Localization of a symmetric monoidal category at a single morphism

Question

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.

Answer 1

I'll try to bend David White's answer towards the actual situation of your question. The outcome is somewhat clumsy and it totally looks like model structures can be eliminated from it, but anyway:

Assume your category C is closed monoidal and locally presentable. Then it is a monoidal model category with cofibrations and fibrations all morphisms and weak equivalences the isomorphisms. This model category is cofibrantly generated: One can take the identity of the initial object as generating trivial cofibration and the set of all morphisms between the objects $G$ from some generating set as generating cofibrations.

This model category then satisfies the hypotheses of Barwick's Thm. 4.46 in this article for a Bousfield localization at the one element set containing $f$. The homotopy category for the localized model structure has the universal property you want and lives in the same universe. You have an adjunction between the homotopy category of the original model structure, which is the category itself, and the localization.

This adjunction is a reflection to an orthogonal subcategory as in Adamek/Rosicky, 1.35-1.38, namely to the full subcategory of all objects from whose point of view $f$ "was already an isomorphism" (i.e. $f$-orthogonal objects; precise definition via a unique-lifting-condition). This is much like in your example (but with the condition on the twist removed and without the domain of f having to be special). If you chase through the proofs, you also get an expression of the reflection functor as a colimit via the small object argument, resp. via Adamek/Rosicky's proof...

Barwick's Prop. 4.47 gives then a criterion for the homotopy category to be closed monoidal again: It suffices that any object $X$ which satisfies the unique right lifting condition with respect to $f$ also satisfies it with respect to $f \otimes G$ for every generating object $G$ (remember the category was locally presentable now) i.e. if $f$ induces an iso $Hom(f,X)$ then $Hom(f \otimes G,X)$ is an iso, too, for every generating object $G$.

edit: Sorry, I am no longer sure that the homotopy category of the Bousfield localization is in fact the localization along $f$ in the sense you asked for: When you localize with respect to an arrow you automatically invert together with it a bunch of other arrows. When you do plain category theory it is somewhat uncontrollable what those other arrows are, it seems to me. When you do Bousfield localization these other arrows are those having the left lifting property with respect to the $f$-local objects. Now I don't see a reason why the class of additionally inverted arrows should be the same in both cases. What Bousfield localization as sketched here probably yields, is the universal colimit preserving functor which inverts $f$.

Answer 2

Mike Shulman's comment is right on the mark. My sense is that these sorts of set theoretic issues were major motivators for the invention of model categories. If you don't want to get into model categories, then the take-away message should be that when you have a model category you can do homotopy theory. In a model category you have a distinguished class $W$ of morphisms (generalizing the homotopy equivalences from topology) which you want to invert. Inverting these morphisms takes you to the homotopy category $Ho(M)$ and the functor $M\rightarrow Ho(M)$ is universal with respect to the property of taking $W$ to isomorphisms. After applying the homotopy relation, this class satisfies a calculus of fractions and this solves the set theoretic issues. The nLab article on calculi of fractions is really great, and you don't need to know anything about model categories to read it. I wrote a bit more on this topic in this answer. Perhaps these references can help you resolve your problem without having to move into the realm of model categories.

If you're okay with moving into the realm of model categories, then I can help you a lot more. Let's resolve your question 1 by assuming the category $C$ in your question is a model category (which I'll denote by $M$ out of habit). Note that this also resolves question 3 because model categories are cocomplete (and complete), and functors between them (so-called Quillen functors) are cocontinuous. Perhaps you're worried because you don't want to assume your map $f$ is a weak equivalence. Well, it turns out the theory has developed a way to deal with that. If you have a map $f$ which is not a weak equivalence (but you wish it were), then you can construct a model category $L_f(M)$ known as the Bousfield Localization of $M$. Here $f$ will be a weak equivalence and will be inverted when you pass to the homotopy category. Also, $M\rightarrow L_f(M)$ is universal with respect to this property.

People have also studied monoidal model categories. A good reference is chapter 4 of Mark Hovey's book ``Model Categories.'' Of note is that you need more than simply a model structure and a monoidal product. There are coherence conditions between these two structures which are covered in Hovey's book. My thesis is concerned with Bousfield localizations of monoidal model categories, and one of my early theorems finds conditions under which $L_f(M)$ is again a monoidal model category. This resolves your question 2. I can email you with a preprint if you want. I had never heard of Day's paper before this moment, but a glance at it suggests my hypothesis is related to his.