Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $λ$-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation of a more discrete nature such as the calculus of finite differences. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. But change action models are very general and do not share the nice properties of Cartesian differential categories. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.
Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential lambda-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation such as the calculus of finite differences or the Boolean differential calculus. On the other hand, change action models have been shown to capture these examples as well as more "exot…
Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freyd-categories and Haskell's Arrows. It is proved that a Cartesian effect category is a Freyd-category where the premonoidal structure is provided by a kind of binary product, called the sequential pr…
Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Pi-types, Sigma-types and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the Bénabou-Hofmann interpretation of Martin-Löf type theory in locally cartesian closed categories yields a biequivalence…
A non-self-contained gathering of notes on category theory, including the definition of locally cartesian closed category, of the cartesian structure in slice categories, or of the pseudo-cartesian structure on Eilenberg-Moore categories. References and proofs are provided, sometimes, to my knowledge, for the first time.
Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $λ$-calculus. An important source of examples of Cartesian differential categories are the coKleisli categories of the comonads of differential categories, where the latter concept provides the categorical semantics of differential linear logic. In this paper, we generalize this construction by introducing Cartesian differential comonads, which are precisely the comonads whose coKl…
We propose a semantics for permutation equivalence in higher-order rewriting. This semantics takes place in cartesian closed 2-categories, and is proved sound and complete.
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by func…
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over Cartesian categories, so that every Cartesian category has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.
In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us to set the whole theory inside the welcoming 2-category of double categories, and to overcome technical problems that were caused by working with left adjoints inside a general bicategory. Cartesian double categories that are also fibrant are of particular interest to us. After describing some important properties of Cartesian and fibrant double categories, …
The goal of this paper is to prove an equivalence between the $(\infty,2)$-category of cartesian factorization systems of $\infty$-categories and that of pointed cartesian fibrations of $\infty$-categories. This generalizes a similar result known for ordinary categories and sheds some light on the interplay between these two seemingly distant concepts.
The present paper gives a generalization of cartesian closed categories, called cartesian closed categories with dependence, whose strict version induces categories with families that support 1-, Sigma- and Pi-types in the strict sense. Consequently, we have obtained a new semantics of dependent type theories that is both categorical and true-to-syntax.
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
It is proved that equalities between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equality in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to model-theoretical methods of normalization.
This note explains how dependent sums and products are interpreted by adjoints of the base change functor in a locally cartesian closed category. An effort is made to unpack all the definitions so as to make the concepts more transparent to new learners.
The existence of adjoints to algebraic functors between categories of models of Lawvere theories follows from finite-product-preservingness surviving left Kan extension. A result along these lines was proved in Appendix 2 of Brian Day's 1970 PhD thesis. His context was categories enriched in a cartesian closed base. A generalization is described here with essentially the same proof. We introduce the notion of cartesian monoidal category in the enriched context. With an advanced viewpoint, we give a result about left extension along a promonoidal module and further related results.
Cartesian differential categories were introduced to provide an abstract axiomatization of categories of differentiable functions. The fundamental example is the category whose objects are Euclidean spaces and whose arrows are smooth maps. Tensor differential categories provide the framework for categorical models of differential linear logic. The coKleisli category of any tensor differential category is always a Cartesian differential category. Cartesian differential categories, besides arising in this manner as coKleisli categories, occur in many different and quite independent ways. Thus, i…
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category. Given a composition of two commutative squares in $\mathcal{C}$, if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian. This covers the result by Mac Lane (1998) for abelian categories and the result by Christensen and Frankland (2022) for triangulated categories.
It has been proved by Bergh and Thaule that the higher mapping cone axiom is equivalent to the higher octahedral axiom for n-angulated categories. In this note, we use homotopy cartesian diagrams to give several new equivalent statements of the higher mapping cone axiom, which are applied to explain the higher octahedral axiom.
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal-Tierney and the straightening construction due to Lurie.
From the analogue of Boehm's Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.
A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to the product functor. And, since functors having a right adjoint preserve co-limits, the result follows immediately. But, when we try to explicitly construct the arrows, things become a bit more involved. In rare cases, it is pretty useful to have the exact arrows that make this isomorphism to hold. The purpose of this note is to develop the explicit proof w…
Using generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over $\mathsf{Set}$, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of $(\mathbb{T},V)$-categories and show that, under suitable conditions, every $(\mathbb{T},V)$-injective category is exponentiable in $(\mathbb{T},V)\text{-}\mathsf{Cat}…
The Faà di Bruno construction, introduced by Cockett and Seely, constructs a comonad $\mathsf{Fa{\grave{a}}}$ whose coalgebras are precisely Cartesian differential categories. In other words, for a Cartesian left additive category $\mathbb{X}$, $\mathsf{Fa{\grave{a}}}(\mathbb{X})$ is the cofree Cartesian differential category over $\mathbb{X}$. Composition in these cofree Cartesian differential categories is based on the Faà di Bruno formula, and corresponds to composition of differential forms. This composition, however, is somewhat complex and difficult to work with. In this paper we provide…
Regular logic is the fragment of first order logic generated by $=$, $\top$, $\wedge$, and $\exists$. A key feature of this logic is that it is the minimal fragment required to express composition of binary relations; another is that it is the internal logic of regular categories. The link between these two facts is that in any regular category, one may construct a notion of binary relation using jointly-monic spans; this results in what is known as the bicategory of relations of the regular category. In this paper we provide a direct axiomatization of bicategories of relations, which we term …
In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $e^x$ from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $e^0 = 1$, $e^{x+y} = e^x e^y$, and $\frac{\partial e^x}{\partial x} = e^x$ all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponent…
In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the dif…
After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every infinity-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved (and detected) by right adjoints t…
We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.
We introduce the new concept of cartesian module over a pseudofunctor $R$ from a small category to the category of small preadditive categories. Already the case when $R$ is a (strict) functor taking values in the category of commutative rings is sufficient to cover the classical construction of quasi-coherent sheaves of modules over a scheme. On the other hand, our general setting allows for a good theory of contravariant additive locally flat functors, providing a geometrically meaningful extension of Crawley-Boevey's Representation Theorem. As an application, we relate and extend some previ…
Every category $\mathcal K$ has a free completion $\mathcal P \mathcal K$ under colimits and a free completion $Σ\mathcal K$ under coproducts. A number of properties of $\mathcal K$ transfer to $\mathcal P \mathcal K$ and $Σ\mathcal K$ (e.g., completeness or cartesian closedness). We prove that $\mathcal P\mathcal K$ is always a pretopos, but, for $\mathcal K$ large, seldom a topos. Moreover, for complete categories $\mathcal K$ we prove that $\mathcal P\mathcal K$ is locally cartesian closed whenever $\mathcal K$ is additive or cartesian closed or dual to an extensive category. We also study …
For a relational Horn theory $\mathbb{T}$, we provide useful sufficient conditions for the exponentiability of objects and morphisms in the category $\mathbb{T}\text{-}\mathsf{Mod}$ of $\mathbb{T}$-models; well-known examples of such categories, which have found recent applications in the study of programming language semantics, include the categories of preordered sets and (extended) metric spaces. As a consequence, we obtain useful sufficient conditions for $\mathbb{T}\text{-}\mathsf{Mod}$ to be cartesian closed, locally cartesian closed, and even a quasitopos; in particular, we provide two …
Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data-structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. In order to reason about trees, we need the notion of path, which can be formalised in the internal logic of any locally cartesian closed pretopos with a natural number object. In such categories, we derive existence results about M-types, leading to stability of locally cartesian closed pretoposes with a natural number object a…
Categories are coreflectively embedded in multicategories via the "discrete cocone" construction, the right adjoint being given by the monoid construction. Furthermore, the adjunction lifts to the "cartesian level": preadditive categories are coreflectively embedded (as theories for many-sorted modules) in cartesian multicategories (general algebraic theories). In particular, one gets a direct link between two ways of considering modules over a rig, namely as additive functors valued in commutative monoids or as models of the theory generated by the rig itself.
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.
We show, using Eckmann-Hilton argument, that the category of 3-computads is not cartesian closed. As a corollary we get that neither the category of all computads nor the category of n-computads, for n>2, do form locally cartesian closed categories, and hence elementary toposes.
This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of $M$-precategories. The procedure can then be iterated, leading to model categories of $(\infty, n)$-categories.
Symmetric 2-rigs: coexponentiability and cartesian closure
Laboratoire J.-A. Dieudonné, Université Côte d’Azur（雅克-阿芒·迪厄多内实验室，蔚蓝海岸大学）；Department of Computer Science and Technology, University of Cambridge（剑桥大学计算机科学与技术系）；Department of Mathematics, The University of Manchester（曼彻斯特大学数学系）
