of the continuation functor is the history functor. 2025 10 21 https://julesh.com/forest/0008/ 0008 /forest/0008/ From actegories to locally graded categories Let be a monoidal category and an -actegory. We construct a locally graded category called , which we regard as a Para construction, as follows: The objects of are objects of For objects of , the hom-object is given by The identity morphism in is given by the unitor The composition of and is given by 2025 10 21 https://julesh.com/forest/0007/ 0007 /forest/0007/ The golden theorem of actegories Let be a symmetric monoidal category. Every -actegory that is also a symmetric monoidal category in a compatible way is necessarily defined by a strong symmetric monoidal functor , up to natural isomorphism. Specifically, if is a symmetric monoidal actegory then where is defined by . This means that in the vast majority of cases, actegories are redundant and strong symmetric monoidal functors suffice. Reference: Actegories for the working amthematician, section 5 2025 10 21 https://julesh.com/forest/0006/ 0006 /forest/0006/ From strong monoidal functors to actegories Let and be monoidal categories, and let be a strong monoidal functor. Then we can define an -actegory structure on by . 2025 10 21 https://julesh.com/forest/0007/ 0007 /forest/0007/ The golden theorem of actegories Let be a symmetric monoidal category. Every -actegory that is also a symmetric monoidal category in a compatible way is necessarily defined by a strong symmetric monoidal functor , up to natural isomorphism. Specifically, if is a symmetric monoidal actegory then where is defined by . This means that in the vast majority of cases, actegories are redundant and strong symmetric monoidal functors suffice. Reference: Actegories for the working amthematician, section 5 2025 10 21 https://julesh.com/forest/0005/ 0005 /forest/0005/ Locally graded categories Let be a monoidal category. A category locally graded in is a category enriched in the monoidal category of presheaves with Day convolution. Unwinding the definition, a category locally graded in is given by the following data: A class of objects For each pair of objects of and each object of , a set of morphisms #\mathcal C (X, Y) (M), which we also write For each object of , an identity morphism in For morphisms and , a composite morphism Coherence isomorphisms 2025 10 21 https://julesh.com/forest/0008/ 0008 /forest/0008/ From actegories to locally graded categories Let be a monoidal category and an -actegory. We construct a locally graded category called , which we regard as a Para construction, as follows: The objects of are objects of For objects of , the hom-object is given by The identity morphism in is given by the unitor The composition of and is given by 2025 10 21 https://julesh.com/forest/0004/ 0004 /forest/0004/ Actegories Let be a monoidal category and a category. An action of on , also called an -actegory structure on , is given by a functor , together with the following coherence data: A natural transformation called the unitor A natural transformation called the actor Coherence equations 2025 10 21 https://julesh.com/forest/0006/ 0006 /forest/0006/ From strong monoidal functors to actegories Let and be monoidal categories, and let be a strong monoidal functor. Then we can define an -actegory structure on by . 2025 10 21 https://julesh.com/forest/0007/ 0007 /forest/0007/ The golden theorem of actegories Let be a symmetric monoidal category. Every -actegory that is also a symmetric monoidal category in a compatible way is necessarily defined by a strong symmetric monoidal functor , up to natural isomorphism. Specifically, if is a symmetric monoidal actegory then where is defined by . This means that in the vast majority of cases, actegories are redundant and strong symmetric monoidal functors suffice. Reference: Actegories for the working amthematician, section 5 2025 10 21 https://julesh.com/forest/0008/ 0008 /forest/0008/ From actegories to locally graded categories Let be a monoidal category and an -actegory. We construct a locally graded category called , which we regard as a Para construction, as follows: The objects of are objects of For objects of , the hom-object is given by The identity morphism in is given by the unitor The composition of and is given by 2025 10 21 https://julesh.com/forest/0003/ 0003 /forest/0003/ Para construction, naively Let be a monoidal category. For objects of , a parametrised morphism in consists of: An object of called the parameter object A morphism of There is a category where: Objects of are objects of Morphisms of are equivalence classes of parametrised morphisms in modulo coherent isomorphisms of the objects The identity morphism on is given by the parameter object and the canonical isomorphism Composition of and is given by the parameter object and the morphism However, in practice it is better to define this category as a quotient of a double category, and in greater generality. 2025 10 21 https://julesh.com/forest/0002/ 0002 /forest/0002/ Para construction The term "parameters construction" or "para construction" is used for several related constructions of categories of parametrised morphisms. For a category , a para construction over it will be referred to as , possibly with subscripts for other data used in the construction. Sometimes we think of these as 1-categories, they are better thought of as double categories. 2025 10 21 https://julesh.com/forest/0003/ 0003 /forest/0003/ Para construction, naively Let be a monoidal category. For objects of , a parametrised morphism in consists of: An object of called the parameter object A morphism of There is a category where: Objects of are objects of Morphisms of are equivalence classes of parametrised morphisms in modulo coherent isomorphisms of the objects The identity morphism on is given by the parameter object and the canonical isomorphism Composition of and is given by the parameter object and the morphism However, in practice it is better to define this category as a quotient of a double category, and in greater generality. 2025 10 21 https://julesh.com/forest/0005/ 0005 /forest/0005/ Locally graded categories Let be a monoidal category. A category locally graded in is a category enriched in the monoidal category of presheaves with Day convolution. Unwinding the definition, a category locally graded in is given by the following data: A class of objects For each pair of objects of and each object of , a set of morphisms #\mathcal C (X, Y) (M), which we also write For each object of , an identity morphism in For morphisms and , a composite morphism Coherence isomorphisms 2025 10 21 https://julesh.com/forest/0008/ 0008 /forest/0008/ From actegories to locally graded categories Let be a monoidal category and an -actegory. We construct a locally graded category called , which we regard as a Para construction, as follows: The objects of are objects of For objects of , the hom-object is given by The identity morphism in is given by the unitor The composition of and is given by 2025 10 20 https://julesh.com/forest/0001/ 0001 /forest/0001/ Test tree This tree exists to test table of contents generation. https://julesh.com/forest/index/ index /forest/index/ Forest Transclusion loop detected, rendering stopped. https://julesh.com/forest/all/ all /forest/all/ Index of all trees in reverse order of creation Transclusion loop detected, rendering stopped. https://julesh.com/forest/macros/ macros /forest/macros/ Macros This tree exists to export Forester macros. 2025 10 21 https://julesh.com/forest/0002/ 0002 /forest/0002/ Para construction The term "parameters construction" or "para construction" is used for several related constructions of categories of parametrised morphisms. For a category , a para construction over it will be referred to as , possibly with subscripts for other data used in the construction. Sometimes we think of these as 1-categories, they are better thought of as double categories. 2025 10 21 https://julesh.com/forest/0003/ 0003 /forest/0003/ Para construction, naively Let be a monoidal category. For objects of , a parametrised morphism in consists of: An object of called the parameter object A morphism of There is a category where: Objects of are objects of Morphisms of are equivalence classes of parametrised morphisms in modulo coherent isomorphisms of the objects The identity morphism on is given by the parameter object and the canonical isomorphism Composition of and is given by the parameter object and the morphism However, in practice it is better to define this category as a quotient of a double category, and in greater generality. 2025 10 21 https://julesh.com/forest/0005/ 0005 /forest/0005/ Locally graded categories Let be a monoidal category. A category locally graded in is a category enriched in the monoidal category of presheaves with Day convolution. Unwinding the definition, a category locally graded in is given by the following data: A class of objects For each pair of objects of and each object of , a set of morphisms #\mathcal C (X, Y) (M), which we also write For each object of , an identity morphism in For morphisms and , a composite morphism Coherence isomorphisms 2025 10 21 https://julesh.com/forest/0008/ 0008 /forest/0008/ From actegories to locally graded categories Let be a monoidal category and an -actegory. We construct a locally graded category called , which we regard as a Para construction, as follows: The objects of are objects of For objects of , the hom-object is given by The identity morphism in is given by the unitor The composition of and is given by 2025 10 24 https://julesh.com/forest/0009/ 0009 /forest/0009/ Colinks Colinks, usually called lenses, are pairs of morphisms with the backwards morphism also having an input in the forwards direction. Composition of colinks is an abstraction of the chain rule of calculus. There are many definitions of colinks that apply with different assumptions on the base category. In each of these cases, for a category with the appropriate structure we will obtain a category whose morphisms are colinks in . 2025 10 24 https://julesh.com/forest/000B/ 000B /forest/000B/ Simple colinks in the category of sets Let be sets. A simple colink is given by: A function called the forwards pass A function called the backwards pass Simple colinks are so-called to distinguish them from indexed colinks, which also exist in the category of sets. Given simple colinks and , there is a compsite colink whose forwards pass is the composite function and whose backwards pass is the function defined by the colink composition law In this case we can directly prove that colink composition is associative as follows: Since the category of sets is cartesian closed, the category of colinks in the category of sets can be constructed either as a category of cartesian colinks or as a category of colinks in a monoidal closed category. 2025 10 24 https://julesh.com/forest/000A/ 000A /forest/000A/ Cartesian colinks Let be a category with finite products, and let be objects of . A cartesian colink is given by: A morphism called the forwards pass A morphism called the backwards pass Given colinks and , there is a composite colink whose forwards pass is the composite morphism and whose backwards pass is the composite It is not obvious from this definition that colink composition is associative. A direct proof is possible but tedious, so we prefer to prove it using machinery of fibred categories. For a category with finite products, there is a category where: Objects of are pairs of objects of , written Morphisms of are colinks in The identity morphism has as forwards pass the identity morphism in , and as backwards pass the projection Composite morphisms are given by colink composition 2025 10 24 https://julesh.com/forest/000F/ 000F /forest/000F/ The simple slice category Let be a category with finite products, and let be an object of . We write for the Kleisli category of the parameters comonad . Concretely, is given as follows: Objects of are objects of Morphisms in are morphisms in The identity morphism in is the projection in The composition of and is given by Reference: Categorical logic and type theory 2025 10 24 https://julesh.com/forest/000H/ 000H /forest/000H/ Products in the simple slice category Let be a category with finite products, and let be an object of . Then the simple slice category has finite products, given by the underlying products in . To prove this, we construct the Kleisli adjunction between and . The left adjoint is given by: On objects, is taken to given by the pairing of the projection and The right adjoint is given by: is identity on objects is taken to given by precomposition with the projection The hom-set isomorphism is proven as follows: Since is a right adjoint it preserves limits. Since has the same objects as and since is identity on objects, this establishes that has finite products given by finite products in . 2025 10 24 https://julesh.com/forest/000I/ 000I /forest/000I/ Simple slices as polynomial categories Let be a category with finite products and an object of . Recall that the simple slice category has finite products. We can universally characterise as the smallest category with finite products that contains and another morphism . Based on the analogous construction of polynomial rings, the simple slice is sometimes called a polynomial category. This is the origin of its notation. Reference: Lambek & Scott, Introduction to higher-order categorical logic 2025 10 24 https://julesh.com/forest/000G/ 000G /forest/000G/ Indexed category structure of simple slices Let be a category with finite products. A morphism of induces a functor where: is identity on objects For a morphism , is given by This yields an indexed category . This structure can also be seen as arising from the comonad morphisms between the parameters comonads, since morphisms of comonads lift contravariantly to functors between Kleisli categories. Since is identity on objects, it also preserves products in simple slice categories. Thus has indexed finite products. 2025 10 25 https://julesh.com/forest/000J/ 000J /forest/000J/ Construction of cartesian colinks from simple slices Let be a category with finite products, and let be the indexed category of simple slices. We define the category of cartesian colinks as the total category of the opposite indexed category: Notice that since the simple slice categories all have the same class of objects, namely the objects of , an object of is a pair of objects of . 2025 10 25 https://julesh.com/forest/000L/ 000L /forest/000L/ The forwards fibration for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. Since is constructed as the total category of an indexed category it is equipped with a fibration called the forwards fibration. Concretely, is given as follows: On objects, On morphisms, takes each cartesian colink to its forwards pass Given an object of and a morphism of , the cartesian lifting is the colink with forwards pass and backwards pass the projection The forwards fibration is closely related to the history functor. They coincide for simple colinks in the category of sets. 2025 10 25 https://julesh.com/forest/000K/ 000K /forest/000K/ Tensor product of cartesian colinks Recall the construction of the category of cartesian colinks from the indexed category of simple slices, Recall that the simple slice categories have finite products given by products in , and that reindexing preserves them. Therefore the opposite categories have finite coproducts given by products in , and reindexing preserves them. This structure defines a symmetric monoidal product in , called the tensor product. In general the tensor product is neither cartesian nor cocartesian. Reference: Shulman, Framed bicategories and monoidal fibrations Concretely, on objects the tensor product is given by pairwise monoidal product: The monoidal unit is denoted , where is the terminal object of . The tensor product of cartesian colinks is fibrewise dual to the cartesian product of cartesian links. 2025 10 25 https://julesh.com/forest/000O/ 000O /forest/000O/ The history functor for cartesian colinks Let be a category with finite products and let be the category of cartesian colinks. We define the history functor to be the covariant functor represented by the tensor unit . Concretely, on objects is given by . factors through the forwards fibration via the functor . For simple colinks in the category of sets, the history functor and the forwards fibration coincide. The dual of the history functor is the continuation functor. 2025 10 25 https://julesh.com/forest/000N/ 000N /forest/000N/ The continuation functor for cartesian colinks Let be a category with finite products, and let be the category of cartesian colinks. We define the continuation functor as the contravariant functor represesented by the tensor unit . Concretely, is given as follows: On objects, Given a cartesian colink with forwards pass and backwards pass , the function takes a morphism to the composite We use the term continuation to refer to a morphism of when regarded as an element of . The dual of the continuation functor is the history functor. References Context Backlinks Related Contributions