ct.category theory – Does the monoidal structure on semisimplicial sets preserve fibrant objects?

The category of semisimplicial sets has the structure of a monoidal category by the geometric product $otimes$, see for example Rourke and Sanderson’s paper ‘$Delta$-sets I: Homotopy Theory’. This geometric product has the property that for the left adjoint $L$ of the forgetful functor $U$ from simplicial sets to semisimplicial sets, we have $L(Xotimes Y)cong L(X)times L(Y)$. Moreover, it is discussed in Sattlers paper on constructive homotopy theory that the forgetful functor preserves Kan complexes, and that the right adjoint $R$ of the forgetful functor preserves Kan complexes. I was hoping to deduce from this that given two semisimplicial Kan complexes $X$ and $Y$, their geometric product $Xotimes Y$ would also be a tensor product, but I failed to do so. Does anyone have a proof, reference or counterexample?

Thanks!

ct.category theory – Is there a monoidal analogue of equalizers?

There are three different kinds of finite limits in categories: terminal objects, binary products, and equalizers. In a category $C$, these define functors $1_{C},times,mathrm{Eq}colonmathrm{Fun}(I,C)to C$ where $I=emptyset,{bullet bullet}$, and ${bulletrightrightarrowsbullet}$ respectively.

Monoidal categories generalise the first two in that we now have a functor $1_{C}$ from $mathcal{C}^{emptyset}=*$ to $mathcal{C}$ and a functor $otimes_C$ from $C^{{bullet bullet}}=C$ to $C$, i.e. functors
begin{align*}
1_C &colon * to mathcal{C}\
otimes_C &colon mathcal{C}timesmathcal{C} to mathcal{C}
end{align*}

together with associativity and unitality natural isomorphisms satisfying compatibility conditions.

What about equalizers? Has the notion of a category $C$ equipped with a functor $rm{Eq}colonrm{Fun}({bulletrightrightarrowsbullet},C)to C$, a unit functor, and unitality/associativity natural isomorphisms satisfying coherence conditions been studied before? Moreover, are there any examples of such structures “found in nature”?

at.algebraic topology – Uses for (Framed) E2 algebras twisted by braided monoidal structure

$newcommand{C}{mathcal{C}}$ $newcommand{g}{mathfrak{g}}$
If $C$ is a monoidal category (not necessarily a symmetric monoidal category), it’s possible to define the notion of an algebra object $A$ in $C$, with multiplication operations $$A^{otimes n} (:= Aotimes_C Aotimes_C cdotsotimes_C A)to A.$$

Similarly, if $C$ is a braided monoidal category (resp., a ribbon category), one can define a notion of $E_2$ DG algebra $A$ (resp., framed $E_2$ DG algebra $A$) “twisted” by $C$, consisting of operations $A^{otimes n}to A$ compatible with braiding. (Note: I actually don’t know a reference for this, but it follows from standard “homotopy field theory” arguments involving the Ran space.)

In particular, if $C$ is a braided monoidal (or ribbon) category coming from an associator on a Lie algebra $g$ (with choice of Casimir), there is a whole category of “associator-twisted” $g$-equivariant $E_2$ (resp., framed $E_2$) algebras.

My question is whether algebras of this type have been encountered before. They feel very CFT-ish, and so I’m particularly curious about physics and knot theory applications. In particular, the framed variant should gove some kind of derived 2D TQFT-style invariants.

Any references would be useful. Thanks!

ct.category theory – Yoneda Lemma for monoidal functors

Let $(mathcal V,otimes,I)$ be a closed symmetric monoidal category, and let $mathcal C$ be a $mathcal V$-enriched category. The (weak) enriched Yoneda Lemma gives us a nice description of the set $Hom(F,G)$ of natural transformations between two $mathcal V$-enriched functors $F,Gcolonmathcal Ctomathcal V$ when $F$ is representable: it is in bijection with the set of maps $I to G(Y)$ in $mathcal V$ where $Y$ is an object representing $F$.

Now suppose that $mathcal C$ itself is a monoidal category, and that our two functors $F$ and $G$ are monoidal functors. Is there a similarly nice description of the set $Hom^otimes(F,G)$ of monoidal natural transformations between the functors, again in the case that $F$ is representable?

My suspicion is that the following might be true (possibly with extra conditions on $mathcal C$). The fact that $F$ is a (lax) monoidal functor induces the structure of a comonoid on the representing object $Y$, and so there is an induced comonoid structure on $G(Y)$. My guess would be that monoidal natural transformations $Fto G$ are in bijection with morphisms of comonoids $Ito G(Y)$, but I can’t prove this in general. (I can prove this in the case that $mathcal V$ is the category of sets with cartesian product, but only for trivial reasons: every map in Set is a morphism of comonoids, and every natural transformation between monoidal Set-valued functors is a monoidal transformation.)

I would be especially interested in any references where this might be addressed.

ct.category theory – Freeness of the action of the ground monoid in a monoidal category

Let $(mathcal{C}, otimes , 1)$ be a monoidal category, and let $mathrm{End}_{mathcal{C}} (1)$ be the ground monoid of $mathcal{C}$ – which is a commutative monoid. If $r_X : X otimes 1 to X$ denotes the right unit constrain of $mathcal{C}$, then $f cdot alpha := r_Y (f otimes alpha) r_X^{-1}$ defines a right action on $hom_{mathcal{C}} (X;Y)$.

Question 1. When is this action free? That is, under what conditions $f cdot alpha = f$ implies $alpha = mathrm{id}_1$?

Question 2. Is it possible to loosen the conditions on $hom_{mathcal{C}} (1;Y)$?

For instance, in the category of $k$-modules this action is free when $k$ is a field, but for a general ring there is torsion and this doesn’t have to be true.

dg.differential geometry – The Swan–Serre theorem as a monoidal equivalence

Let $X$ be a compact Hausdorff The well-known Swan–Serre theorem gives an equivalence between the continuous vector bundles over a compact Hausdorff space $X$, and finitely-generated projective $C(X)$-modules. Both the category of vector bundles and the category of projective modules have evident monoidal structures. With respect to these structures, is the Swan–Serre equivalence a monoidal equivalence? I would guess that this is the case but I cannot find a reference.

Symmetric monoidal structure(s) on the $infty$-category of dg-categories

Let $k$ be a commutative ring with $1$, and let $mathsf{dgCat}_k$ be the category of $k$-linear dg-categories, as defined in (1, Section 2). We may equip $mathsf{dgCat}_k$ with the Morita model structure (2, Théorème 2.27), which we will denote by $mathsf{dgCat}_k^textrm{Mor}$.

The category $mathsf{dgCat}_k$ is symmetric monoidal: if $mathcal{C},mathcal{D}inoperatorname{Obj}(mathsf{dgCat}_k)$, then we define $mathcal{C}otimes_{mathsf{dgCat}_k}mathcal{D}$ to be the $k$-linear dg-category with objects
$$
operatorname{Obj}(mathcal{C}otimes_{mathsf{dgCat}_k}mathcal{D}) := operatorname{Obj}(mathcal{C})timesoperatorname{Obj}(mathcal{D})
$$

and morphism complexes
$$
left(mathcal{C}otimes_{mathsf{dgCat}_k}mathcal{D}right)((X,Y),(X’,Y’))_* := mathcal{C}(X,X’)_*otimes_k mathcal{D}(Y,Y’)_*.
$$

While the tensor product defined above does not respect the Morita model category structure$^1$, we may derive it to obtain a symmetric monoidal model category structure on $mathsf{dgCat}_k^{textrm{Mor}}$, which we will denote by $left(mathsf{dgCat}_k^textrm{Mor},otimes_{mathsf{dgCat}_k^textrm{Mor}}^Lright)$ (see (2, Remarque 2.40)). Explicitly,
$$
mathcal{C}otimes_{mathsf{dgCat}_k^textrm{Mor}}^Lmathcal{D} simeq Q(mathcal{C})otimes_{mathsf{dgCat}_k}mathcal{D},
$$

where $Q$ is a cofibrant replacement functor for the $mathsf{dgCat}_k^{textrm{Mor}}$.

Cohn has shown (3, Corollary 5.5) that the underlying $infty$-category as defined in (4, Definition 1.3.4.15)$^2$ of $mathsf{dgCat}_k^textrm{Mor}$ is equivalent to the $infty$-category of small idempotent-complete $k$-linear stable $infty$-categories:
$$
N(mathsf{dgCat}_k^textrm{Mor})(W^{-1})simeqoperatorname{Mod}_{operatorname{Perf}(Hk)}((mathcal{Cat}_infty^textrm{perf})^otimes).
$$

In fact (3, Corollary 5.7), it is also equivalent to the $infty$-category of compactly-generated presentable $k$-linear stable $infty$-categories with functors that preserve colimits and compact objects:
$$
N(mathsf{dgCat}_k^textrm{Mor})(W^{-1})simeqoperatorname{Mod}_{Hktextrm{-}operatorname{Mod}}((mathcal{Pr}_{textrm{st},omega}^L)^otimes).
$$

We also have symmetric monoidal structures on both these $infty$-categories. The Lurie tensor product (4, Proposition 4.8.1.15) induces a symmetric monoidal structure on $operatorname{Mod}_{operatorname{Perf}(Hk)}((mathcal{Cat}_infty^textrm{perf})^otimes)$ and $operatorname{Mod}_{Hktextrm{-}operatorname{Mod}}((mathcal{Pr}_{textrm{st},omega}^L)^otimes)$ (I believe this follows from (4, Theorem 3.3.3.9), at least in the case of $operatorname{Mod}_{Hktextrm{-}operatorname{Mod}}((mathcal{Pr}_{textrm{st},omega}^L)^otimes)$), and the symmetric monoidal structure $otimes^L_{mathsf{dgCat}_k^textrm{Mor}}$ induces a symmetric monoidal structure on $N(mathsf{dgCat}_k^textrm{Mor})(W^{-1})$ (see (4, Example 4.1.7.6)).

My question: is the equivalence above an equivalence of symmetric monoidal $infty$-categories? In other words, does the symmetric monoidal model category structure on $mathsf{dgCat}_k^textrm{Mor}$ induce the expected symmetric monoidal structure given by the Lurie tensor product on the underlying $infty$-category of dg-categories? If the symmetric monoidal structures do not coincide, what is the relationship between them (if any)?


Footnotes.

  1. The linked MO question deals with the Dwyer-Kan model structure on $mathsf{dgCat}_k$ (see (1, Section 2) and (2, Théorème 1.8)), but the issue is present in the Morita model structure as well.
  2. It appears Cohn does not restrict to cofibrant objects as in the cited definition, but (4, Remark 1.3.4.16) implies that this is not a problem, as the Morita model structure on $mathsf{dgCat}_k$ admits functorial fibrant-cofibrant factorizations of morphisms.

This question is very much related to my question 2 here. In some sense, both are about the same underlying question/confusion, but I wanted to ask a more focused question about the symmetric monoidal structures in particular. I also hope that the (perhaps excessive) citations will be useful to anyone attempting to make their way through this literature and find precise definitions and references for the first time.


(1) Toën, B. The homotopy theory of dg-categories and derived Morita theory.

(2) Tabuada, G. Théorie homotopique des DG categories.

(3) Cohn, L. Differential Graded Categories are k-linear Stable Infinity Categories.

(4) Lurie, J. Higher Algebra. (September 18, 2017 version).

homological algebra – The derived $infty$-category of sheaves on a site is closed symmetric monoidal

Let $X$ be a quasicompact semiseparated scheme. I am trying to recover the (closed) symmetric monoidal structure on $mathcal{D}(mathrm{QCoh}(X))$, the derived $infty$-category of quasicoherent sheaves on $X$. As such I would like to use the flat model structure on $mathrm{Ch}(mathrm{QCoh}(X))$ from Gillespie’s Kaplansky classes and derived categories and conclude with Lurie, Higher algebra, 4.1.7.6 and 4.1.7.10., which say that the underlying $infty$-category of a simplicial symmetric monoidal model category is symmetric monoidal. But, as far as I understand $mathrm{Ch}(mathrm{QCoh}(X))$ is not simplicial. Is there a way to recover the result ? I would be more interested in references which I could cite.

In the case of $X=mathrm{Spec}(R)$ with $R$ a commutative ring, there might be a workaround going as : $mathcal{D}(R)$ identifies with the $infty$-category of $HR$-module spectra so is symmetric monoidal because the $infty$-category of spectra is ; but I must admit that this sort of argument is too advanced for me, and I have yet to hunt for precise references of those statements. Similarly I thik section 2 of Lurie’s Derived algebraic geometry VII should do the trick in general. The main problem I have with that reference is that I don’t understand if it does correctly specialize to my case.

ct.category theory – In a rigid monoidal category, why is $W^*otimes V^*$ a left dual of $V otimes W$?

My approach: We want to produce a coevaluation map $c: mathbf{1} rightarrow (V otimes W) otimes (W^* otimes V^*)$ and an evaluation map $e : (W^* otimes V^*) otimes (V otimes W) rightarrow mathbf{1} $ such that the maps

$$r_{Votimes W} circ (1_{Votimes W} otimes e) circ a_{Votimes W, W^* otimes V^*, Votimes W} circ cotimes 1_{Votimes W} circ l_{Votimes W}^{-1}$$

$$l_{W^*otimes V^*} circ (eotimes 1_{W^*otimes V^*} ) circ a_{W^*otimes V^*, V otimes W, W^*otimes V^*}^{-1} circ 1_{W^*otimes V^*}otimes c circ r_{W^*otimes V^*}^{-1}$$

are the identity maps on $V otimes W$ and $W^* otimes V^*$, respectively. Naturally one would want to use the maps $e_V, c_V, e_W, c_W$, so I suggest the following definition of $e$, suppressing the associativity constraint:

$$ e = W^* otimes V^* otimes V otimes W xrightarrow{1_W^* otimes e_V otimes 1_V^*} W^*otimes mathbf{1} otimes W cong W^* otimes W xrightarrow{ev_W} mathbf{1}$$

and similarly

$$ c = mathbf{1} xrightarrow{c_V} V^*otimes V cong (V^* otimes 1) otimes V xrightarrow{1_{V^*} otimes e_W otimes 1_V } V otimes W otimes W^* otimes V^*$$

It can be checked that these give $W^* otimes V^*$ the structure of a left dual by verifying the definition. Then, since all left duals are unique up to unique isomorphism, it follows that $(V otimes W)^* cong W^*otimes V^*$.