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1answer
75 views

Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?
2
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2answers
148 views

When is the adjoint to a monoidal functor monoidal?

Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is lax monoidal if it is equipped with maps $1_{\mathcal D} \to F(1_{\mathcal C})$ and $F(X) ...
6
votes
1answer
143 views

Rectifying the definition of a closed category

The definition of a closed category I'm using is here. Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal ...
13
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2answers
578 views

Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
6
votes
1answer
215 views

Categorical definition of infinite symmetric product

Let $(C,\otimes,I)$ be a symmetric monoidal category with coequalizers and directed colimits. Fix some object $X$ and morphism $\tau\colon I\to X.$ Using $\tau$ one can construct a sequence of ...
9
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2answers
364 views

A cosmos where coproduct injections are not monic

The injections (coprojections) of a coproduct in a category are very often monomorphisms. For instance, this happens in any extensive category (essentially by definition) and also in any category ...
1
vote
0answers
100 views

group completion of topological monoid

Let $M$ be a topological monoid. The group completion of $M$ is defined as $\Omega BM$, where $BM$ is the classifying space of $M$. In Lecture Notes in Math. 533, The homology of C n+1 spaces, n>=0, ...
8
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2answers
304 views

Reference for an unbiased definition of a symmetric monoidal category

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor $$ \mathcal{C}\times ...
5
votes
1answer
177 views

What is the monoidal equivalent of a locally cartesian closed category?

If a closed monoidal category is the monoidal equivalent of a Cartesian closed category, is there an analogous equivalent for locally cartesian closed categories? Is there a standard terminology or ...
1
vote
0answers
85 views

Does the Boardman-Vogt tensor product of operads commute with their W-construction

I have absolutely no idea whether this is true or not but it could well be useful for me in the future if it is. If we have topological operads $\mathcal{P}$ and $\mathcal{Q}$ and we let $W$ denote ...
4
votes
2answers
268 views

Kan extensions in the $2$-category of monoidal categories

Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of ...
0
votes
0answers
53 views

What is a fusion system/category without duals or an identity

What do you call a fusion system or fusion category without duals or an identity? For fusion systems I am using Definition 3.1 from On Arithmetic Modular Categories. I want to remove axioms (i) and ...
1
vote
0answers
102 views

How does one compute a colimit of monoidal categories?

The question is in the title. I'm also happy to get answers about (your favorite adjective) monoidal categories. Here's a guess: In order to compute a colimit of monoids we can push everything down ...
2
votes
0answers
70 views

Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...
3
votes
0answers
107 views

Proofs in monoidal categories [closed]

I have to do some pretty ugly proofs in monoidal categories. Basically, I have some long identities that I would like to prove. A random example: $$(a\otimes b)\circ (c\otimes d) \circ q = q $$ Are ...
1
vote
1answer
104 views

Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105 Unfortunately I am struggling to make the algorithm work on ...
0
votes
0answers
38 views

How is the monoidal product defined on the functor category between symmetric monoidal dagger cats

I have found a quote in a paper by Abramsky and Heunen If C and D are symmetric monoidal dagger categories, then so is the category [C, D] of functors F : C → D that preserve the dagger. ...
4
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0answers
127 views

When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?

Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal ...
1
vote
1answer
91 views

Lax monoids where only the unit triangle is lax

I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper ...
6
votes
1answer
239 views

In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective? Recall that an object ...
12
votes
2answers
351 views

Counterexample for associativity of smash product

In Section 1.7 of Parametrized Homotopy Theory by May and Sigurdsson it is stated that the smash product of pointed topological spaces is not associative (which is just another hint that ...
7
votes
2answers
277 views

How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?

Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original ...
2
votes
1answer
163 views

Reference for “multi-monoidal categories”

I have attempted to find a definition of a monoidal category which incorporates $n$-fold tensor products instead of just binary tensor products. Definition. A "multi-monoidal category" consists of ...
4
votes
1answer
88 views

Singularity-free isotopies between string diagrams for monoidal categories

Given a monoidal category, it is a theorem of Joyal and Street that an equation between string diagrams is provable from the axioms if and only if there is a recumbent isotopy that relates them. The ...
8
votes
2answers
311 views

Why does a tetracategory with one object, one 1-morphism and one 2-morphism give a symmetric monoidal category

According to the periodic table of k-tuply monoidal n-categories, it should be the case that a tetracategory (= weak 4-category) with one object, one 1-morphism and one 2-morphism is effectively ...
3
votes
3answers
234 views

Tensor product over a monoid in a monoidal category

nLab article on tensor product says: "Given two objects in a monoidal category $(C,\otimes)$ with a right and left action, respectively, of some monoid $A$, their tensor product over $A$ is the ...
2
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0answers
234 views

What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?

I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...
2
votes
1answer
457 views

Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$. Questions: are there definitions of image and kernel for a ...
7
votes
1answer
112 views

Does an equivalence of fusion categories depend on choice of simple objects within isomorphism classes?

Let $C$ be a fusion category with simple objects $X_1,...,X_n $, and let $Y_1,...,Y_n$ be objects with each $Y_i$ isomorphic to $X_i$. Is there a monoidal auto-equivalence $F:C \rightarrow C $ which ...
3
votes
1answer
369 views

Category of modules over commutative monoid in symmetric monoidal category

Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes: In many cases, ...
7
votes
0answers
138 views

When is Rep(U_q(g)) invariant under q -> -q and why?

Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...
5
votes
2answers
281 views

What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
2
votes
0answers
113 views

Pseudomodules, “general coherence theorem”

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...
5
votes
2answers
363 views

A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea. Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
3
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0answers
93 views

symmetric monoidal dagger endofunctor categories

Take an arbitrary category $C$, and consider $End(C)$, any endofunctor category consisting of some or all endofunctors of $C$. I have several related questions: What restrictions must we impose on ...
6
votes
2answers
238 views

String diagrams for bimodules over noncommutative algebras?

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...
2
votes
1answer
189 views

Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T ...
2
votes
1answer
116 views

Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...
3
votes
1answer
143 views

How to construct a free 2-group on a groupoid?

Let G be a groupoid. I'm wondering how to construct the free 2-group on G. By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$ equipped with a functor ...
3
votes
1answer
124 views

Strictifying strong monoidal functors

Let $C_1$ and $C_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C_1 \rightarrow C_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal ...
10
votes
1answer
377 views

Categorical interpretation of disjoint cycle notation for tracing permutations

For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$ A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...
2
votes
1answer
225 views

Inducing a Monoidal Structure using an Equivalence of Categories [closed]

Given an equivalence of categories $C \equiv D$, such that $C$ has a monoidal structure, is it clear that we can use the equivalence to induce a monoidal structure on $D$. Is there a standard ...
11
votes
1answer
165 views

Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories

Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all ...
2
votes
1answer
163 views

When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...
4
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2answers
168 views

symmetric monoidal double categories?

Let me preface this by saying that I don't know much category theory. I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing ...
3
votes
2answers
155 views

Why is a braided left autonomous category also right autonomous?

In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that: Lemma 4.17 ([23, Prop. 7.2]). A braided ...
6
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1answer
193 views

Making additive envelopes of monoidal categories monoidal

I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$. (This is defined as the category with ...
2
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0answers
75 views

A question about braiding represented as pseudofunctors

I just asked a very similar question here (About symmetry, braids, and pseudo-functors.), still now I dont get a answere. I hope to express myself more clearly and correctly this time. Let ...
0
votes
3answers
295 views

Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...
12
votes
1answer
262 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...