15
votes
Accepted
Trace in the category of propositional statements
To start with, I just want to make sure no one gets the impression that the categorical notion of trace was introduced by the paper you linked to; however "semi-famous" it might or might not be, it's ...
- 66.7k
15
votes
Accepted
Inequivalent compact closed symmetric monoidal structures on the same category
A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G \times G \times G \to M$, one can ...
- 53.3k
12
votes
If a $\otimes$-idempotent object has a dual, must it be self-dual?
If $C$ is not assumed to be symmetric, then the answer to questions 1 and 2 is no. Let $p^* \dashv p_* \dashv p^!$ be a fully faithful adjoint triple with $p_* : A \to B$. (For instance, $A= \mathrm{...
- 66.7k
11
votes
Accepted
Why are enriched (co)ends defined like that?
The $\newcommand{\V}{\mathcal{V}}\newcommand{\D}{\mathcal{D}}\newcommand{\C}{\mathcal{C}}$universal property of being an “initial/terminal cowedge” — even strengthened to “$\V$-initial/-terminal” — is ...
- 21.3k
10
votes
Accepted
What are the advantages of various "models" for the motivic stable homotopy category
The injective model structure is monoidal (satisfies the pushout product axiom), see Hornbostel's paper "Localizations in motivic homotopy theory", Thm 1.9 and Lemma 1.10. The projective model ...
- 30.3k
9
votes
Accepted
A result on symmetric closed monoidal categories
This is a special case of:
Let $F\dashv G$ be an adjunction. If there exists an isomorphism $id \cong GF$, then the unit $id \to GF$ is an isomorphism.
Indeed, $[-,A]$ is left adjoint to itself as ...
- 15.8k
8
votes
Accepted
A question about the Tannaka-Krein reconstruction of finite groups
I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a ...
- 28.1k
8
votes
Accepted
How is the morphism of composition in the enriched category of modules constructed?
I’ll change notation slightly, writing $\newcommand{\V}{\mathcal{V}}\V(A,B)$ and $\newcommand{\AV}{{}_{A}\! \V}\AV(M,N)$ respectively for the $\V$-hom-objects and $A$-module-homomorphism objects ...
- 21.3k
7
votes
Inequivalent compact closed symmetric monoidal structures on the same category
The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to ...
- 30.3k
7
votes
Krein's theorem in the Tannaka-Krein duality
In the comments you ask:
The question is the following: if I have an "abstract" category (or even a subcategory in the category of vector spaces), how can I understand that this is the category of ...
- 118k
7
votes
Mac Lane's proof of coherence for symmetric monoidal categories
I think that this is a great question, since it is quite common to just say at this point "each relation between the transpositions follows from the relations in the group presentation of the ...
- 63.1k
7
votes
Accepted
Symmetric monoidal structure on algebras
This is worked out in Higher Algebra, example 3.2.4.4.
Concretely, $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})^\otimes$ is defined as follows: it is the simplicial set over $\mathrm{Fin}_\ast$ such that ...
- 16.5k
6
votes
Accepted
Symmetric monoidal category with trivial switch morphisms
Coincidentally, terminology for such categories has been introduced very recently:
John C. Baez and Jade Master. Open Petri Nets. Nov 2018. arXiv:1808.05415
More precisely, the authors refer to a ...
- 3,393
6
votes
A question about the Tannaka-Krein reconstruction of finite groups
Although you already have one sufficient answer, I thought I would add here that Etingof and Gelaki, "Isocategorical groups" considered exactly this question. For finite groups, they gave a complete ...
- 3,230
6
votes
Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent to a simplicial such?
Yes, this is proved in https://arxiv.org/abs/1506.01475 by Thomas Nikolaus and Steffen Sagave.
If you need a specific monoidal left Quillen equivalence, you can upgrade Dugger's result to the ...
- 37.8k
6
votes
Accepted
If a $\otimes$-idempotent object has a dual, must it be self-dual?
EDIT: It looks like I was too hasty to delete this answer. I think I have fixed the gap that Anton noted in the comments and in his answer. As Anton pointed out in his answer, the trace loop that I ...
- 1,593
6
votes
Accepted
Adding inverses to a symmetric monoidal category (Reference?)
Maybe you could be interested in the recent PhD thesis:
Une introduction élémentaire au 2-groupe de Grothendieck
by C. Drugmand, 2016, UCL, Louvain-la-Neuve.
http://hdl.handle.net/2078.1/176774
6
votes
Accepted
Formal completion of a quotient stack
I will assume $X$ is smooth for simplicity, but it is probably not needed. Given the stack $X/G$, there are two completions one may consider:
Completing along $\mathrm{B}G\rightarrow X/G$ one obtains ...
- 3,357
6
votes
Accepted
Functors that preserve monoids
A simple class of counterexamples can be found among thin small strict monoidal categories, i.e. preordered monoids. These are just sets equipped with a preorder and a monoid structure such that the ...
- 63.1k
6
votes
Accepted
Base change for module categories? ($E_\infty$-modules in $\mathrm{Cat}$)
This is indeed the case, and follows from various things in Higher Algebra.
The $\infty$-category of $\infty$-categories $\mathrm{Cat}_\infty$ is presentable, and moreover the cartesian product on it ...
- 409
5
votes
Accepted
Monoidal tensor product which preserves directed limits
Public Service Announcement!
It's very confusing -- I'd daresay incorrect -- to say "directed limit" to mean "limit indexed by a cofiltered diagram". Actually, historically the term "directed limit" ...
- 63.9k
5
votes
Accepted
Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper
I think that the best thing in this direction is the paper "Permutative categories, multicategories and algebraic K-theory" by Elmendorff and Mandell.
I have only skimmed this so I may well not be ...
- 56.9k
5
votes
Accepted
Tensor product of unit and co-unit in a closed compact category
The definition of compact closed category merely says that for every object $A$ there exists an object $A^\star$ and a unit $\eta_A$ and counit $\varepsilon_A$ satisfying the snake equations. That is, ...
- 3,937
5
votes
Accepted
Day convolution for bimonoidal categories
Regarding Q2: probably there is a way to avoid going deep into coherence conditions: instead of proving by hand the equivalence between promonoidal structures on $C$ and biclosed monoidal structures ...
- 13.6k
5
votes
Accepted
The change-of-monoid adjunction between categories of modules induced by a morphism of monoids
Certainly overkill, but this result is a special case of `change of operads' since the given morphism of monoids induces a morphism from the operad $O$ for left $A$-modules to the operad $P$ for left $...
- 30.3k
5
votes
Accepted
Box tensor product in the correspondence category
The point is that an arbitrary correspondence can be factored as a composition of pullback, tensor, and compactly-supported pushforward correspondences.
For each $i' \in f^{-1}(j)$ we have a pullback ...
- 148k
4
votes
Symmetric monoidal category with trivial switch morphisms
In my thesis I have named objects $x$ whose switch map $x \otimes x \to x \otimes x$ is the identity symtrivial (since I could not find any term in the literature). It was then used by others as well, ...
- 63.1k
4
votes
The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors
I am not aware of Kock's works.
Nevertheless Kelly provides the definition of its tensor product in the next page: it defines its tensor product $\mathcal A \otimes_{\mathcal F} \mathcal B$ as the $\...
- 3,349
4
votes
If a $\otimes$-idempotent object has a dual, must it be self-dual?
Ok, I have no idea how to fix the invariant of string diagrams, but I have an explicit counterexample. Consider the 2-category of distributors $Dist$: its objects are small categories, its morphisms ...
- 4,780
4
votes
Inequivalent compact closed symmetric monoidal structures on the same category
The large class of examples I have in mind, though I am not sure if it meets your compactness requirement (definition?) are the Lax tensor products on $n$-Cat. It has been constructed in the cases $n=...
- 19.6k
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