Is there a left adjoint to the forgetful functor from cartesian monoidal categories to symmetric monoidal categories? And if so: how can it be described explicitely.

In more detail: Given a symmetric monoidal category $\mathbb V$ i am looking for a monoidal functor

$$\chi:\mathbb V \to \mathcal C(\mathbb V)$$

into a cartesian monoidal category $\mathcal C(\mathbb V)$ such that every other monoidal functor

$$\varphi:\mathbb V \to \mathbb C $$

into a cartesian monoidal category $\mathbb C$ factors 'uniquely' over $\chi$.

  • $\begingroup$ what is a cartesian monoidal category? Do you mean a (symmetrical) monoidal closed one? $\endgroup$ – Buschi Sergio Jun 2 '13 at 21:31
  • 2
    $\begingroup$ "A cartesian monoidal category (usually just called a cartesian category), is a monoidal category whose monoidal structure is given by the category-theoretic product (and so whose unit is a terminal object)." -- ncatlab.org/nlab/show/cartesian+monoidal+category $\endgroup$ – Gerrit Begher Jun 3 '13 at 8:15
  • 1
    $\begingroup$ You look for a $2$-adjoint, right? See also the comments in Chris' answer. You don't want that $\hom(V,U(C))$ and $\hom(C(V),C)$ are isomorphic, but rather that they are equivalent. $\endgroup$ – Martin Brandenburg Jul 15 '13 at 20:48

Yes, there is a left adjoint. Both the category $\textbf{Mon}$ of (symmetric) monoidal categories, and the category $\textbf{Cart}$ of Cartesian categories, are algebraic over the category $\textbf{Cat}$ of categories. That is, there are monads on $\textbf{Cat}$ whose categories of algebras are $\textbf{Mon}$ and $\textbf{Cart}$, respectively.

More interestingly, there is also a right adjoint, that sends a monoidal category $\textbf{C}$ to the category of comonoids in $\textbf{C}$. This is a classic result by Thomas Fox.

  • 7
    $\begingroup$ Shouldn't Mon and Cart be treated as 2-categories as opposed to 1-categories? Or is there some magic working behind the scenes that ensures that issues with higher coherences do not arise? $\endgroup$ – Dmitri Pavlov Jun 3 '13 at 0:28
  • 1
    $\begingroup$ Dmitri, definitely they are 2-cateogries. I'm pretty sure that 1-categorically those algebraicity statements are false. E.g. associators correspond to associativity conditions in algebraic 2-theory, cartesian structure is encoded in adjunction between diagonal and multiplication, etc. But formally this doesn't seem to change anything, classical adjunction proofs can be repeated 2-categorically. $\endgroup$ – Anton Fetisov Jun 3 '13 at 11:44
  • 1
    $\begingroup$ I should have specified that Fox uses strict monoidal functors as the morphisms. As Dmitri points out and Anton mentions, lax (or even strong) monoidal functors require 2-categories. $\endgroup$ – Chris Heunen Jun 3 '13 at 14:11
  • 1
    $\begingroup$ Thanks for the link; how can said monads - and the left adjoint - be described explicitly? Any reference? $\endgroup$ – Gerrit Begher Jun 4 '13 at 16:19
  • 2
    $\begingroup$ As usual with these abstract proofs you can trace them back and find explicit constructions - which are not really useful. Here, when your monoidal category is given by generators and relations, you just replace $\otimes$ by $\times$ in each relation. A basic example is the ($2$-)free symmetric monoidal category on one object, which is the permutation groupoid. Then the left adjoint maps it to the free cartesian category on one object, which should be $\mathsf{FinSet}^{\mathsf{op}}$. Such things get interesting when you really work out specific examples where no presentation is given. $\endgroup$ – Martin Brandenburg Jul 15 '13 at 21:03

Edit I had developed a demonstration of the inexistence of the 'left adjoint funtor, this demonstration shows an error (sneaky) in the final, then (since it was long) I removed this part as unuseful.

EDIT I did this ultimate proof:


1) We call a category with finite product (and final object, or empty product) cartesian, i.e. a category with a finite product monoidal structure, call a functor cartesian is it preserve finite product (and final object).

2) Given a category $\mathcal{C}$ its cartesian-closure (finite product completion) $Ct(\mathcal{C})$, is the (generalization) of the category $Pro(\mathcal{C})$ where we considering finite discrete diagrams (instead co-direct), its objets are the finite family $(X_i)_{i\in I}$ and its morphisms are $(f, \hat{f} ) : (X_i)_{i\in I } \to (Y_j)_{j\in J} $ with $\hat{f}: J \to I $ and $f=(f_j: X_{\hat{f}(j) } \to Y_j)_{j\in J }$ by composition $(g, \hat{g} )\circ (f, \hat{f} ) : (X_i)_{i\in I } \to (Y_j)_{j\in J}\to (Z_k)_{k\in K} $ as $\hat{f} \circ \hat{g}$ and $(g_k \circ f_{\hat{g}(k)})$. we have a full embedding $\iota: \mathcal{C}\to Ct(\mathcal{C}): X \mapsto (X)$ and we identify $X$ ith $\iota(X)= (X)$, we have that $(X_i)_{i\in I}= \times_{i\in I} X_i $. Then for $F: \mathcal{C} \to \mathcal{A}$ with $\mathcal{A}$ cartesian there exist unique (but isomorphism) $F': C(\mathcal{C}) \to \mathcal{A}$ such that $F=F'\circ \iota$. We have that $C(C(\mathcal{C}))\cong C(\mathcal{C})$ and if $\mathcal{C}$ is cartesian exist $r: C(\mathcal{C}) \to \mathcal{C}$ with $1= r\circ \iota$.\

3) Given a category $\mathcal{C}$, and a class $\Sigma$ of new arrow between object of $\mathcal{C}$, the phrase "add new isomorphisms $\Sigma$ to $\mathcal{C}$" means: add $\Sigma$ to the graph of $\mathcal{C}$, consider the congruences inducted by the composition of morphisms of $\mathcal{C}$ everywhere its possible, and consider the fraction category respect the class of morphisms $\Sigma$. The problem is that in general the Hom-classes of morphisms is a proper class, or in other terms then the new category belong to a more large set-universe.

End of Premises

Let $\textbf{V}$ monoidal. Let $\iota: \textbf{V} \to Ct(\textbf{V}) $ the cartesian closure of the underling category (we exclude the final object given by the empty family, we'll see how get the final object in more useful way). Given a strict-monoidal functor $F: \textbf{V} \to \mathscr{A}$ with $\mathscr{A}$ cartesian, exist a (iso)unique cartesian functor $F': Ct(\textbf{V}) \to \mathscr{A}$ with $F= F' \circ W$, and for each such $F$ if $X \cong A_1 \otimes \ldots \otimes A_n $ then $F(X) \cong F(A_1) \times \ldots \times F(A_n)$, and also our reflector $\chi$ (see original question above) must have this property .

Then for each isomorphism $X \cong A_1 \otimes \ldots \otimes A_n $ in $\textbf{V}$, we have to add to $C(\textbf{V})$ a formal isomorphism $X \cong A_1 \times \ldots \times A_n$, we get a new (large) category $\widehat{\mathcal{C}}$ that still have the universal property for functors like $F$ before, then also its cartesian closure $C(\widehat{\mathcal{C}})$ have this property. Now, observe that the finite product of $C(\widehat{\mathcal{C}})$ are identified with the old finite product of $C(\mathcal{C})$ (identify singletons (or double singletons) with the object, and a finite family od finite families as a global finite family). Now make the quotient for the congruence generated by the identification of the natural projection $I\times I \to I$ by $I\times I \cong I\otimes I \cong I$, then follow that $I$ is the final object.

In fact from $A \cong A\otimes I\cong A\times I\to I$ follow a morphism $A \to I$, given $f, g: A \to I$ the couple of these are just a morphism $A \to I \times I \cong I\otimes I \cong I$ but the projections $I\times I \to I$ are both the composition $I\times I \cong I\otimes I\cong I$, then $f=g$.

Observing that the final $\widehat{\mathcal{C}}$ (with the final congruence) is in generale a LARGE category, then cannot be a result that come from theorem of existence of (left) adjoint based on properties of monadic functors or algebraic theory properties.

In the case of monoidal functors (original question), I claim that exist a left inverse:

to the (monoidal simmetric) category $\textbf{V}$ add a new object $e$ the arrow identity $1_e: e \to e$, for each $X \in \textbf{V}$ a new arrow $e^X: X \to e $ and a arrows family $(e_f: e \to A)_{f : I \to A }$ and let $\textbf{V}_e$ the category generated by all these new arrows: the morphisms are concatenation of some arrows of $\textbf{V}$ and some new arrows, where we compose arrows of $\textbf{V}$ whenever is possible, and the composition is for concatenation, in this category consider the congruence that identify all endomorphisms $e \to e$ to $1_e$, and $e^B\circ g\cong e^A $ and $e_{g\circ f}\cong g\circ e_f$ for $g: A\to B$, and let $\widehat{ \textbf{V}} := \textbf{V}_e/ \cong $ the quotient category.

Consider the category named "theory of $ \widehat{ \textbf{V}}$-algebras" (see [LPAC] p. 146): the sorts are the object of $\widehat{\textbf{V}}$ and operators (of sort $B$): $A_1 \times \ldots \times A_n \to B$ are (formal labels of) morphisms $f: A_1\otimes \ldots \otimes A_n \to B $ in $\widehat{ \textbf{V}}$ where indicate the sequence of sorts as a (formal) product, and where the monoidal tensor is thinked with brackets (all moved to the left). Briefly this category has for object the finite sequences of objects of $\textbf{V}$ (write as formal product): $X_1\times \ldots \times X_n $, the product of these objects is defined for sequences concatenation. Morphism are defined (as formal strings) for induction as follow:

operators labels as above $f: A_1 \times \ldots \times A_n \to B$ (for $f: A_1\otimes \ldots \otimes A_n \to B $ in $\widehat{ \textbf{V}}$) are morphisms of $C(\textbf{V})$.

(formal) projections $\pi_k : A_1 \times \ldots \times A_n \to A_k$ are morphisms of $C(\textbf{V})$.

Give morphisms $h: Y_1 \times \ldots \times Y_n \to X$ and morphisms $k_i: X_i \to Y_i $ $1 \leq i \leq n$ then the formal string $h(k_1,\ldots, k_n)$ is a morphisms of $C(\textbf{V})$.

Morphisms $A \to (X_1\times\ldots \times X_n)$ are just sequences $h_i: A \to X_i$ $1 \leq i \leq n$ of morphisms.

Now, let $C(\widetilde{V})$ the full subcategory with object the product $X_1\times \ldots \times X_n $ where $X_i\in \textbf{V}$ $1\leq i\leq n$, plus the object $e$ and define $X\times e= e\times X= X$ for any $X\in C(\widetilde{V})$ (then $e\times\ldots \times e= e$), and define the compositions $e \xrightarrow{(e_{f_1},\ldots, e_{f_n})} A_1\times\ldots \times A_n \xrightarrow{g} B$ as $e_{g\circ (f_1\odot \ldots \odot f_n)}$ where $f_1\odot \ldots \odot f_n: I \cong I\otimes\ldots \otimes I \xrightarrow{(f_1,\ldots, f_n)}A_1\otimes\ldots \otimes A_n $

. We have that $e$ is a final object of $C(\widetilde{V})$ and this category has finite product.

The monoidal functor $\chi: \textbf{V}\to C(\textbf{V})$ map objects and morphisms in themselves, with $\widetilde{\chi}_{A, B}: \chi(A, B) \to \chi(A) \times \chi(B)$ given by $1_{A\otimes B}$, and $\chi_o: e \to \chi(I)$ given by $e_{1_I}$. For $F: \textbf{V}\to \mathcal{C}$ monoidal, where $\mathcal{C}$ is cartesian, $\widetilde{F}: C(\textbf{V})\to \mathcal{C}: X_1\times\ldots \times X_n \mapsto FX_1\times\ldots \times FX_n $ is the only cartesian functior such that $\widetilde{F}\circ \chi = F$.


[LPAC] : "locally presentable and accessible categories". Adámek , Rosický. Cambridge University Press 1994


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.