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Let $C$ be a symmetric monoidal category, and $f : x \to y$ be a morphism in $C$. I would like to construct the localization $C_f$ explicitly, which solves the universal property $$\mathrm{Hom}_{\otimes}(C_f,D) = \{F \in \mathrm{Hom}_{\otimes}(C,D) : F(f) \text{ iso}\}.$$

I am not interested in a general existence proof or alike; instead I would like to exhibit $C_f$ as an explicit full reflective subcategory of $C$, thereby also showing its existence. The reason is that I want to actually compute something in these localizations which a priori does not simply follow from the universal property.

There is a general construction of a localization of a plain category with respect to arbitrary sets of morphisms, which can be found on page 6 of Gabriel-Zisman's Calculus of fractions and homotopy theory. Thus, an object of the localization $C_f$ is an object of $C$, and a morphism is a class of a finite sequence of the form $f_1 s^{-1} f_2 \cdots s^{-1} f_n$ (perhaps without $f_1$ or $f_n$), where the sources and targets should fit, subject to the obvious cancellation rules. But this might not define a (small) set of morphisms, right?

Question 1. Which conditions have to be imposed on $C$ and $f$ so that $C_f$ exists (without leaving the universe)?

I know the basics about left/right multiplicative systems (as in Gabriel-Zisman, Weibel, Kashiwara-Schapira, etc.), saturations etc., but I could not find any answer to this question in the literature.

EDIT: As Theo points out, there is no set-theoretic problem if we localize a category at just one single morphism. But when $C$ is monoidal, there is no reason why the tensor product $C \times C \to C$ extends to a tensor product $C_f \times C_f \to C_f$, because this means that for every $x \in C$ the invertibility of $f$ forces the invertibility of $x \otimes f$ and $f \otimes x$ in the language of categories, which is unplausible. Instead we should better localize at all morphisms $x \otimes f$ and $f \otimes x$, where $x$ runs through all objects of $C$; this is a monoidal class in the language of Day's paper "A Note on Monoidal Localisation". But now there are set-theoretic problems in the description of $C_f$ above. On the other hand, we repair this easily if $C$ has a small colimit-dense subcategory, which happens to be the case when $C$ is presentable.

Question 2. Which conditions have to be imposed on $C$ and $f$ so that we can write down explicitly the localization $C_f$ in the $2$-category of symmetric monoidal categories? How does it look like?

Question 3. Actually I am interested in cocomplete symmetric monoidal categories and cocontinuous symmetric monoidal functors between them. How does the localization look like in this context?

Let me mention a special case where everything works out: Let $\mathcal{L} \in C$ be an object whose symmetry $\mathcal{L}^{\otimes 2} \to \mathcal{L}^{\otimes 2}$ is the identity and $f : 1_C \to \mathcal{L}$ a morphism (imagine a global section of a line bundle on a scheme). Let $C_f \subseteq C$ the full subcategory consisting of those $M \in C$ such that $M \otimes f : M \to M \otimes \mathcal{L}$ is an isomorphism. If $C$ is cocomplete, the inclusion $C_f \subseteq C$ has a left adjoint: It maps $M \in C$ to the colimit of $M \to M \otimes \mathcal{L} \to M \otimes \mathcal{L}^{\otimes 2} \to \dotsc$. Using this left adjoint, one can define tensor products and colimits in $C_f$ (one may cite Day's reflection theorem here) and verify easily that $C \leadsto C_f$ is the localization in the context of cocomplete symmetric monoidal categories.

One can also verify that if $\mathcal{L}$ is a line bundle on a scheme $X$ and $f \in \Gamma(X,\mathcal{L})$ is a global section, then we really have $\mathrm{Qcoh}(X)_f = \mathrm{Qcoh}(X_f)$, so this categorical localization is compatible with the scheme theoretic localization.

However, for other applications, I need more general morphisms $f$. This motivated my question. I am pretty sure that this should be standard in category theory, therefore the reference request tag.

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I am not interested in a general existence proof or alike; instead I would like to exhibit $C_f$ as an explicit full reflective subcategory of $C$, thereby also showing its existence. The reason is that I want to actually compute something in these localizations which a priori does not simply follow from the universal property.

There is a general construction of localizations a localization of a plain category with respect to arbitrary sets of morphisms, which can be found on page 6 of the book by Gabriel-ZismanGabriel-Zisman's Calculus of fractions and homotopy theory. Thus, an object of the localization $C_f$ as a plain category is an object of $C$, and a morphism is a class of a finite sequence of the form $f_1 s^{-1} f_2 \cdots s^{-1} f_n$ (perhaps without $f_1$ or $f_n$), where the sources and targets should fit, subject to the obvious cancellation rules. But this might not define a (small) set of morphisms, right?

I know the basics about left/right multiplicative systems (as in Gabriel-Zisman, Weibel, Kashiwara-Schapira, etc.), saturations etc., but I could not find any answer to this question in the literature.

Question 2

EDIT: As Theo points out, there is no set-theoretic problem if we localize a category at just one single morphism. Which conditions have But when $C$ is monoidal, there is no reason why the tensor product $C \times C \to be imposed on C$ extends to a tensor product $C_f \times C_f \to C_f$, because this means that for every $x \in C$ and the invertibility of $f$ so that forces the invertibility of $C_f$ becomes symmetric monoidal x \otimes f$and$C f \leadsto C_f$satisfies otimes x$ in the universal property with respect to symmetric monoidal language of categoriesstated above?

According to , which is unplausible. Instead we should better localize at all morphisms $x \otimes f$ and $f \otimes x$, where $x$ runs through all objects of $C$; this is a monoidal class in the language of Day's paper "A Note on Monoidal Localisation"perhaps we should demand that some right multiplicative system associated . But now there are set-theoretic problems in the description of $C_f$ above.

Question 2. Which conditions have to be imposed on $C$ and $f$ is so that we can write down explicitly the localization $C_f$ in the $2$-category of symmetric monoidal , but I don't know how to make this explicit.categories? How does it look like?

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Let $C$ be a symmetric monoidal category, and $f : x \to y$ be a morphism in $C$. I would like to construct the localization $C_f$ explicitly, which solves the universal property $$\mathrm{Hom}_{\otimes}(C_f,D) = \{F \in \mathrm{Hom}_{\otimes}(C,D) : F(f) \text{ iso}\}.$$

I am not interested in a general existence proof or alike; instead I would like to exhibit $C_f$ as an explicit full reflective subcategory of $C$, thereby also showing its existence.

There is a general construction of localizations with respect to arbitrary sets of morphisms, which can be found on page 6 of the book by Gabriel-Zisman. Thus, an object of the localization $C_f$ as a plain category is an object of $C$, and a morphism is a class of a finite sequence of the form $f_1 s^{-1} f_2 \cdots s^{-1} f_n$ (perhaps without $f_1$ or $f_n$), where the sources and targets should fit, subject to the obvious cancellation rules. But this might not define a (small) set of morphisms, right?

Question 1. Which conditions have to be imposed on $C$ and $f$ so that $C_f$ exists (without leaving the universe)?

I know the basics about left/right multiplicative systems, saturations etc., but I could not find any answer to this question in the literature.

Question 2. Which conditions have to be imposed on $C$ and $f$ so that $C_f$ becomes symmetric monoidal and $C \leadsto C_f$ satisfies the universal property with respect to symmetric monoidal categories stated above?

According to Day's paper "A Note on Monoidal Localisation" perhaps we should demand that some right multiplicative system associated to $f$ is monoidal, but I don't know how to make this explicit.

Question 3. Actually I am interested in cocomplete symmetric monoidal categories and cocontinuous symmetric monoidal functors between them. How does the localization look like in this context?

Let me mention a special case where everything works out: Let $\mathcal{L} \in C$ be an object whose symmetry $\mathcal{L}^{\otimes 2} \to \mathcal{L}^{\otimes 2}$ is the identity and $f : 1_C \to \mathcal{L}$ a morphism (imagine a global section of a line bundle on a scheme). Let $C_f \subseteq C$ the full subcategory consisting of those $M \in C$ such that $M \otimes f : M \to M \otimes \mathcal{L}$ is an isomorphism. If $C$ is cocomplete, the inclusion $C_f \subseteq C$ has a left adjoint: It maps $M \in C$ to the colimit of $M \to M \otimes \mathcal{L} \to M \otimes \mathcal{L}^{\otimes 2} \to \dotsc$. Using this left adjoint, one can define tensor products and colimits in $C_f$ (one may cite Day's reflection theorem here) and verify easily that $C \leadsto C_f$ is the localization in the context of cocomplete symmetric monoidal categories.

One can also verify that if $\mathcal{L}$ is a line bundle on a scheme $X$ and $f \in \Gamma(X,\mathcal{L})$ is a global section, then we really have $\mathrm{Qcoh}(X)_f = \mathrm{Qcoh}(X_f)$, so this categorical localization is compatible with the scheme theoretic localization.

However, for other applications, I need more general morphisms $f$. This motivated my question. I am pretty sure that this should be standard in category theory, therefore the reference request tag.

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2 Fixed a typo, corrected name of a reference
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