cartesian bicategory

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#Contents#
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## Idea

A **cartesian bicategory** is a [[bicategory]] with properties that make it behave like a bicategory of generalized [[relations]].

Examples of cartesian bicategories include

 * the bicategory [[Rel]] of sets and relations,
 * more generally, the [[bicategory of relations]] of a  [[regular category]],
 * the bicategory [[Span]] of sets and spans,
 * more generally, the bicategory of [[spans]] in a finitely [[complete category]],
 * the bicategory [[profunctor|Prof]] of (small) categories and profunctors,
 * more generally, the bicategory of internal categories and [[modules]] ([[profunctors]]) in a finitely complete category $C$,
 * as an alternative generalization, the bicategory of enriched categories and modules over a _cartesian_ monoidal category $V$.

Non-examples include

 * the locally-discrete bicategory [[Set]] of sets and functions,
 * the bicategory [[Cat]] of categories and functors,
 * the bicategory of profunctors and enriched in a _noncartesian_ monoidal category $V$.

Consider for example the bicategory [[Rel]]. This bicategory admits a symmetric monoidal structure given by the cartesian product. Because the cartesian product is _not_ the categorical product, this symmetric monoidal structure is _not_ [[cartesian monoidal category|cartesian monoidal]]. However, the cartesian product restricts to the locally-discrete sub-bicategory [[Set]] of sets and functions, on which it _is_  the categorical product. Note that a relation is the graph of a function if and only if it possesses a right adjoint (which is given by the opposite relation). In a general bicategory, a 1-cell is called _map-like_ if it possesses a right adjoint, and these are the 1-cells playing the role that functions play in $Rel$.

Generalizing this situation, a cartesian bicategory $B$ is a symmetric monoidal bicategory for which the monoidal product restricts to the categorical product on the sub-bicategory of map-like 1-cells, subject to natural compatibility conditions. Remarkably, the structure of such a tensor is [[property-like structure|property-like]] (is essentially unique when it exists). Thus the definition can be formulated directly in terms of the bicategory $B$ without mentioning a symmetric monoidal structure at all. Not only is this technically convenient (as the precise definition of a symmetric monoidal bicategory is quite technical), but it is moreover a very natural perspective to take.

Like allegories, cartesian bicategories provide a convenient abstract setting in which to study the calculus of relations, but unlike allegories they embrace higher examples like $Span$ and $Mod$. The original idea, as explained by Carboni and Walters in their paper, clearly had these various examples in mind but was initially confined to the locally posetal case; in part this was due to the lack at the time (circa 1987) of a complete definition of symmetric monoidal bicategories. In more recent years a full bicategorical treatment has emerged, due principally to Carboni, Kelly, Verity, and Wood. The alternative treatment we give below has not yet appeared in the literature as far as this author ([[Todd Trimble]]) is aware.

## Technical preliminaries

Note: All compositions will be written in the traditional order, in which application proceeds from right to left.

We work with familiar notions of the theory of bicategories (which, for reasons of consonance with 2-terminology, we also call 2-categories) but in some cases under new names. We calculate with pasting diagrams in 2-categories as if they were strict 2-categories.

Our notion of morphism between 2-categories has gone under various names: "homomorphism" in the sense of Bénabou, also known as "pseudofunctor" or weak 2-functor, where the structural constraints are isomorphisms. Here they are simply called **2-functors**.

Each 2-category $B$ gives rise to a hom 2-functor $hom: B^{op} \times B \to Cat$, which we denote by $B(-, -)$, with the contravariant argument in the first place as is customary.

Given 2-functors $F, G: B \to C$, it is for our purposes convenient to use the [[lax natural transformation|lax notion of morphism]] $\theta: F \to G$ between 2-functors for which the structural cells $\theta \cdot f$ (for 1-cells $f: a \to b$ in $B$) point in the direction

$$\array{
F a & \overset{\theta a}{\to} & G a\\
F f \downarrow & \overset{\theta \cdot f}{\Rightarrow} & \downarrow G f\\
F b & \underset{\theta b}{\to} & G b
}$$

These are called "oplax transformations" by some authors such as Bénabou and "lax transformations" by other authors such as Johnstone; on this page we will simply call them (2-)**transformations**. A transformation is **strong** if the structural cells $\theta \cdot f$ are isomorphisms.

There is a well-known notion of morphism between transformation which has been called [[modification]]. We retain this usage, but as an aside we counsel against inventing a new term (e.g., "perturbation" between modifications) every time a new level of morphism is reached -- a more uniform terminology is called for. The term "transfor" (due to Sjoerd Crans) has been tentatively adopted elsewhere on this site; modifications may then be called **2-transfors**.

In any case, given 2-categories $B$ and $C$, there is a 2-category of 2-functors, strong transformations and modifications from $B$ to $C$; this will be denoted $Hom_s(B, C)$. We have also the 2-category of 2-functors, transformations, and modifications; this will be denoted $Hom_l(B, C)$.

### 2-adjunctions between 2-categories

A **2-adjunction** $F \dashv G$ consists of 2-functors $F: B \to C$, $G: C \to B$, and an adjoint equivalence

$$C(F -, -) \simeq B(-, G -)$$

in the 2-category $Hom_s(B^{op} \times C, Cat)$. In more elementary terms, the data of a 2-adjunction is given by strong transformations

$$\eta: 1_B \to G F \qquad \varepsilon: F G \to 1_C$$

and invertible modifications (called _triangulators_)

$$\array{
G & \overset{\eta G}{\to} & G F G & & & & F & \overset{F\eta}{\to} & F G F\\
1_G \downarrow & \overset{s}{\Rightarrow} & \downarrow G \varepsilon & & & & 1_F \downarrow & \overset{t}{\Leftarrow} & \downarrow \varepsilon F\\
G & \underset{1_G}{\to} & G & & & & F & \underset{1_F}{\to} & F
}$$

such that the _triangulator coherence conditions_ hold: there are pasting diagram equalities

$$\array{
1_B & \overset{\eta}{\to} & G F & \overset{1_{G F}}{\to} & G F\\
\eta \downarrow & \overset{\eta \cdot \eta}{\Rightarrow} & \downarrow G F \eta & & \downarrow 1_{G F}\\
G F & \underset{\eta G F}{\to} & G F G F & \overset{G t}{\Rightarrow} & G F & & & = & 1_{\eta}\\
1_{G F} \downarrow & & \overset{s F}{\Rightarrow} & G \varepsilon F \searrow & \downarrow 1_{G F} \\
G F & \underset{1_{G F}}{\to} & G F & \underset{1_{G F}}{\to} & G F
}$$

$\, $

$$\array{
F G & \overset{1_{F G}}{\to} & F G & \overset{1_{F G}}{\to} & F G\\
1_{F G} \downarrow & F \eta G \searrow & \overset{t G}{\Rightarrow} & & \downarrow 1_{F G}\\
F G & \overset{F s}{\Rightarrow} & F G F G & \overset{\varepsilon F G}{\to} & F G & & & = & 1_{\varepsilon}\\
1_{F G} \downarrow & & F G \varepsilon \downarrow & \overset{\varepsilon \cdot \varepsilon}{\Rightarrow} & \downarrow \varepsilon\\
F G & \underset{1_{F G}}{\to} & F G & \underset{\varepsilon}{\to} & 1
}$$

### Lax adjunctions between 2-categories

A **lax** adjunction is defined in the same way as a 2-adjunction, except that we relax (remove) the assumptions that $\eta$ and $\varepsilon$ are strong and that $s$, $t$ are invertible; the triangulator coherence conditions are still in effect. In that case, for objects $b$ of $B$ and $c$ of $C$, there is a local adjunction between hom-categories

$$(L: B(b, G c) \to C(F b, c)) \dashv (R: C(F b, c) \to B(b, G c))$$

given by $L(g: b \to G c) = (F b \overset{F g}{\to} F G c \overset{\varepsilon c}{\to} c)$ and $R(f: F b \to c) = (b \overset{\eta b}{\to} G F b \overset{G f}{\to} G c)$. The unit of $L \dashv R$ at $g: b \to G c$ is given by the pasting

$$\array{
b & \overset{g}{\to} & G c & \overset{1_{G c}}{\to} & G c\\
\eta b \downarrow & \overset{\eta \cdot g}{\Leftarrow} & \downarrow \eta G c & \overset{s c}{\Leftarrow} & \downarrow 1_{G c}\\
G F b & \underset{G F g}{\to} & G F G c & \underset{G \varepsilon}{\to} & G c
}$$

and the counit of $L \dashv R$ at $f: F b \to c$ is given by the pasting

$$\array{
F b & \overset{F \eta b}{\to} & F G F b & \overset{F G f}{\to} & F G c\\
1_{F b} \downarrow & \overset{t b}{\Leftarrow} & \downarrow \varepsilon F b & \overset{\varepsilon \cdot f}{\Leftarrow} & \downarrow \varepsilon c\\
F b & \underset{1_{F b}}{\to} & F b & \underset{f}{\to} & c
}$$

The triangular equations for $L \dashv R$ follow from the triangulator coherence conditions. (Warning: it is not generally true that a lax adjunction induces an adjoint pair in $Hom_l(B^{op} \times C, Cat)$. Cf. lemma 1 below.)

### The 2-category $Map(B)$

Let $B$ be a 2-category. Following Carboni and Walters, we define a *map* in $B$ to be a 1-cell in $B$ that possesses a right adjoint. It is useful to think of them as playing the role of "functional relations" in an ambient 2-category of relations. We let $Map(B)$ be the locally full sub-2-category whose 0-cells are those of $B$ and whose 1-cells are the maps in $B$. We observe that every 2-functor $F: B \to C$ restricts to a 2-functor $F|: Map(B) \to Map(C)$, and by the following proposition, every transformation $\theta: F \to G$ restricts to a strong transformation $\theta$ in $Hom_s(Map(B), C)$:

**Proposition 1:** If $f: b \to c$ is a map in $B$, then $\theta \cdot f$ is invertible.

**Proof:** Let $f^*$ denote a right adjoint of $f$. It is easily verified that the diagram

$$\array{
F b & \overset{F f}{\to} & F c & \overset{\theta c}{\to} & G c & \overset{1_{G c}}{\to} & G c\\
1_{F b} \downarrow & \Rightarrow & F f^* \downarrow & \overset{\theta \cdot f^*}{\Rightarrow} & \downarrow G f^* & \Rightarrow & \downarrow 1_{G c}\\
F b & \underset{1_{F b}}{\to} & F b & \underset{\theta b}{\to} & G b & \underset{G f}{\to} & G c
}$$

pastes to give the inverse $(\theta \cdot f)^{-1}$, where the left and right squares are the unit and counit of adjunctions $F f \dashv F f^*$, $G f \dashv G f^*$.

A transformation $\theta: F \to G: B \to C$ is **map-valued** if each 1-cell $\theta b$ is a map in $C$. By the previous proposition, a map-valued transformation $\theta$ in $Hom_l(B, C)$ restricts to a strong transformation $\theta|: F| \to G|$ in $Hom_s(Map(B), Map(C))$.

* **Warning:** In general it is imprecise to identify maps in a 2-category with the "functional relations". That is, for some of the standard examples of cartesian bicategories, such as $B = Prof$, we cannot simply identify $Map(B)$ with $Cat$: 1-cell equivalences are only Morita equivalences, not $Cat$-equivalences. A better adapted framework is one of a bicategory with a proarrow equipment, or a [[framed bicategory]]. Eventually this article will be rewritten with this remark in mind...

## Definition of cartesian structure and basic results

A **cartesian structure** on a bicategory $B$ consists of the following data:

* 2-functors $\otimes: B \times B \to B$ and $I: \mathbf{1} \to B$, where $\mathbf{1}$ is the terminal 2-category;

* Map-valued transformations
$$\delta: 1_B \to \otimes \Delta, \qquad \pi: \Delta \otimes \to 1_{B \times B}, \qquad \varepsilon: 1_B \to I !$$
where $\Delta: B \to B \times B$ is the diagonal 2-functor and $!: B \to \mathbf{1}$ is the unique 2-functor;

* Invertible modifications
$$\array{
\otimes & \overset{\delta \otimes}{\to} & \otimes \Delta \otimes & & & & \Delta & \overset{\Delta \delta}{\to} & \Delta \otimes \Delta & & & & I & \overset{\varepsilon I}{\to} & I ! I\\
\mathllap{1_{\otimes}} \downarrow & \overset{s}{\Rightarrow} & \downarrow \mathrlap{\otimes \pi} & & & & \mathllap{1_{\Delta}} \downarrow & \overset{t}{\Leftarrow} & \downarrow \mathrlap{\pi \Delta} & & & & \mathllap{1_{I}} \downarrow & \overset{u}{\Rightarrow} & \downarrow \mathrlap{I \cdot id}\\
\otimes & \underset{1_{\otimes}}{\to} & \otimes & & & & \Delta & \underset{1_{\Delta}}{\to} & \Delta & & & & I & \underset{1_I}{\to} & I
}$$
satisfying the triangulator coherence conditions. $\Box$

The crucial observation is that, by proposition 1, this data restricts to give 2-adjunctions

$$(Map(B) \overset{\Delta}{\to} Map(B) \times Map(B)) \dashv (Map(B) \times Map(B) \overset{\otimes}{\to} Map(B))$$

\linebreak

$$(Map(B) \overset{!}{\to} \mathbf{1}) \dashv (\mathbf{1} \overset{I}{\to} Map(B))$$

making the restriction of $\otimes$ to $Map(B)$ a _2-product_ on $Map(B)$ with 2-terminal object $I$. Naturally this finite 2-product structure on $Map(B)$ is property-like: is uniquely determined up to equivalence which is uniquely specified up to unique isomorphism.

A little later we will argue that cartesian structure on the full bicategory $B$ is also property-like; the main thing is to see that the way in which $\otimes$ acts on 1-cells is essentially uniquely determined.

### Symmetric monoidal structure via cartesian structure

We now argue that a cartesian bicategory $B$ carries a symmetric monoidal structure whose tensor product is

$$\otimes: B \times B \to B$$

The proof is pretty easy to sketch, once the fact is admitted that the induced 2-product structure makes $Map(B)$ a symmetric monoidal bicategory. This fact about 2-products, while never in dispute, was given a complete proof only in the past few years by the late Max Kelly, and we freely take advantage of it below.

First, given objects $a, b, c$ of $B$, we may regard them as objects of $Map(B)$, where there is an associativity constraint

$$\alpha_{a, b, c}: (a \otimes b) \otimes c \to a \otimes (b \otimes c)$$

on $Map(B)$ which is definable by exploiting 2-universal properties of the 2-product. The associativity thus has an expression in terms of $\otimes$, $\delta$, and $\pi$ which are globally defined on $B$, hence $\alpha$ is globally defined as a transformation on $B$. By similar considerations, the symmetry and unit constraints on $Map(B)$ also extend to globally defined transformations on $B$. We argue in a moment that these constraints are strong (adjoint) equivalences.

Symmetric monoidal structure on $B$ also demands various structural modifications (such as a Yang-Baxter modification $R_{\bullet |\bullet, \bullet}$) satisfying various coherence conditions, but the components of such modifications ($R_{a|b, c}$, say) are defined by regarding their components $a, b, c$ as objects of $Map(B)$ and using the corresponding modifications there. Again, each such modification on $Map(B)$ is defined in terms of 2-adjunction data $\otimes$, $I$, $\delta$, $\pi$, $\varepsilon$, $s$, $t$, $u$ which are globally defined on $B$, so each such structure is a modification on $B$. Various coherence conditions on the modifications must be checked, but the conditions hold at every choice of objects of $B$ by regarding them as objects of $Map(B)$ and using the symmetric monoidal structure there, so the conditions hold on $B$.

Now we check that the structural transformations $\alpha$ are strong adjoint equivalences. Indeed, when regarded as being defined on $Map(B)$, the constraint $\alpha$ is an equivalence and so has a left adjoint (with invertible unit and counit) $\alpha^-$ with components

$$\alpha_{a, b, c}^{-}: a \otimes (b \otimes c) \to (a \otimes b) \otimes c$$

and just like $\alpha$, $\alpha^-$ is definable in terms of global structure on $B$, making $\alpha^-$ a transformation on $B$. Then $\alpha$, and by similar reasoning the symmetry and unit constraints, are strong transformations on $B$ by the following lemma:

**Lemma 1:** If $\alpha$ is a right adjoint in $Hom_l(C, B)$, then the transformation $\alpha$ is strong. Consequently, if $\alpha^- \dashv \alpha$ is an adjoint equivalence, so that both $\alpha^- \dashv \alpha$ and $\alpha \dashv \alpha^-$, then $\alpha^- \dashv \alpha$ is a strong adjoint equivalence in $Hom_s(C, B)$.

**Proof:** Only the first statement requires proof. Given $r: c \to d$ in $C$, let $ev_c: Hom_l(C, B) \to B$ denote the 2-functor which evaluates at $c$ (keep in mind that the 1-cells in $Hom_l(C, B)$ are transformations which are oplax in the sense of Bénabou), and let $ev_r: ev_c \to ev_d$ denote the evident transformation; this is _lax_ in the sense of Bénabou. Then, by dualizing proposition 1, $ev_r(\alpha) = \alpha \cdot r$ is an isomorphism if $\alpha$ is a _right_ adjoint. Since $\alpha \cdot r$ is an isomorphism for all 1-cells $r$ in $C$, it follows that $\alpha$ is strong. $\Box$

This completes the argument that the symmetric monoidal structure on $Map(B)$ extends to $B$.

### Comparison with Carboni-Walters

Let us begin unpacking the terse definition of cartesian bicategory so that it becomes more recognizable. For the sake of notational convenience, we use the fact that as a monoidal bicategory, a cartesian bicategory may be strictified and replaced by a (monoidally biequivalent) $Gray$ monoid $B$. That is to say, if $\otimes_G$ denotes the tensor product of the symmetric monoidal closed category $Gray$, then there is firstly a biequivalence $B \times B \simeq B \otimes_G B$, and we may transfer the data of the cartesian structure across this biequivalence to get a $Gray$ monoid multiplication

$$\otimes: B \otimes_G B \to B$$

together with a corresponding diagonal

$$B \overset{\Delta}{\to} B \times B \simeq B \otimes_G B$$

which we again denote by $\Delta$. In that case, given a 1-cell $r: a \to b$ in $B$, we may write structure cells $\delta \cdot r$ for the transformation $\delta: 1_B \to \otimes \Delta$ in the form

$$\array{
a & \overset{\delta a}{\to} & a \otimes a\\
r \downarrow & \overset{\delta \cdot r}{\Rightarrow} & \downarrow r \otimes r\\
b & \underset{\delta b}{\to} & b \otimes b
}$$

where to be definite we make the convention that $r \otimes r := (r \otimes 1)(1 \otimes r)$.

An object $c$ of $B$, viewed as an object of a 2-category $Map(B)$ with 2-products, naturally carries a 2-comonoid (or pseudocomonoid) structure; the unit of the 2-adjunction $\Delta \dashv \otimes$ is the diagonal map

$$\delta c: c \to c \otimes c,$$

i.e., the 2-comonoid comultiplication. The unit of the 2-adjunction $! \dashv I$ is the projection to the 2-terminal object

$$\varepsilon c: c \to I,$$

i.e., the 2-comonoid counit. (It is more usual to denote a (2-)terminal object by the symbol $1$, and from here on we will follow that practice, writing the projection as $\varepsilon c: c \to 1$, and forget $I$.)

Spelling this out a little further, let us turn attention to the transformation $\pi: \Delta \otimes \to 1_{B \otimes_G B}$. The components of the transformation take the form

$$\pi (a, b) = \langle \pi_1, \pi_2 \rangle: (a \otimes b, a \otimes b) \to (a, b)$$

where, as a consequence of how the 2-product $\otimes$ on $Map(B)$ is defined, we have the following formulas for the projections $\pi_1$, $\pi_2$:

$$\pi_1 = (a \otimes b \overset{1_a \otimes \varepsilon}{\to} a \otimes 1 \cong a); \qquad \pi_2 = (a \otimes b \overset{\varepsilon \otimes 1_b}{\to} 1 \otimes b \cong b$$

Thus, the unit and counit data $\delta$, $\pi$ are expressible in terms of the symmetric monoidal 2-category structure and the comonoid structures on the objects therein.

Finally, for a given 1-cell $r: c \to d$ in $B$, the structure cells

$$\array{
c & \overset{\delta c}{\to} & c \otimes c & & & & c & \overset{\varepsilon c}{\to} & 1\\
r \downarrow & \overset{\delta \cdot r}{\Rightarrow} & \downarrow r \otimes r & & & & r \downarrow & \overset{\varepsilon \cdot r}{\Rightarrow} & \downarrow id_1\\
d & \underset{\delta d}{\to} & d \otimes d & & & & d & \underset{\varepsilon d}{\to} & 1
}$$

exhibit $r: c \to d$ as a colax morphism of 2-comonoids; if $r$ is a map, then $\delta \cdot r$ and $\varepsilon \cdot r$ are isomorphisms and $r$ becomes a strong morphism of 2-comonoids. This may be appreciated more fully by considering specific examples such as $B = Rel$ (where maps are functions, which become comonoid morphisms by virtue of naturality of diagonal and projection maps), and $B = Mod$.

Carboni and Walters define a cartesian bicategory to be a symmetric monoidal bicategory in which each object carries a 2-comonoid structure (for which the comultiplication and counit are _maps_), and each 1-cell is a colax morphism between the corresponding comonoid structures. However, spelling this approach out in full detail leads to a rather largish definition; it seems more efficient to approach the definition by taking advantage of some high-level 2-categorical algebra as we have done here, and deriving the structures envisaged by Carboni and Walters as a consequence.

### Local cartesian products

The definition of cartesian structure _a fortiori_ involves _lax_ adjunctions (in the sense explained earlier)

$$\Delta \dashv_{lax} \otimes: B \times B \to B \qquad ! \dashv_{lax} I: \mathbf{1} \to B$$

and by our earlier discussion of lax adjunctions, this means we have local 1-adjunctions of the form

$$(B(a, b \otimes c) \stackrel{\lambda}{\to} (B \times B)(\langle a, a \rangle, \langle b, c \rangle)) \dashv (B \times B)(\langle a, a \rangle, \langle b, c \rangle)) \stackrel{\rho}{\to} B(a, b \otimes c))$$

Now let $b = c$, and use the fact that we have an adjunction

$$(b \stackrel{\delta b}{\to} b \otimes b) \dashv (b \otimes b \stackrel{\delta b^{\ast}}{\to} b)$$

to obtain an adjoint pair where

$$B(a, b) \stackrel{B(a, \delta b)}{\to} B(a, b \otimes b) \stackrel{\lambda}{\to} B \times B(\langle a, a \rangle, \langle b, b \rangle)$$

is left adjoint to

$$B \times B(\langle a, a \rangle, \langle b, b \rangle) \stackrel{\rho}{\to} B(a, b \otimes b) \stackrel{B(a, \delta b^\ast)}{\to} B(a, b).$$

The first of these arrows is just the diagonal functor

$$B(a, b) \stackrel{\delta_{loc}}{\to} B(a, b) \times B(a, b)$$

on the local hom-category $B(a, b)$. Therefore the second arrow, being right adjoint to the first, provides a product functor on $B(a, b)$. Similarly, each local hom-category carries a terminal object $1 \to \hom(a, b)$. We may summarize this by saying that cartesian bicategories are _locally cartesian_.

### Essential uniqueness of cartesian structure

So far we have seen that for a cartesian bicategory $B$,

* $Map(B)$ carries 2-products,

* Local hom-categories carry products.

It is clear that 2-product data on $Map(B)$ and local product data on local hom-categories are uniquely determined up to appropriate notions of equivalence, since they are determined by appropriate universal properties.
Now we show that these data actually determine the whole of the cartesian structure. We may therefore conclude that a cartesian structure on a bicategory must be essentially unique, if it exists.

To explain how this works, it helps to consider some simple examples of cartesian bicategories such as $Rel$. First, let us reconstruct

$$\otimes: B \times B \to B$$

from the data above. Suppose given two 1-cells $r: a \to b$, $s: c \to d$ in $B$, e.g., relations between sets, or even just subsets $r \subseteq b$, $s \subseteq d$. The tensor product $r \otimes s$ is intuitively a "rectangle", like $r \times s \subseteq b \times d$ in the case of $Rel$, obtained by "pulling back" $r$ along the projection $\pi_1: b \times d \to b$, pulling back $s$ along the projection $\pi_2: b \times d \to d$, and taking the intersection -- more generally, taking a local product. By formalizing this procedure, we are led to the general formula

$$r \otimes s \cong (\pi_{1, b, d}^{\ast} r \pi_{1, a, c}) \times_{loc} (\pi_{2, b, d}^{\ast} s \pi_{2, a, c})$$

where $r$ and $s$ play the role of mere placeholders.

It is a theorem (whose proof we defer) is that in a cartesian bicategory, we may reconstruct the functor $- \otimes -$ by the formula

$$(\pi_{1}^{\ast} (-) \pi_{1}) \times_{loc} (\pi_{2}^{\ast} (-) \pi_{2})$$

where each of the displayed data is manifestly part of the 2-product structure on $Map(B)$ or the local cartesian structure.

The other data we are required to reconstruct are structure cells like $\delta r$, $\pi \langle r, s\rangle$ which pertain to the transformations $\delta$, $\pi$. As an example, consider $\delta r$. This is a 2-cell

$$\delta r: (\delta b)r \to (r \otimes r)(\delta a)$$

which is mated to a 2-cell

$$r \to (\delta b)^\ast (r \otimes r) \delta = r \times_{loc} r.$$

This 2-cell is precisely the local diagonal $r \to r \times r$, manifestly part of the local cartesian structure. Thus the structure cells for $\delta$ are uniquely determined, and similarly for the transformations $\pi$ and $\varepsilon$.

This completes the sketched proof of essential uniqueness.

## Cartesian bicategories and hyperdoctrines

Each cartesian bicategory $\mathbf{B}$ gives rise to a bifibration, or in other language a (weak) 2-functor

$$\mathbf{B}(i-, i-): Map(\mathbf{B})^{op} \times Map(\mathbf{B}) \to Cat$$

or in other words a $Cat$-valued bimodule over $Map(\mathbf{B})$, for which each $\mathbf{B}(i f, i c): \mathbf{B}(i b, i c) \to \mathbf{B}(i a, i c)$ has a left adjoint

$$\mathbf{B}(f^\ast, i c): \mathbf{B}(i a, i c) \to \mathbf{B}(i b, i c)$$

and similarly each $\mathbf{B}(i c, i f): \mathbf{B}(i c, i a) \to \mathbf{B}(i c, i b)$ has a right adjoint $\mathbf{B}(i c, f^\ast)$.

Thus, each cartesian bicategory gives rise to an $Ob(\mathbf{B})$-indexed family $\mathbf{B}(i-, c)$ of what we shall call ([_faute de mieux_]) generalized coherent hyperdoctrines:

+-- {: .un_def}
######Definition
Let $C$ be a 2-category with 2-products. A **generalized coherent hyperdoctrine** over $C$ is a 2-functor
$$P: C^{op} \to FPCat$$
to the 2-category of categories with finite products, such that for each 1-cell $f$ in $C$, the functor $P(f)$ has a left adjoint.
=--

This is called a generalized coherent hyperdoctrine because it deals with a generalized form of coherent logic (involving finite "conjunctions", i.e., local finite cartesian products and existential quantifiers $\exists_f \dashv P(f)$). It is actually a weak form of coherent logic because we have not included finite "disjunctions", and we have said nothing yet about appropriate Beck-Chevalley conditions. (More will be said on this in a [section to follow](#frobenius).) It is generalized in the sense that the base category $C$ is actually a 2-category.

If the symmetric monoidal bicategory $\mathbf{B}$ is _compact_ (e.g., if $\mathbf{B}$ is of type $Rel_C$ for $C$ a regular category, or $Span_C$ for $C$ a finitely complete category, or the bicategory of small categories and profunctors between them), then without loss of generality, we may restrict attention to the single hyperdoctrine

$$\mathbf{B}(i-, 1): Map(\mathbf{B})^{op} \to FPCat$$

## Frobenius conditions
{#frobenius}

A major class of examples of cartesian bicategories, including $Rel$, $Span$, and the bicategory of profunctors between groupoids, have the properties that

* Each object is self-dual (in the sense of compact closed bicategories),

* The bicategory of maps is locally groupoidal.

These arise as follows. In any cartesian bicategory and for each object $b$, there is a canonical 2-cell

$$Frob_b: (\delta b)(\delta b)^\ast \to (1_b \otimes \delta b^\ast)(\delta b \otimes 1_b)$$

mated to the coassociativity isomorphism for $\delta$. We say the **Frobenius condition** holds if $Frob_b$ is an isomorphism for each $b$. This is equivalent to demanding that a similar canonical 2-cell

$$Frob_{b}': (\delta b)(\delta b)^\ast \to (\delta b^\ast \otimes 1_b)(1_b \otimes \delta b)$$

be an isomorphism for each $b$.

## References

* [[Aurelio Carboni]], [[Bob Walters]], _Cartesian Bicategories, I_, Journal of Pure and Applied Algebra, Volume 49, Issues 1–2, November 1987,  (<a href="https://doi.org/10.1016/0022-4049(87)90121-6">doi:10.1016/0022-4049(87)90121-6</a>)

* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, Theory and Applications of Categories, Vol. 19, 2008, No. 6, pp 93-124. ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921), [tac:19-06](http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html))

[[!redirects cartesian bicategories]]
[[!redirects Cartesian bicategory]]
[[!redirects Cartesian bicategories]]


cartesian category

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#### Category theory
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The term _cartesian category_ can mean either of

* [[cartesian monoidal category]]
* [[finitely complete category]]

The first of these is the traditional meaning, while the second is advocated in the [[Elephant]].  It is probably best to avoid the ambiguous term.

Compare [[locally cartesian category]], which is unambiguous.  (There is also some discussion of the issue there.)

[[!redirects cartesian category]]
[[!redirects cartesian categories]]
[[!redirects Cartesian category]]
[[!redirects Cartesian categories]]


cartesian closed (infinity,1)-category

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#### Monoidal categories
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#Contents#
* table of contents
{:toc}

## Definition

An [[(∞,1)-category]] with [[(∞,1)-product]]s $\times$ which is a [[closed monoidal (∞,1)-category]] with respect to $\otimes := \times$ is called a **cartesian closed $(\infty,1)$-category.

## Examples

Every [[(∞,1)-topos]] is a cartesian closed $(\infty,1)$-category. See the section Closed monoidal structure.

## Related concepts

* [[cartesian closed category]], [[locally cartesian closed category]]

* [[cartesian closed functor]], [[locally cartesian closed functor]]

* [[cartesian closed model category]], [[locally cartesian closed model category]]

* **cartesian closed (∞,1)-category**, [[locally cartesian closed (∞,1)-category]]

[[!redirects cartesian closed (∞,1)-category]]

[[!redirects cartesian closed (∞,1)-categories]]
[[!redirects cartesian closed (infinity,1)-categories]]


cartesian closed 2-category

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#### Limits and colimits
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#Contents#
* table of contents
{:toc}

## Idea

The notion of *cartesian closed 2-category* is the analog in [[2-category theory]] (hence the [[categorification]]) of the notion of *[[cartesian closed category]]* in [[category theory]]:

A **cartesian closed 2-category** is a [[2-category]], $\mathcal{B}$, with [[2-limit|finite products]] and a cartesian closed structure: For any $A$, $B$ in $\mathcal{B}$, there is an [[exponential object]] $B^A$, an evaluation [[1-morphism]], $eval_{A, B}: B^A \times A \to B$, and for every $X$ an [[adjoint equivalence]] between $\mathcal{B}(X, B^A)$ and $\mathcal{B}(X \times A, B)$.

The concept was introduced in ([Makkai 96](#Makkai96)). There is no connection to the concept of [[cartesian bicategory]].

## Examples
 {Examples}

The following 2-categories are cartesian closed:

* the  2-categories [[Cat]]($\mathcal{C}$) of [[internal categories]] or [[Grpd]]$(\mathcal{C})$ of [[internal groupoids]] [[internalization|internal to]] a [[cartesian closed category]] $\mathcal{C}$ (e.g. [Niefield & Pronk 2019](#NiefieldPronk19));

* the 2-category of generalized [[species]] ([FGHW](#FGHW));

* the 2-category of cartesian [[distributors]];

* the 2-category of [[operads]].

## Related concepts

* [[cartesian closed category]]

## References

* {#Makkai96} [[Michael Makkai]], _Avoiding the axiom of choice in general category theory_, Journal of Pure and Applied Algebra, 108(2):109 – 173, 1996.

On cartesian closed 2-categories of [[internal categories]] or [[internal groupoids]]:

* {#NiefieldPronk19} [[Susan B. Niefield]], [[Dorette A. Pronk]], *Internal groupoids and exponentiability*, Cahiers de topologie et géométrie différentielle catégoriques, [**LX** 4 (2019)](http://cahierstgdc.com/index.php/volume-lx/) ([pdf](http://cahierstgdc.com/wp-content/uploads/2019/10/Niefield-Pronk-LX-4.pdf))

* [[Enrico Ghiorzi]], Section 1.2 of: *Complete internal categories* ([arXiv:2004.08741](https://arxiv.org/abs/2004.08741))

Discussion of the example of generalised [[species]]:

* {#FGHW} M. Fiore, [[Nicola Gambino]], [[Martin Hyland]], G. Winskel, _The cartesian closed bicategory of generalised species of structures_, ([pdf](http://www1.maths.leeds.ac.uk/~pmtng/Publications/generalised-species.pdf))

Discussion of [[syntax]] for cartesian closed 2-categories in [[type theory]]:

* [[Marcelo Fiore]], Philip Saville, _A type theory for cartesian closed bicategories_, ([arXiv:1904.06538](https://arxiv.org/abs/1904.06538))

* Philip Saville, _Cartesian closed bicategories: type theory and coherence_, ([PhD thesis](http://homepages.inf.ed.ac.uk/psaville/thesis-for-print.pdf), [arXiv:2007.00624](https://arxiv.org/abs/2007.00624))

[[!redirects cartesian closed bicategory]]
[[!redirects cartesian closed bicategories]]
[[!redirects cartesian closed 2-categories]]


cartesian closed category

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#### Category theory
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# Contents
* automatic table of contents goes here
{: toc}

## Definition

A __cartesian closed category__ (sometimes: __ccc__) is a category with [[finite products]] which is [[closed monoidal category|closed]] with respect to its [[cartesian monoidal category|cartesian monoidal structure]].

The [[internal hom]] $[S,X]$ in a cartesian closed category is often called [[exponential object|exponentiation]] and is denoted $X^S$.

A _cartesian closed functor_ between cartesian closed categories $C$, $D$ is a functor $F \colon C \to D$ that is [[product-preserving functor|product-preserving]] and that is also _exponential-preserving_, meaning that the canonical map

$$F(a^b) \to F(a)^{F(b)},$$

corresponding to the composite

$$F(a^b) \times F(b) \cong F(a^b \times b) \stackrel{F(eval_{a, b})}{\to} F(a),$$

is an isomorphism.

## Examples

* Any [[topos]] or [[quasitopos]], such as [[Set]], is cartesian closed.

  See also [[closed monoidal structure on presheaves]].

* [[Cat]] is also cartesian closed. (Cf. e.g. p. 98 of [[Categories Work|Mac Lane]], 2nd ed.,  or the remark below Definition 4.3.9 in [[Category Theory in Context|Riehl]].)

* Many [[nice category of spaces|nice categories of topological spaces]] are also cartesian closed, particularly the [[convenient category of topological spaces|convenient categories of spaces]].

## Some basic consequences

A category is cartesian closed if it has finite products and if for any two objects $X$, $Y$, there is an object $Y^X$ (thought of as a "space of maps from $X$ to $Y$") such that for any object $Z$, there is a bijection between the set of maps $Z \to Y^X$ and the set of maps $Z \times X \to Y$, and this bijection is natural in $Z$.

There is an evaluation map $ev_{X, Y}: Y^X \times X \to Y$, which by definition is the map corresponding to the identity map $Y^X \to Y^X$ under the bijection. Using the naturality, it may be shown that the bijection $\hom(Z, Y^X) \to \hom(Z \times X, Y)$ takes a map $\phi: Z \to Y^X$ to the composite

$$Z \times X \stackrel{\phi \times 1_X}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y$$

and the universal property of $Y^X$ may be phrased thus: given any $\psi: Z \times X \to Y$, there exists a unique map $\phi: Z \to Y^X$ for which $\psi = ev_{X, Y} \circ (\phi \times 1_X)$.

Taking $Z = 1$ (the terminal object), maps $1 \to Y^X$ (or "points" of $Y^X$) are in bijection with maps $X \stackrel{\pi_{2}^{-1}}{\to} 1 \times X \to Y$. So the "underlying set" of $Y^X$, namely $\hom(1, Y^X)$, is identified with the set of maps from $X$ to $Y$. Let us denote the point corresponding to $f: X \to Y$ by $[f]: 1 \to Y^X$. Then, by definition,

$$(1 \times X \stackrel{[f] \times 1_X}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y) = (1 \times X \stackrel{\pi_2}{\to} X \stackrel{f}{\to} Y)$$

The following lemma says that the internal evaluation map $ev_{X, Y}$ indeed behaves as an evaluation map at the level of underlying sets.

+-- {: .num_lemma}
######Lemma
Given a map $f: X \to Y$ and a point $x: 1 \to X$, the composite
$$1 \stackrel{\langle [f], x \rangle}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y$$
is the point $f x: 1 \to Y$.
=--

+-- {: .proof}
######Proof
We have
$$\array{
1 \stackrel{\langle [f], x \rangle}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} X & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{[f] \times x}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y\\
 & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{1 \times x}{\to} 1 \times X \stackrel{[f] \times 1_X}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y\\
 & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{1 \times x}{\to} 1 \times X \stackrel{\pi_2}{\to} X \stackrel{f}{\to} Y\\
 & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{\pi_2}{\to} 1 \stackrel{x}{\to} X \stackrel{f}{\to} Y\\
 & = & 1 \stackrel{x}{\to} X \stackrel{f}{\to} Y
}$$
where the penultimate equation uses naturality of the projection map $\pi_2$.
=--

+-- {: .num_defn}
######Definition
**Internal composition** $c_{X, Y, Z}: Z^Y \times Y^X \to Z^X$ is the unique map such that
$$(Z^Y \times Y^X \times X \stackrel{c \times 1_X}{\to} Z^X \times X \stackrel{ev_{X, Z}}{\to} Z) = (Z^Y \times Y^X \times X \stackrel{1 \times ev_{X, Y}}{\to} Z^Y \times Y \stackrel{ev_{Y, Z}}{\to} Z)$$
=--

One may show that internal composition behaves as the usual composition at the underlying set level, in that given maps $g: Y \to Z$, $f: X \to Y$, we have

$$(1 \stackrel{\langle [g], [f] \rangle}{\to} Z^Y \times Y^X \stackrel{c_{X, Y, Z}}{\to} Z^X) = [g f]: 1 \to Z^X$$

## Properties

* In a cartesian closed category, the product functors $A \times -$ have [[right adjoints]], so they preserve all [[colimits]].  In particular, a cartesian closed category that has finite [[coproducts]] is a [[distributive category]].

* The [[internal logic]] of cartesian closed categories is [[minimal logic]] or the [[typed lambda-calculus]].

### Inheritance by reflective subcategories

In showing that a given category is cartesian closed, the following theorem is often useful (cf. A4.3.1 in the [[Elephant]]):

+-- {: .num_theorem}
###### Theorem
If $C$ is cartesian closed, and $D\subseteq C$ is a [[reflective subcategory]], then the reflector $L\colon C\to D$ preserves finite [[products]] if and only if $D$ is an [[exponential ideal]] (i.e. $Y\in D$ implies $Y^X\in D$ for any $X\in C$).  In particular, if $L$ preserves finite products, then $D$ is cartesian closed.
=--

### Exponentials of cartesian closed categories

The following observation was taken from a post of Mike Shulman at [MathOverflow](http://mathoverflow.net/questions/104152/exponentials-in-functor-categories/104178#104178).

If $\mathcal{C}$ is small and $\mathcal{D}$ is complete and cartesian closed, then $\mathcal{D}^{\mathcal{C}}$ is also complete and cartesian closed. Completeness is clear since limits in $D^C$ are computed pointwise. As for cartesian closure, we can compute exponentials in essentially the same way as for presheaves, motivated by $\mathcal{D}$-enriched category theory:
$$
G^F(x) = \int_{y\in \mathcal{C}} \prod_{\mathcal{C}(x,y)} G(y)^{F(y)}.
$$
Then we can compute

$$\array{
  \mathcal{D}^{\mathcal{C}}\left(H,G^F\right)
  &=& \int_{x\in \mathcal{C}} \mathcal{D}\left(H(x), G^F(x)\right)\\
  &=& \int_{x\in \mathcal{C}} \mathcal{D}\left(H(x), \int_{y\in \mathcal{C}} \prod_{\mathcal{C}(x,y)} G(y)^{F(y)}\right)\\
  &=& \int_{x\in \mathcal{C}} \int_{y\in \mathcal{C}} \prod_{\mathcal{C}(x,y)} \mathcal{D}\left(H(x), G(y)^{F(y)}\right)\\
  &=& \int_{y\in \mathcal{C}} \mathcal{D}\left( \int^{x\in \mathcal{C}} \sum_{\mathcal{C}(x,y)} H(x), G(y)^{F(y)}\right)\\
  &=&  \int_{y\in \mathcal{C}} \mathcal{D}\left(H(y), G(y)^{F(y)}\right)\\
  &=&  \int_{y\in \mathcal{C}} \mathcal{D}\left(H(y)\times F(y), G(y)\right)\\
  &=& \mathcal{D}^{\mathcal{C}}(H\times F, G).
}$$

Here the antepenultimate step uses the [[co-Yoneda lemma]].  This appears to involve colimits in $\mathcal{D}$ as well, but the existence of these colimits is not actually an extra assumption: the co-Yoneda lemma tells us that $\int^{x\in \mathcal{C}} \sum_{\mathcal{C}(x,y)} H(x)$ _exists and is isomorphic to_ $H(y)$.

Similarly, the above argument can be interpreted to say that even if $\mathcal{D}$ is not complete, then the exponential $G^F$ in $\mathcal{D}^{\mathcal{C}}$ exists if and only if the particular limits above exist, and in that case they are isomorphic.

A more abstract argument using [[comonadic functor|comonadicity]] and the [[adjoint triangle theorem]], which also applies to [[locally cartesian closed categories]], can be found in Theorem 2.12 of

* Street and Verity, *The comprehensive factorization and torsors*, [TAC](http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html)

and is reproduced in the setting of closed monoidal categories at [[closed monoidal category]].

### Functional completeness theorem

+-- {: .num_theorem #coKleisliOfComonadIsCartesianClosed}
###### Theorem
Let $C$ be a [[cartesian closed category]], and $c$ an object of $C$. Then the [[Kleisli category]] of the [[comonad]] $c \times - \colon C \to C$, denoted $C[c]$, is also a cartesian closed category.
=--

+-- {: .num_remark}
###### Remark

The [[Kleisli category of a comonad|Kleisli category of the comonad]] $c \times -$ on $C$ is canonically [[equivalence of categories|equivalent]] to the [[Kleisli category]] of the [[monad]] $(-)^c$ on $C$.

=--

+-- {: .proof}
###### Proof

of Theorem \ref{coKleisliOfComonadIsCartesianClosed}

Let $a$ and $b$ be objects of $C[c]$. Claim: the product $a \times b$ in $C$, considered as an object of $C[c]$, is the product of $a$ and $b$ in $C[c]$, according to the following series of natural bijections:

$$\array{
\underline{z \to a \qquad z \to b} & C[c] \\
\underline{c \times z \to a, c \times z \to b} & C \\
\underline{c \times z \to a \times b} & C \\
z \to a \times b & C[c]
}$$

Claim: the exponential $a^b$ in $C$, considered as an object of $C[c]$, is the exponential of $a$ and $b$ in $C[c]$, according to the following series of natural bijections:

$$\array{
\underline{z \to a^b} & C[c] \\
\underline{c \times z \to a^b} & C \\
\underline{c \times z \times b \to a} & C \\
z \times b \to a & C[c]
}$$

where in the last step, we used the prior claim that the product of objects in $C$, when viewed as an object of $C[c]$, gives the product in $C[c]$.
=--

Observe that the object $c$ in $C[c]$ has an element $e \colon 1 \to c$, corresponding to the canonical isomorphism $c \times 1 \cong c$ in $C$. We call this element the "generic element" of $c$ in $C[c]$, according to the following theorem. This theorem can be viewed as saying that in the [[doctrine]] of cartesian closed categories, $C[c]$ is the result of freely adjoining an arrow $1\to c$ to $C$.

+-- {: .num_theorem}
###### Theorem
**(Functional completeness)**

Let $C$ and $D$ be cartesian closed categories, and let $F \colon C \to D$ be a cartesian closed functor. Let $c$ be an object of $C$, and let $t: F(1) \to F(c)$ be an element in $D$. Then there exists an extension of $F$ to a cartesian closed functor $\tilde{F} \colon C[c] \to D$ that takes the generic element $e \colon 1 \to c$ in $C[c]$ to the element $t$, and this extension is unique up to isomorphism.
=--

+-- {: .proof}
###### Proof (sketch)
On objects, $\tilde{F}(a) = F(a)$. Let $f \colon a \to b$ be a map of $C[c]$, i.e., let $g \colon c \times a \to b$ be the corresponding map in $C$, and define $\tilde{F}(f)$ to be the composite

$$F(a) \cong 1 \times F(a) \stackrel{t \times 1}{\to} F(c) \times F(a) \cong F(c \times a) \stackrel{F(g)}{\to} F(b)$$

It is straightforward to check that $\tilde{F}$ is a cartesian closed functor and is, up to isomorphism, the unique cartesian closed functor taking $e$ to $t$.
=--

## Related concepts

* [[exponential object]], [[mapping space]], [[internal hom]]

* [[convenient category of topological spaces]]

\linebreak

* **cartesian closed category**, [[locally cartesian closed category]]

  * [[cartesian doubly closed monoidal category]]

* [[cartesian closed functor]], [[locally cartesian closed functor]]

* [[cartesian closed model category]], [[locally cartesian closed model category]]

* [[cartesian closed 2-category]]

* [[cartesian closed (∞,1)-category]], [[locally cartesian closed (∞,1)-category]]

* [[cartesian closed enriched category]], [[locally cartesian closed enriched category]]

## References

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §IV.2 of: *Closed Categories*, in: *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

Textbook accounts:

* [[Saunders MacLane]], §IV.6 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Def. 6.1.5 in: *Categories and Structures*, Vol. 2 of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

Discussion with focus on [[mapping spaces]] ([[compact-open topology]]) in [[topological spaces]] and [[compactly generated topological spaces]]:

* [[Susan Niefield]], *Cartesianness*, PhD thesis, Rutgers 1978 ([proquest:302920643](https://www.proquest.com/docview/302920643))

* [[Susan Niefield]], _Cartesianness: topological spaces, uniform spaces and affine schemes_, J. Pure Appl. Algebra __23__ (1982) 147-167 (doi:10.1016/0022-4049(82)90004-4)

See also the references at *[[exponential object]]*.

Establishing the [[syntax-semantics duality|syntax/semantics]] [[relation between category theory and type theory|relation between]] cartesian closed categories and [[simply-typed lambda calculi]]:

* {#LambekScott86} [[Joachim Lambek]], [[Philip J. Scott]], Part I of: *Introduction to higher order categorical logic*, Cambridge Studies in Advanced Mathematics **7** (1986) [[ISBN: 0-521-24665-2](https://www.cambridge.org/ae/academic/subjects/mathematics/logic-categories-and-sets/introduction-higher-order-categorical-logic?format=PB&isbn=9780521356534), [pdf](https://raw.githubusercontent.com/Mzk-Levi/texts/master/Lambek%20J.%2C%20Scott%20P.J.%20Introduction%20to%20Higher%20Order%20Categorical%20Logic.pdf)]

A proof that a cartesian closed category is [[poset|posetal]] if it satisfies an equality of morphisms not satisfied in a free cartesian closed category:

* Kosta Dosen, and Zoran Petric. _The maximality of the typed lambda calculus and of cartesian closed categories_ (1999), ([arXiv:math/9911073](https://arxiv.org/abs/math/9911073))

[[!redirects cartesian closed]]
[[!redirects cartesian closed category]]
[[!redirects cartesian closed categories]]
[[!redirects closed cartesian category]]
[[!redirects closed cartesian categories]]
[[!redirects Cartesian closed category]]
[[!redirects Cartesian closed categories]]
[[!redirects cartesian-closed category]]
[[!redirects cartesian-closed categories]]
[[!redirects Cartesian-closed category]]
[[!redirects Cartesian-closed categories]]

[[!redirects cartesian closed monoidal category]]
[[!redirects cartesian closed monoidal categories]]


cartesian closed enriched category

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# Cartesian closed enriched categories

* table of contents
{: toc}

## Definition

Let $V$ be a [[monoidal category]] with [[products]] and $C$ a $V$-[[enriched category]] with [[products]], in the enriched sense that we have a [[enriched natural transformation|$V$-natural]] [[natural isomorphism|isomorphisms]] of [[hom-objects]] in $V$:

$$
  C(Z, X\times Y)
  \;\cong\;
  C(Z,X) \times C(Z,Y)
  \,.
$$

We say $C$ is **$V$-cartesian-closed** if each $V$-functor $(X\times -) \colon C \to C$ has a [[enriched adjunction|$V$-enriched]] [[right adjoint]].

## Relation to ordinary cartesian closedness

If $C$ is $V$-cartesian-closed, then its [[underlying ordinary category]] $C_0$ is [[cartesian closed category|cartesian closed]] in the usual sense, since $V$-enriched right adjoints have underlying ordinary right adjoints.

The converse is true in some cases, such as the following:

* When $V=Set$, trivially.

* More generally, whenever the underlying-set functor $V(I,-) : V\to Set$ is [[conservative functor|conservative]], since the morphism of hom-objects $C(X,Y^Z) \to C(X\times Y,Z)$ induced by the evaluation morphism $Y^Z\times Y \to Z$ has invertible image in $Set$, hence is itself invertible if $V(I,-)$ is conservative.

* When $V$ is cartesian monoidal and $C=V$: a cartesian closed category is automatically enriched-cartesian-closed over itself.  In other words, the defining isomorphisms $V_0(X\times Y,Z) \cong V_0(X, Z^Y)$ induce, by the [[Yoneda lemma]], isomorphisms of [[exponential objects]] $Z^{X\times Y} \cong (Z^Y)^X$.

However, the converse is false in general.  [[counterexample|Counterexamples]] can be found in [this mathoverflow discussion](#MO).

## Related concepts

* [[cartesian closed category]]

* [[locally cartesian closed enriched category]]

* [[monoidal enriched category]]

* [[cartesian closed model category]]

## References

* {#MO} [Enriched cartesian closed categories on MO](https://mathoverflow.net/q/322085)

* {#MO2} [Simplicial cartesian closed categories on MO](https://mathoverflow.net/q/322917)

[[!redirects cartesian closed enriched category]]
[[!redirects cartesian closed enriched categories]]
[[!redirects enriched cartesian closed category]]
[[!redirects enriched cartesian closed categories]]
[[!redirects simplicially cartesian closed category]]
[[!redirects simplicially cartesian closed categories]]


cartesian closed functor

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#### Monoidal categories
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#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn}
###### Definition

A **cartesian closed functor** is a [[functor]] $F\colon \mathcal{C}\to \mathcal{D}$ between [[cartesian closed categories]] which preserves both [[products]] and [[exponential objects]]/[[internal homs]] (all the structure of cartesian closed categories).

More precisely, if $F\colon \mathcal{C}\to \mathcal{D}$ preserves products, then the canonical [[morphisms]] $F(A\times B) \to F A \times F B$ (for all [[objects]] $A,B \in \mathcal{C}$) are [[isomorphisms]], and we therefore have canonical induced morphisms $F[A,B] \to [F A, F B]$ --- the [[adjuncts]] of the composites $F[A,B] \times F A \xrightarrow{\cong} F([A,B] \times A) \to F B$.  $F$ is **cartesian closed** if these maps $F[A,B] \to [F A, F B]$ are also isomorphisms.

=--

+-- {: .num_remark}
###### Remark

When cartesian closed categories are identified with [[cartesian monoidal categories]] that are also [[closed monoidal category|closed monoidal]], a cartesian closed functor can be identified with a [[strong monoidal functor]] which is also [[strong closed functor|strong closed]].

=--

## Properties

+-- {: .num_prop #FrobeniusReciprocity}
###### Proposition
**(Frobenius reciprocity)**

Let $R : \mathcal{C} \to \mathcal{D}$ be a [[functor]] between [[cartesian closed categories]] with a [[left adjoint]] $L$. Then $R$ is cartesian closed precisely if the [[natural transformation]]

$$
  (L \pi_1, \epsilon_A L \pi_2)
  :
  L(B \times R(A))
  \to
  L(B) \times A
$$

is an [[isomorphism]].

=--

+-- {: .proof}
###### Proof

The above natural transformation is the [[mate]] of the exponential comparison natural transformation $R[A,B] \to [R A, R B]$ under the composite adjunctions
$$
\mathcal{C}
\underoverset{R}{L}{\leftrightarrows}
\mathcal{D}
\underoverset{[R A, -]}{- \times R A}{\leftrightarrows}
\mathcal{D}
$$
and
$$
\mathcal{C}
\underoverset{[A,-]}{A\times -}{\leftrightarrows}
\mathcal{C}
\underoverset{R}{L}{\leftrightarrows}
\mathcal{D}
$$

=--

This is called the **[[Frobenius reciprocity]]** law.  It is discussed, for instance, as ([Johnstone, lemma 1.5.8](#Johnstone)). More generally for closed monoidal categories (not necessarily cartesian monoidal) it is discussed in "[[Wirthmüller contexts]]" in

Let still $R$ and $L$ be as above.

+-- {: .num_cor}
###### Corollary

If $R$ is [[full and faithful]] and $L$ preserves binary [[products]], then $R$ is cartesian closed.

=--

For instance ([Johnstone, corollary A1.5.9](#Johnstone)).

## Examples
 {#Examples}

+-- {: .num_prop}
###### Proposition

For $\mathcal{C}$ a [[locally cartesian closed category]] and $f : X_1 \to X_2$ a [[morphism]], the [[base change]]/[[pullback]] functor between the [[slice categories]]

$$
  f^* : \mathcal{C}_{/X_2} \to \mathcal{C}_{/X_1}
$$

is cartesian closed.

In particular the [[inverse image]] functor of an [[étale geometric morphism]] between [[toposes]] is cartesian closed and hence constitutes a cartesian [[Wirthmüller context]].

=--

+-- {: .proof}
###### Proof

The functor $f^*$ has a [[left adjoint]]

$$
  \sum_f : \mathcal{C}_{/X_1} \to \mathcal{C}_{/X_2}
$$

given by postcomposition with $f$ (the [[dependent sum]] along $f$). Therefore by prop. \ref{FrobeniusReciprocity} it is sufficient to show that
for all $(A \to X_2) \in \mathcal{C}_{/X_2}$ and $(B \stackrel{b}{\to} X_1) \in \mathcal{C}_{/X_1}$ that

$$
  B \times_{X_1} f^* A \simeq B \times_{X_2} A
$$

in $\mathcal{C}$. But this is the [[pasting law]] for pullbacks in $\mathcal{C}$, which says that the two consecutive pullbacks on the left of

$$
  \array{
    B \times_{X_1} f^* A &\to& f^* A &\to& A
    \\
    \downarrow && \downarrow && \downarrow
    \\
    B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2
  }
  \;\;\;
  \simeq
  \;\;\;
  \array{
    (b \circ f)^* A &\to&  &\to& A
    \\
    \downarrow &&  && \downarrow
    \\
    B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2
  }
$$

are isomorphic to the direct pullback along the composite on the right.

=--

## Related concepts

* [[strong functor]]

* [[cartesian closed category]], [[locally cartesian closed category]]

* **cartesian closed functor**, [[locally cartesian closed functor]]

* [[cartesian closed model category]], [[locally cartesian closed model category]]

* [[cartesian closed (∞,1)-category]], [[locally cartesian closed (∞,1)-category]]

## References

For instance section A1.5 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

Also

* H. Fausk, P. Hu, [[Peter May]],  _Isomorphisms between left and right adjoints_, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. ([TAC](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf))
 {#May05}

[[!redirects cartesian closed functors]]


cartesian differential category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

One source of cartesian [[differential category|differential categories]]
is [[tangent bundle categories]]:
in such a category, the [[full subcategory]] of [[objects]]
with a trivial (in the appropriate sense)
tangent bundle is a cartesian differential category.

## Definition

We present the revised definition of a cartesian differential category
due to [[Geoff Cruttwell]], [see below](#Cruttwell).

The adjective “cartesian” refers to the existence of finite products.

A prototypical example of a carteisan differential category
is the category whose objects are open subsets of finite-dimensional real vector (or affine) spaces
and morphisms are smooth maps.
Below, if $U\subset R^n$ is such an object, then $L_0(U)=R^n$ is the tangent space at any point of $U$.

\begin{definition}
A __(generalized) cartesian differential category__
is a category $C$ with finite products
equipped with the following data:

* for any object $X\in C$ a commutative monoid
$$L(X)=(L_0(X),+_X\colon L_0(X)\times L_0(X)\to L_0(X),0_X\colon 1\to L_0(X))$$
(the tangent space at any point)
such that $L(L_0(X))=L(X)$ and $L(X\times Y)=L(X)\times L(Y)$;

* for any morphism $f\colon X\to Y$ in $C$, a morphism $D(f):L_0(X)\times X \to L_0(Y)$ (the derivative of $f$) such that

  * $D(+_X)=+_X\circ p_1$, $D(0_X)=0_X\circ p_1$ (where $p_1$ projects onto the first factor of a product);

  * (the derivative is a linear map)
$$D(f)\circ (a +_X b,c)=D(f)\circ (a,c) +_X D(f)\circ (b,c)$$
and
$$D(f)(0_X,a)=0_X$$
(where $a,b,c\colon A\to L_0(X)$ are morphisms in $C$);

  * (to be finished…)

\end{definition}

## References

*  [[Rick Blute]], [[Robin Cockett]], and [[R.A.G. Seely]],  _Cartesian differential categories,_ Theory and Applications of Categories,22, pg. 622–672, 2008 ([tac:22-23](http://www.tac.mta.ca/tac/volumes/22/23/22-23abs.html))

* {#Cruttwell}[[Geoff Cruttwell]], _Cartesian differential categories revisited,_ Mathematical Structures in Computer Science, Vol. 27 (1), pg. 70-91 (first published online in 2015), [doi:10.1007/s10485-019-09572-y](https://doi.org/10.1007/s10485-019-09572-y)

## Related concepts

* [[differential category]]

[[!redirects cartesian differential categories]]


Cartesian fibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

Cartesian fibrations are one of the types of [[fibrations of quasi-categories]].

A _Cartesian fibration_ of quasi-categories -- or more generally of [[simplicial set]]s -- is a morphism that generalizes the notion of [[Grothendieck fibration]] from [[category theory]] to [[(∞,1)-category theory]], specifically with [[(∞,1)-categories]] incarnated as [[quasi-categories]]:

It is an [[inner fibration]] that lifts also all right _outer_ [[horn]] inclusions whose last edge is a [[cartesian morphism]], and such that there is a sufficient supply of [[cartesian morphisms]].

An [[inner fibration]] $p : C \to D$ may be thought of as a family of [[(infinity,1)-categories]] $(C_d)_{d \in D}$ which is [[functorial]] in $d$ only in the sense of [[correspondences]].  Then the condition of $p$ being a cartesian fibration ensures that the family is actually [[functorial]].
More precisely, if an [[(∞,1)-functor]] $p : C \to D$ is a Cartesian fibration, then it is possible to interpret its value over any [[morphism]] $f : d_1 \to d_2$ in $D$ as an [[(∞,1)-functor]] $p^{-1}(f) : p^{-1}(d_2) \to p^{-1}(d_1)$ between the [[fibers]] $p^{-1}(d_2)$ and $p^{-1}(d_1)$ over its source and target [[objects]].

This way a Cartesian fibration $p : C \to D$ determines and is determined by an [[(∞,1)-functor]] $D^{op} \to (\infty,1)Cat$ into the
[[(∞,1)-category of (∞,1)-categories]]. This is the content of the [[(∞,1)-Grothendieck construction]].

Cartesian fibrations over $S$ are the fibrant objects in the [[model structure on marked simplicial over-sets]] over $S$.

## Definition

+-- {: .un_defn}
###### Definition

A morphism $p : X \to Y$ in [[sSet]] is a
**Cartesian fibration** (or **Grothendieck fibration**) if

* it is an [[inner fibration]]

* for every edge $f : x \to y$ of $Y$ and for every lift $\hat {y}$ of
  $y$ (that is: $p(\hat{y})=y$)
  there is a [[cartesian morphism|Cartesian edge]]
  $\hat{f} : \hat{x} \to \hat{y}$ in $X$ lifting $f$
  (that is: such that $p(\hat f) = f$)

The morphism is a **cocartesian fibration** (or **Cartesian opfibration**, **Grothendieck opfibration**) if the [[opposite quasi-category|opposite]] $p^{op} : X^{op} \to Y^{op}$ is a Cartesian fibration.

=--

This is [[Higher Topos Theory|HTT, def. 2.4.2.1]].

+-- {: .un_defn}
###### Definition

Given cartesian fibrations $p : X \to C$ and $q : Y \to C$, let $Map_C^{cart}(p,q) \subseteq Map_C(p,q)$ and $Fun_C^{cart}(p,q) \subseteq Fun_C(p,q)$ be the full subsimplicial sets spanned by the functors that preserve cartesian morphisms. We call such functors **cartesian functors**.

Dually, we make the analogous definition of cocartesian functor.

=--

+-- {: .un_remark}
###### Remark

The $Map_C^{cart}$ are the mapping spaces in the (cartesian) model structure on marked simplicial sets over $C$.

=--

## Properties {#Properties}

### General properties

+-- {: .un_prop}
###### Proposition

We have:

* Every [[isomorphism]] of [[simplicial sets]] is a Cartesian fibration.

* The composite of two Cartesian fibrations is again a Cartesian fibration.

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop. 2.4.2.3 ]].

=--

+-- {: .un_prop}
###### Proposition

Every Cartesian fibration is a fibration in the [[Andre Joyal|Joyal]]-[[model structure for quasi-categories]].

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop. 3.3.1.7]].

=--

### Behaviour under pullback {#BehaviourUnderPullback}

Since a Cartesian fibration is in particular an [[inner fibration]] and since inner fibrations are stable under [[pullback]] in [[sSet]], it follows that for $p : E \to C$ a Cartesian fibration, the fiber $E_x$ over every point $x \in C_0$ is a [[quasi-category]]

$$
  \array{
    E_x &\to& E
    \\
    \downarrow && \downarrow
    \\
    \{x\} &\to& C
  }
  \,.
$$

The difference between inner fibrations and Cartesian fibrations is that only for Cartesian fibrations is it generally guaranteed that these fibers over the points are _functorially_ related over the morphisms in $C$. This is the content of the [[(∞,1)-Grothendieck construction]].

But moreover:

+-- {: .un_prop}
###### Proposition

The pullback of a Cartesian fibration in [[sSet]] is again a Cartesian fibration.

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop. 2.4.2.3]].

We know from the discussion at [[inner fibration]] that the pullback is an inner fibration. It remains to check if it has enough [[Cartesian morphism]]s. By [[Higher Topos Theory|HTT, prop 2.4.1.3]] we have that in a [[pullback]] diagram

$$
  \array{
    E' &\stackrel{q}{\to}& E
    \\
    {}^{\mathllap{p'}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    C' &\stackrel{k}{\to}& C
  }
$$

a morphism $f \in E'$ is $p'$-Cartesian if $q(f)$ is $p$-Cartesian. Since the morphisms of $E'$ are pairs of morphisms $(\gamma, \hat f) \in C'_1 \times E_1$  and since by assumption $p$ is a Cartesian fibration, there is for $\gamma \in C'_1$ and $y \in E'_0$ such that $p'(y)$ is the target of the morphism $\gamma$ a Cartesian lift $\hat f \in E$ of $k(\gamma)$ such that $q(y)$ is the target of $\hat f$. Hence a Cartesian lift $(\gamma, \hat f)$ of $\gamma$ in $E'$ having $y$ as target.

=--

We can test locally if a morphism is a Cartesian fibration:

+-- {: .un_prop}
###### Proposition

An [[inner fibration]] $p : X \to Y$ is Cartesian precisely if for each $n \leq 2$ and for every $n$-[[simplex]] $\sigma : \Delta[n] \to Y$ the [[sSet]]-[[pullback]] $\sigma^* p : X \times_Y \Delta[n] \to \Delta[n]$ in

$$
  \array{
    X \times_Y \Delta[n] &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \Delta[n] &\stackrel{\sigma}{\to}& Y
  }
$$

is a Cartesian fibration.

=--

+-- {: .proof}
###### Proof

This is  [[Higher Topos Theory|HTT, cor. 2.4.2.10]].

=--

+-- {: .un_prop}
###### Proposition

The [[pullback]] in [[sSet]] of a weak equivalence in the [[Andre Joyal|Joyal]]-[[model structure for quasi-categories]] along a Cartesian fibration is again a Joyal-weak equivalence

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop 3.3.1.3]]

=--

+-- {: .un_lemma}
###### Lemma

Equivalences in $sSet_{Joyal}$ are stable under pullback along Cartesian fibrations:

if

$$
  \array{
    X \times_S T &\to & X
    \\
    \downarrow && \downarrow
    \\
    T &\stackrel{\simeq}{\to}& S
  }
$$

is a [[pullback]] square in [[sSet]] with $T \to S$ a weak equivalence in $sSet_{Joyal}$ and $X \to S$ a Cartesian fibration, then $X \times_S T \to X$ is also a weak equivalence.

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop. 3.3.1.3]].

=--

The following proposition asserts that the ordinary pullback (in [[sSet]]) of Cartesian fibrations already models the correct [[homotopy pullback]].

+-- {: .un_prop}
###### Proposition

Let

$$
  \array{
    X &\to& X'
    \\
    \downarrow && \downarrow
    \\
    S &\to & S'
  }
$$

be an [[pullback]] [[diagram]] in [[sSet]] of quasi-categories, where $X' \to S'$ is a Cartesian fibration. Then this is already a [[homotopy pullback]] diagram with respect to the [[model structure for quasi-categories]].

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop 3.3.1.4]]. We factor the bottom morphism as $ S \stackrel{\simeq}{\to} T \to S' $ into a weak equivalence and a fibration in $sSet_{Joyal}$. Then the right square in

$$
  \array{
    X &\to& T \times_{S'} X'& \to& X'
    \\
    \downarrow && \downarrow && \downarrow
    \\
    S &\to & T &\to& S'
  }
$$

is the ordinary pullback over a fibrant replacement of the original diagram hence is a homotopy pullback. The claim follows thus if $X \to T \times_{S'} X'$ is a weak equivalence, which it is by one of the above lemmas.

=--

### Straightening and unstraightening

There is a [[Quillen equivalence]]
between the [[model category]] of cartesian fibrations
and the [[model category]] of [[presheaves]] valued in [[quasicategories]].
See the article [[straightening functor]] for more information.

### Relation to other kinds of fibrations

The notion of [[right fibration]] is a special case of that
of Cartesian fibration:

+-- {: .un_prop}
###### Proposition

The following are equivalent:

* $p : X \to Y$ is a Cartesian fibration and every edge in $X$ is
  $p$-Cartesian

* $p$ is a [[right fibration]];

* $p$ is a Cartesian fibration and every fiber is a [[Kan complex]].

=--

+-- {: .proof}
###### Proof

This is  [[Higher Topos Theory|HTT, prop. 2.4.2.4]].

=--

There are also the "categorical fibrations", the fibrations in the Joyal [[model structure for quasi-categories]] on $sSet$. These turn out not to have much of an intrinsic category theoretic meaning. By the following proposition one can understand the notion of Cartesian fibration as  a suitable refinement of the notion of categorical fibration

+-- {: .un_prop}
###### Proposition

Every Cartesian fibration $p : X \to Y$ in [[sSet]] is a fibration with respect to the [[Andre Joyal|Joyal]] [[model structure for quasi-categories]] on [[sSet]].

=--

+-- {: .proof}
###### Proof

This is  [[Higher Topos Theory|HTT, prop. 3.3.1.7]].

=--

However, when the base is an $\infty$-groupoid, then the difference between Cartesian fibrations and categorical fibrations disappears:

+-- {: .un_prop}
###### Proposition (over $\infty$-groupoids)

Let $p : X \to Y$ be a morphism of [[simplicial set]]s where $Y$ is a [[Kan complex]]. Then the following are equivalent:

1. $p$ is a Cartesian fibration

1. $p$ is a coCartesian fibration

1. $p$ is a [[Joyal model structure on simplicial sets|categorical fibration]]

=--

+-- {: .proof}
###### Proof

This is  [[Higher Topos Theory|HTT, prop. 3.3.1.8]].

=--

There is however another [[model category]] structure, which does model Cartesian fibrations.

+-- {: .un_prop}
###### Proposition

Let $sSet^+/S$ be the [[overcategory]] of the category of [[marked simplicial set]]s over $S$, equipped with the [[model structure on marked simplicial over-sets]].

An object $X \to S$ is fibrant in that model category precisely if

* the underlying morphism of simplicial sets $X \to S$ is a Cartesian fibration;

* the marked edges in $X$ are precisely the [[Cartesian morphism]]s.

=--

+-- {: .proof}
###### Proof

This is  [[Higher Topos Theory|HTT, prop. 3.1.4.1]].

=--

### Adjoints to the inclusion into (∞,1)Cat/C

The inclusion of the cartesian fibrations and cartesian functors in the category $(\infty,1)Cat_{/C}^{cart} \subseteq (\infty,1)Cat_{/C}$ has both a left and a right adjoint.

The left adjoint is given by the construction of "free fibrations"

+-- {: .un_prop}
###### Proposition

For maps $p : X \to C$ and cartesian fibrations $q : Y \to C$, there are a natural equivalences
$$
  Fun_C(p, q) \simeq Fun_C^{cart}((C \downarrow p), q)
$$
Dually, for maps $p : X \to C$ and cocartesian fibrations $q : Y \to C$, there are natural equivalences
$$
  Fun_C(p, q) \simeq Fun_C^{cocart}((p \downarrow C), q)
$$

=--

+-- {: .proof}
###### Proof

The cartesian case for mapping spaces is theorem 4.11 of [Gepner-Haugseng-Nikolaus](#GepnerHaugsengNikolaus15). For the cocartesian case,

$$
  Fun_C(p,q) \simeq Fun_{C^{op}}(p^{op}, q^{op})^{op}
  \simeq \Fun_{C^{op}}^{cart}((C^{op} \downarrow p^{op}), q^{op})^{op}
  \simeq \Fun_C^{cocart}((p \downarrow C), q)
$$

=--

On the other side,

+-- {: .un_prop}
###### Proposition

The inclusions $(\infty,1)Cat_{/C}^{cart} \subseteq (\infty,1)Cat_{/C}$
and  $(\infty,1)Cat_{/C}^{cocart} \subseteq (\infty,1)Cat_{/C}$
have right adjoints.

=--

+-- {: .proof}
###### Proof

We handle the cartesian case. The functor tensor product $- \otimes_C C_{/\bullet}$
is the left adjoint part of an adjunction between $Fun(C^{op}, (\infty,1)Cat)$ and $(\infty,1)Cat$. Since it sends $1 \mapsto C$, we also get an adjunction
between $Fun(C^{op}, (\infty,1)Cat)$ and $(\infty,1)Cat_{/C}$

By the description of the [[(∞,1)-Grothendieck construction]] as a [[lax (∞,1)-colimit]], the left adjoint part is an equivalence $Fun(C^{op}, (\infty,1)Cat) \to (\infty,1)Cat_{/C}^{cart}$

=--

## Examples

### Projections from comma categories

+-- {: .un_prop}
###### Proposition

Given functors $F : A \to C$ and $G : B \to C$ between quasi-categories,
the projection $(F \downarrow G) \to A$ is a Cartesian fibration
and the projection $(F \downarrow G) \to B$ is a coCartesian fibration.

In particular, if $C$ is a quasi-category, then $eval_0 : C^{\Delta^1} \to C$
is a Cartesian fibration and $eval_1 : C^{\Delta^1} \to C$ is a coCartesian fibration.

=--

+-- {: .proof}
###### Proof

Consider the diagram of pullback squares

$$
  \array{
    (F \downarrow G) &\to& (id_C \downarrow G) &\to& B
    \\
    \downarrow && \downarrow && \downarrow G\!\!
    \\
    (F \downarrow id_C) &\to& C^{\Delta^1} &\xrightarrow{1}& C
    \\
    \downarrow && \downarrow 0\!\!
    \\
    A &\xrightarrow{F}& C
  }
$$

[[Higher Topos Theory|HTT, prop. 2.4.7.12]] states the projection $(id_C \downarrow G) \to C$ is a Cartesian fibration.  Then $(F \downarrow G) \to A$ is as well, since fibrations are preserved by pullback. Dually for the other projection.

=--

The fiber of $eval_1 : C^{\Delta^1} \to C$ at an object $X \in C$ is the slice category $C_{/X}$. So, by the [[(infinity,1)-Grothendieck construction]],
this fibration corresponds to the slice functor
$C \to (\infty,1)Cat : X \mapsto C_{/X}$, where a morphism $X \to Y$
induces the composition map $C_{/X} \to C_{/Y}$.

### Cartesian fibrations over one morphism and Adjunctions

By the [[(infinity,1)-Grothendieck construction]] a Cartesian fibration $K \to \Delta[1]$ corresponds to a morphism $\Delta[1]^{op} \to (\infty,1)Cat$, hence to an [[(infinity,1)-functor]] $f : C \to D$.

Obtaining this through the straightening functor above is tedious, but there is a more immediate way to characterize $f$:

+-- {: .un_def}
###### Definition

For $p : K \to \Delta[1]$ a Cartesian fibration with specified equivalences

$$
  h_0  : C \stackrel{\simeq}{\to} p^{-1}(0)
$$

and

$$
  h_1 : D \stackrel{\simeq}{\to} p^{-1}(1)
$$

an [[(infinity,1)-functor]] $f : D \to C$ is **associated** to $p$ if there exists a commuting diagram

$$
  \array{
    D \times \Delta[1] &&\stackrel{s}{\to}&& K
    \\
    & \searrow && \swarrow
    \\
    && \Delta[1]
  }
$$

such that

* $s|_{D \times \{1\}} = h_1$;

* $s|_{D \times \{0\}} = h_0 \circ f$

* for every object $x$ of $D$ we have that $s|_{\{x\} \times \Delta[1]}$ is
  a p-[[Cartesian morphism]] of $K$.

=--

This is [[Higher Topos Theory|HTT, def. 5.2.1.1]].

If $p : K \to \Delta[1]$ is both a Cartesian fibration as well as a coCartesian fibration, then it determines $(\infty,1)$-functors in both directions

$$
  C
    \stackrel{\leftarrow}{\to}
  D
  \,.
$$

Such a pair is a pair of [[adjoint (infinity,1)-functor]]s.

### Cartesian fibrations over simplices {#CartOverSimplex}

... for the moment see [[Higher Topos Theory|HTT, section 3.2.2]] ...

### The universal Cartesian fibration

for the moment see

* [[universal fibration of (∞,1)-categories]].

## Related concepts

* [[Kan fibration]], [[anodyne morphism]]

* [[right/left Kan fibration]], [[right/left anodyne map]]

* [[inner fibration]]

* **Cartesian fibration**, [[Cartesian fibration of dendroidal sets]]

* [[coCartesian fibration of (∞,1)-operads]]

* [[straightening functor]]

## References

* [[Jacob Lurie]], section 2.4.2 in _[[Higher Topos Theory]]_

* {#MazelGee15} [[Aaron Mazel-Gee]], _A user's guide to co/cartesian fibrations_ ([arXiv:1510.02402](http://arxiv.org/abs/1510.02402))

* {#GepnerHaugsengNikolaus15} [[David Gepner]], [[Rune Haugseng]], [[Thomas Nikolaus]], _Lax colimits and free fibrations in $\infty$-categories_ ([arXiv:1501.02161](http://arxiv.org/abs/1501.02161))

Discussion [[category internal to an (infinity,1)-topos|internal to]] any [[(∞,1)-topos]]:

* [[Louis Martini]], *Cocartesian fibrations and straightening internal to an ∞-topos* [[arXiv:2204.00295](https://arxiv.org/abs/2204.00295)]

[[!redirects cartesian fibration]]
[[!redirects coCartesian fibration]]
[[!redirects co-cartesian fibration]]
[[!redirects cocartesian fibration]]
[[!redirects Cartesian fibrations]]
[[!redirects cartesian fibrations]]
[[!redirects coCartesian fibrations]]
[[!redirects co-cartesian fibrations]]
[[!redirects cocartesian fibrations]]


Cartesian fibration of dendroidal sets

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Cartesian fibration of dendroidal sets_ is the generalization from [[simplicial sets]] to [[dendroidal sets]] of the notion of [[Cartesian fibration]]. Accordingly, it is models the notion of [[Grothendieck fibration]] for [[(∞,1)-operads]].
Its [[operad|1-operadic]] analog is the notion of _[[fibration of multicategories]]_.

## Definition

(...)

## Properties

### Relation to $\infty$-algebra

Let $P$ be an [[(∞,1)-operad]], incarnated as a [[dendroidal set]]. For instance the [[homotopy coherent dendroidal nerve]] of a [[topological operad]]/[[simplicial operad]].

Then coCartesian fibrations over $P$ are equivalent to [[∞-algebra over an (∞,1)-operad|∞-algebras]] over $P$ in [[(∞,1)Cat]]:

$$
  coCart_P \simeq Alg_P(Cat_{(\infty,1)})
  \,.
$$

This is ([Heuts, theorem 0.1](#HeutsA)).

## Related concepts

* [[model structure for dendroidal Cartesian fibrations]]

## References

* [[Gijs Heuts]], _Algebras over infinity-operads_ ([arXiv:1110.1776](http://arxiv.org/abs/1110.1776))
  {#HeutsA}

* [[Gijs Heuts]], _An infinite loop space machine for infinity-operads_ ([arXiv:1112.0625](http://arxiv.org/abs/1112.0625))

[[!redirects Cartesian fibrations of dendroidal sets]]

[[!redirects dendroidal Cartesian fibration]]
[[!redirects dendroidal Cartesian fibrations]]


cartesian functor

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## Definition

The phrase **cartesian functor** has been used in category theory with several different (but related) meanings.

* A functor between [[fibered categories]] has been called __cartesian__ if it sends [[cartesian morphisms]] into cartesian morphisms.

* A functor between categories with finite [[products]] has been called __cartesian__ if it [[preserved limit|preserves]] finite products (that is, it is a [[strong monoidal functor]] between [[cartesian monoidal categories]]).

* A functor between [[finitely complete categories]] has been called __cartesian__ (notably in [[Sketches of an Elephant]]) if it preserves finite limits.  This is the case just when the induced functor on [[codomain fibrations]] is cartesian in the first sense.

Because of this ambiguity, it is perhaps always better to use a more precise term such as "(strong) morphism of fibrations", "cartesian monoidal functor" or "finite product preserving functor", and "finitely continuous functor" or "lex functor".

## Related concepts

* [[monoidal functor]]

* [[strong monoidal functor]]

* **cartesian functor**

* [[lax monoidal functor]]

* [[oplax monoidal functor]]

* [[bilax monoidal functor]]

* [[Frobenius monoidal functor]]

* [[braided monoidal functor]]

* [[symmetric monoidal functor]]

* [[morphism of fibrations]]

[[!redirects cartesian functors]]
[[!redirects Cartesian functor]]
[[!redirects Cartesian functors]]
[[!redirects cartesian monoidal functor]]
[[!redirects cartesian monoidal functors]]


cartesian logic

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## Idea

_Cartesian logic_ or _finite limit logic_ is the [[internal logic]] of [[finitely complete categories]] (which the [[Elephant]] calls [[cartesian categories]]).

An important property is that every cartesian theory has an initial model. It follows that model reduct functors between cartesian theories have left adjoints: broadly, the free algebra constructions supplied by universal algebra for algebraic theories are also available for cartesian theories.

## Definition

Various definitions and names for the logic can be found in the references.
The [[Elephant]] definition amounts to saying that a cartesian theory is a [[first-order theory]] in which the only connectives used are $=$, $\top$, $\wedge$ and existential quantifiers over provably [[subterminal]] formulae, that is, $\exists x:A. \phi$ can only be used if
$$\phi(x),\phi(y) \vdash x = y$$
is provable. Unlike other first-order theories, this side-condition makes the order of presentation of axioms matter, as an axiom might require another in order to prove that a formula involved is subterminal.

[Palmgren and Vickers](#PalmgrenVickers07) showed that in a geometric logic of partial terms (with a non-reflexive equality), the cartesian theories are precisely those that can be presented using only $\wedge$, $\top$ and $=$.

## Semantics in Categories with Finite Limits

Cartesian logic can be given models interpreting formulae as [[subobjects]] in a category with [[finite limits]]. See [[internal logic]] for more details. The most interesting component of this is the existential quantifiers. As with [[regular logic]], the existential quantifier is interpreted as an [[image factorization]], but the subterminality side condition ensures the image exists even in an arbitrary category with finite limits. A formula $\phi(x)$ with $x:A$ in a context $\Gamma$ is interpreted as a subobject $\phi \to \Gamma \times A$, and the subterminality side condition is satisfied in the internal logic if and only if the first projection $\Gamma \times A \to \Gamma$ is mono, in which case the composition $\phi \to \Gamma \times A \to \Gamma$ is already the inclusion into its image.

## Related concepts

[[!include categories and logic - table]]

## References

Cartesian logic was introduced in the early seventies by [[John Isbell]], [[Peter Freyd]] and [[Michel Coste]] (cf. [Johnstone 1979](#Johnstone79)). A standard source is Johnstone ([2002](#Johnstone02)).
[Palmgren and Vickers](#PalmgrenVickers07) gave a new proof of the initial model theorem, valid in weak foundational settings.

* {#AdamekRosicky94}[[Jiří Adámek]], [[Jiří Rosický]], _Locally presentable and accessible categories_ ,  Cambridge UP 1994.

* {#Freyd02}[[Peter Freyd]], _Cartesian Logic_ , Theor. Comp. Sci. **278** (2002) pp.3-21.

* {#Johnstone79}[[Peter Johnstone]], _A Syntactic Approach to Diers' Localizable Categories_ , pp.466-478 in Springer LNM **753** Heidelberg 1979.

* {#Johnstone02}[[Peter Johnstone]], _Sketches of an [[Elephant]] II_ , Oxford UP 2002. (Around D1.3.4 p.833)

* {#PalmgrenVickers07}[[Erik Palmgren]], [[Steve Vickers]], _Partial Horn logic and cartesian categories_, Ann Pure Appl Logic **145 (3)** (2007) pp.314-353.

[[!redirects finite-limit logic]]


cartesian model category

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## Idea

A __cartesian model category__ (alias __cartesian closed model category__) is a [[cartesian closed category]] that is equipped with the structure of a [[monoidal model category]] in a compatible way, which combines the axioms for a [[monoidal model category]] and an [[enriched model category]].

## Definition

\begin{definition}
A **cartesian closed model category** (following [Rezk 09, 2.2](#Rezk09) and [Simpson 12](#Simpson12)) is

* a [[cartesian closed category]]

* equipped with a [[model category]]-[[mathematical structure|structure]]

that satisfies

1. the following equivalent axioms:

   * **[[pushout-product axiom]]**

     For $f \colon X \to Y$ and $f' \colon X' \to Y'$ [[cofibrations]], the induced morphism

     $$
       (Y \times X')
         \overset{X \times X'}{\coprod}
       (X \times Y')
          \longrightarrow
       Y \times Y'
     $$

     is a cofibration that is a [[weak equivalence]] if at least one of $f$ or $f'$ is;

   * **[[pullback-power axiom]]**

     For $f \colon X \to Y$ a [[cofibration]] and $f' \colon A \to B$ a [[fibration]], the induced morphism

     $$
       [Y,A]
         \longrightarrow
       [X,A] \underset{[X,B]}{\prod} [Y,B]
     $$

     is a fibration, and a weak equivalence if at least one of $f$ or $f'$ is.

1. the **unit axiom**:

   The [[terminal object]] is [[cofibrant object|cofibrant]].

\end{definition}

## Examples
 {#Examples}

Cartesian monoidal model categories include:

* the standard [[Quillen model structure on topological spaces]] on [[compactly generated topological space|compactly generated]] [[weakly Hausdorff topological spaces]]

  (see [there](classical+model+structure+on+topological+spaces#TopologicalEnrichment))

* the [[fine model structure on topological G-spaces]]

  (see [there](fine+model+structure+on+topological+G-spaces#MonoidalModelCategoryStructure))

* the [[classical model structure on simplicial sets]]

* the [[canonical model structure on categories]] (and [[canonical model structure on groupoids|that on groupoids]])

  (see [there](canonical+model+structure+on+Cat#CartesianMonoidalModelStructure))

* the [[model structure for complete Segal spaces]]

  (see [there](model+structure+for+complete+Segal+spaces#CartesianMonoidalModelStructure))

* the model structure for [[Theta-spaces]]

  (see [there](Theta-space#CartesianMonoidalAndEnrichedStructure))

## Related concepts

* [[cartesian closed category]], [[locally cartesian closed category]]

* [[cartesian closed functor]], [[locally cartesian closed functor]]

* [[locally cartesian closed model category]]

* [[cartesian closed enriched category]]

* [[cartesian closed (∞,1)-category]] [[locally cartesian closed (∞,1)-category]]

## References

* {#Rezk09} [[Charles Rezk]], _A cartesian presentation of weak $n$-categories_, Geom. Topol. 14(1): 521-571 (2010). ([arXiv:0901.3602](http://arxiv.org/abs/0901.3602), [doi:10.2140/gt.2010.14.521](https://projecteuclid.org/journals/geometry-and-topology/volume-14/issue-1/A-cartesian-presentation-of-weak-ncategories/10.2140/gt.2010.14.521.full))

* {#Simpson12} [[Carlos Simpson]],  Chapter 10 of: _[[Homotopy Theory of Higher Categories]]_, Cambridge University Press 2011 ([hal:00449826](https://hal.archives-ouvertes.fr/hal-00449826/document), [ISBN:9780521516952](https://www.cambridge.org/in/academic/subjects/mathematics/geometry-and-topology/homotopy-theory-higher-categories-segal-categories-ini-categories-and-beyond?format=HB))

[[!redirects cartesian model categories]]

[[!redirects cartesian monoidal model category]]
[[!redirects cartesian monoidal model categories]]

[[!redirects cartesian closed model category]]
[[!redirects cartesian closed model categories]]

[[!redirects cartesian closed monoidal model category]]
[[!redirects cartesian closed monoidal model categories]]

[[!redirects cartesian model structure]]
[[!redirects cartesian model structures]]

[[!redirects cartesian closed model structure]]
[[!redirects cartesian closed model structures]]

[[!redirects cartesian closed monoidal model structure]]
[[!redirects cartesian closed monoidal model structures]]


cartesian monad

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# Cartesian monads
* table of contents
{: toc}

##Idea##

A _cartesian monad_ is a [[monad]] on a [[locally cartesian category]] that preserves [[pullbacks]] and whose unit and multiplication are [[cartesian natural transformations]].

## Motivation through generalised multicategories

Ordinary [[categories]] can be defined as [[monad|monads]] in the [[bicategory]] of [[spans]] of [[sets]]. Multicategories can be defined in a similar way. (A [[multicategory]] is like an ordinary category where each morphism has a list of objects as its domain, and a single object as its codomain; think [[Vect|vector spaces and multilinear maps]]).

To see how a multicategory $C$ can be defined as a monad in some appropriate bicategory, let $C_0$ be the set of objects of $C$, and notice that the domain of a morphism of $C$--a finite list of objects--is an element of $T C_0$, where $T$ is the [[free monoid]] monad. In this way the data for $C$ can be conveniently organized in the diagram
\[\begin{matrix}
&&C_1&&\\
&\overset{d}\swarrow&
&\overset{c}\searrow&\\
T C_0 &&&& C_0.
\end{matrix}\]

[[Tom Leinster]] built on the idea of [[generalized multicategory|generalized multicategories]], where the domain of a morphism can have a more general, higher dimensional shape than just a list. This is accomplished by considering a category $\mathcal{E}$ other than $\mathrm{Set}$, and a monad $T$ on $\mathcal{E}$, and mimicking the above construction. So the data for a $T$-multicategory is a diagram in $\mathcal{E}$ like the one above.

To state the structure required on the data for a $T$-multicategory, we want to define a bicategory in which the above span is an [[endomorphism]]. Then a $T$-multicategory will be such a span together with structure making it a monad in that bicategory. The bicategory is $\mathcal{E}_{(T)}$, the bicategory of $T$-spans in $\mathcal{E}$. Its objects are the objects of $\mathcal{E}$, and its morphisms are spans
\[\begin{matrix}
&&M&&\\
&\overset{d}\swarrow&
&\overset{c}\searrow&\\
T E &&&& E'.
\end{matrix}\]

This won't in general be a bicategory without a few extra assumptions. Identity spans are defined using the unit $\eta: \mathrm{Id}\to T$. Composition of spans is defined using [[pullbacks]] and the multiplication $\mu: T^2\to T$, so the category $\mathcal{E}$ must at least have pullbacks--usually it will be [[finitely complete category|finitely complete]]. The associativity and unit $2$-cells are defined using the universal property of the pullbacks. However, these $2$-cells won't in general be invertible. In fact, it turns out that requiring the monad $T$ to be _cartesian_ is exactly what is needed to ensure that the coherence $2$-cells are isomorphisms, and hence that $T$-spans do in fact form a bicategory. Maybe this should be the "fundamental theorem of cartesian monads".

Extending Bénabou's observation that a [[small category]] is a monad in the bicategory of spans of sets, [[Burroni]] defined  $T$-multicategories as _monads in the bicategory_ $\mathcal{E}_{(T)}$ from above. When $T$ is the [[identity monad]] on $\mathrm{Set}$, $T$-multicategories reduce to small categories, and when $T$ is the free monoid monad on $\mathrm{Set}$, $T$-multicatories are exactly ordinary small multicategories.

As an indication of how this theory is useful as a language for higher categories, take $T$ to be the free [[strict ∞-category]] monad on the category of [[globular set]]s. Then $T$-multicategories with exactly one object are called [[globular operad]]s, and Leinster defines one such globular operad (the initial "globular operad with contraction") for which the algebras are [[weak ∞-categories]].

## Definition

+-- {: .num_defn}
###### Definition

Let $(T, \mu, \nu)$ be a
[[monad]] on a category $C$.
Specifically, $T: C \to C$ is a functor,
and $\mu: T^2 \to T$ and
$\nu: \mathrm{Id}_C \to T$ are natural
transformations, satisfying unital and associative
axioms making $T$ a monoid in the (strict)
monoidal category $\mathrm{End}(C)$.
This monad is cartesian if

* the category $C$ has all [[pullback]]s,
* the functor $T$ preserves pullbacks,
* the natural transformations $\mu$ and $\nu$
  are [[cartesian natural transformation|cartesian]].
  Recall that a natural transformation $\alpha: S \to T$
  between functors $C \to D$
  is cartesian if for each map $f: A \to B$
  in $C$, the naturality square
  \[\array{
    S A & \overset{S f}{\to} & S B \\
    \alpha_A \downarrow & & \downarrow \alpha_B \\
    T A & \underset{T f}{\to} & T B
  }\]
  is a pullback.

=--

+-- {: .num_remark}
###### Remark

There is some slight inconsistency in the use of
the word _cartesian_ in category theory.
Sometimes, a category is called cartesian if it [[finitely complete category|has all finite limits]];
similarly, a functor is called cartesian if it [[exact functor|preserves all finite limit]]s.
In most examples of cartesian monads, the category
$C$ has a [[terminal object]], and hence finite limits.
However, the functor $T$ almost never preserves
terminal objects.
For example, the free monoid monad on
$\mathrm{Set}$ is cartesian, as can be checked
directly, but $T 1 \simeq \mathbb{N}$ is not
a terminal object.
In this sense, a cartesian monad is really *locally* cartesian.

=--

## Examples and Non-Examples

* The [[free monoid]] monad $(-)^*: Set \to Set$ is cartesian.

* The [[free category]] monad acting on [[quivers]] is cartesian.

* The free strict $\omega$-[[strict omega-category|category]] monad acting on [[globular set]]s, $T: Set^{G^{op}} \to Set^{G^{op}}$, is cartesian.

* The [[free strict monoidal category]] monad on $\mathrm{Cat}$ is cartesian.

* The [[free symmetric strict monoidal category]] monad on $\mathrm{Cat}$ -- where all coherence cells are required to be identities _except_ the symmetries $A\otimes B \cong B \otimes A$ -- is cartesian.

BUT:

* The [[free commutative monoid]] monad on $\mathrm{Set}$ is **NOT** cartesian.

* The [[free strict symmetric monoidal category]] monad on $\mathrm{Cat}$ -- where all the coherence cells are required to be identities _including_ the symmetry isomorphisms $A \otimes B \cong B \otimes A$ -- is **NOT** cartesian. In fact this is exactly the free commutative monoid monad on $\mathrm{Cat}$.

## Properties

### Relation to operads

To every non-symmetric [[operad]], hence to every [[multicategory]] is associated a cartesian monad, such that the corresponding [[algebras over an operad]] coincide with the corresponding [[algebras over a monad]].

## Related pages

* Every [[p.r.a. monad]] is cartesian; these are sometimes called "strongly cartesian monads".

## References

* [[Tom Leinster]], _Higher Operads, Higher Categories_
  ([arXiv:math.CT/0305049](http://arxiv.org/abs/math.CT/0305049)),
  section 4.1
  {#Leinster}

* [[Albert Burroni]], _$T$-catégories (catégories dans un triple)_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 12 no. 3 (1971), p. 215-321 ([numdam](http://www.numdam.org/item?id=CTGDC_1971__12_3_215_0))

* [blog comment](http://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html#c025402) giving the Motivation above

* MO, _[Monad arising from operad](http://mathoverflow.net/questions/66313/monad-arising-from-operad)_

* {#BMW} [[Clemens Berger]], [[Paul-André Melliès]], [[Mark Weber]], _Monads with Arities and their Associated Theories_ (2011) ([arXiv:1101.3064](http://arxiv.org/abs/1101.3064))

## Discussion ##

Some past discussion about the term 'cartesian' has been moved to [[locally cartesian category]].

[[!redirects cartesian monads]]
[[!redirects Cartesian monad]]
[[!redirects Cartesian monads]]


cartesian monoidal (infinity,1)-category

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## Idea

A _cartesian monoidal (∞,1)-category_ is a [[symmetric monoidal (∞,1)-category]] whose [[tensor product]] is given by the categorical [[product]].
This is dual to the notion of [[cocartesian monoidal (∞,1)-category]].

In the special case that the underlying [[(∞,1)-category]] is equivalent to just a [[1-category]], then this is equivalently a [[cartesian monoidal category]].

## Definition

(...)

([Lurie, def. 2.4.0.1](#Lurie))

(...)

## Properties

### Coalgebra objects
 {#CoalgebraObjects}

 Every object in a Cartesian monoidal $\infty$-category is canonically a [[comonoid object]] via the [[diagonal map]] (just as in the 1-categorical case [here](comonoid#InACartesianMonoidalCategory)).

See also at _[[(infinity,n)-category of correspondences]]_ the section _[Via coalgebras](%28infinity%2Cn%29-category+of+correspondences#DefinitionViaCoalgebras)_.

## References

Section 2.4 of

* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_

[[!redirects cartesian monoidal (infinity,1)-categories]]

[[!redirects cartesian monoidal (∞,1)-category]]
[[!redirects cartesian monoidal (∞,1)-categories]]

[[!redirects Cartesian monoidal (infinity,1)-category]]
[[!redirects Cartesian monoidal (infinity,1)-categories]]

[[!redirects Cartesian monoidal (∞,1)-category]]
[[!redirects Cartesian monoidal (∞,1)-categories]]

[[!redirects cartesian monoidal infinity,1-category]]
[[!redirects cartesian monoidal infinity,1-categories]]

[[!redirects Cartesian monoidal infinity,1-category]]
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