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Underlying functor of tensor product in a closed and symmetric monoidal category.

I will follow, for terminology and notation, G. M. Kelly, Basic Concepts of Enriched Category Theory. For sake of a self-contained exposition, I will try to write here all the needed concepts. Let ...
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32 views

Module and bimodule categories equivalent to a 2-functor

Let $A$ be a tensor category (i.e. monoidal category). A (right) module category of $A$ is a category $M$ with a coherent action $\mu\colon M\otimes A\to M$. Denote $BA$ be the one-object bicategory ...
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1answer
41 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 ...
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Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
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0answers
27 views

Coherence theorem for symmetric monoidal categories

What's the formal statement for the coherence theorem for symmetric monoidal categories? I've seen there's some notion of permutation around, but I can't get my head around the thing that "all ...
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1answer
124 views

What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
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2answers
123 views

Relations between monoids and modules?

What is the relation between monoids and modules? Are they completely different algebraic structures, or is there a kind of inclusion relation like "elements of a module are also elements of a ...
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2answers
35 views

Distributivity in linear monoidal categories

Let $\mathcal{C}$ be a linear monoidal category, that is a monoidal category (with tensor product $\otimes$) enriched over $\mathbf{Vect}$. Now as far as I can tell the axioms for a linear category ...
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1answer
83 views

Is dualizablility of an object equivalent to tensoring with that object having a left adjoint?

Let $C$ be a closed symmetric monidal category. There is hence an adjunction $$ -\otimes X\colon C\leftrightarrows C\colon Map(X,-) $$ involving the internal Hom $Map(-,-)$ for every object $X$ of $C$ ...
5
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1answer
49 views

Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...
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35 views

Characterization of certain maps in $Hom(A \otimes A^{*}, A \otimes A^{*})$

Let $(M, \otimes, I)$ be a symmetric monoidal category and let $(A, B, \eta, \epsilon)$ be a dual pair in $M$. Consider maps $i_{A}: Hom(A, A) \rightarrow Hom(A \otimes B, A \otimes B)$, $i_{B}: ...
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71 views

At a closed monoidal category, how can I derive a morphism $C^A\times C^B\to C^{A+B}$?

Let $A$, $B$ and $C$ be objects of a closed monoidal category which is also bicartesian closed. How can I derive a morphism $C^A\times C^B\to C^{A+B}$? $(-)\times (-)$ denotes the product, $(-)+(-)$ ...
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0answers
66 views

Every monoidal category is strictly equivalent to a strict monoidal category.

I am reading Joyal and Street's article "braided tensor categories". The following theorem is proved (Theorem 1.2). Let $ C $ be a category. Let $FC$ and $F_sC$ be respectively the free monoidal ...
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0answers
36 views

Recovering Ordered Monoid Operation from the Order

I have a partially ordered set $(X, \preceq)$ with the following properties: $X$ has a minimum. I'll name it $1$. For every $x \in X$, the principal filter ${\uparrow} x$ is order isomorphic to $X$. ...
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63 views

Natural transformation defined by one element

Let $C$ be a self-enriched category, a CCC, and $F : C \to C$ an endofunctor with strength, that is, $F$ comes with a natural transformation $$st_{A,B} : A \times F B \to F (A \times B)$$ such that ...
3
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1answer
72 views

Monoid as a one-object category… for monoidal categories

A monoid can be seen as a one-object category. Is there analogous thing for monoids in a monoidal category $(M, \otimes, I)$? Can I form some kind of one-object category from a monoid in $M$?
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1answer
64 views

Inverse of Braiding in Monoidal Category is Braiding

In Mac Lane's Categories for the Working Mathematician in section $\mathrm{XI}.1$ about symmetric monoidal categories we find that a braided monoidal category $M$ is a monoidal category ...
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1answer
52 views

What is required for a monoidal category to have products/coproducts?

What is required for a monoidal category to have products/coproducts? If it helps the particular category I am interested in also has zero morphisms.
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1answer
53 views

$\textbf{C}$-Monoids and products

i have a question about $\textbf{C}$-Monoids. We can make a new category $\textbf{Mon(C)}$ from the category $\textbf{C}$, namely the category of all $\textbf{C}$-monoids. A $\textbf{C}$-monoid is a ...
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0answers
18 views

A coherence question in rigid monoidal categories

Let $\mathscr{C}$ a monoidal simmetrical closed category and $t_{A, B}: [A, B] \otimes B \to A$, $u_{X, Y}: X \to [Y, X \otimes Y]$ the co-unity and unity of the adjunction $(A \otimes B, C) ...
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63 views

On the definition of algebra .

Let $\mathscr{C}$ a monoidal category with monoidal product $A\circ B$. Is defined the bicategory of bimodules $Mod(\mathscr{C})$ on $\mathscr{C}$ (see [Gray] p. 46). Its objects are ...
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1answer
46 views

On the monoidal product of comodules

Let $\mathcal{V}$ a symmetric monoidal category. A commutative comonoid is a triple $(A, \delta, \epsilon)$ whit $\delta: A \to A\otimes A$, $\epsilon: A\to I$ by the dual of monoid axioms. Given two ...
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2answers
70 views

Understanding associators as natural transformations

Reading Baez and Stay's "Rosetta Stone," and trying to understand the definition of monoidal category on page 12, I read that a monoidal category requires a natural isomorphism called the associator, ...
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Generalization of analytic functors

A functor $F\colon \bf Sets\to Sets$ is said to be analytic if it results from the left Kan extension of a functor $f\colon \mathbf{Bij}(\mathbb N)\to \bf Sets$ (the "species" of the functors $F$) ...
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1answer
72 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
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1answer
129 views

Grothendieck group of a symmetric monoidal category is a lambda ring?

I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a ...
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3answers
194 views

Theory of promonads

I'm led to define a promonad in $\bf D$ as a monoid in the category of endo-profunctors of a category $\bf D$, where the product of two profunctors is their composition as profunctors: $$ F\odot G := ...
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1answer
54 views

Rooted trees morphisms and categories

If I take the monoidal category freely generated by a single object $A$ and a morphism $f: A \otimes A \to A$, I end up with the monoidal category whose morphisms are forests of binary rooted trees. ...
2
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1answer
98 views

Is there a categorical construction of the general linear group?

This question is related to the answer of Qiaochu in this one. Since the object $X=\mathbb{F}_2^2$ generates the category of vector spaces of dimension $2^n$ over $\mathbb{F}_2$, and since we know ...
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116 views

Nonsymmetric monoidal product on $[\mathbb N,\mathbf{Sets}]$

Let $\mathbf P$ be the category with objects the natural numbers and $\hom(m,n)=Sym(n)$ if $n=m$, and the empty set otherwise. It is symmetric monoidal wrt the sum of natural numbers, and has $0$ as a ...
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2answers
195 views

Associativity of Day convolution

I'm trying to follow Day's argument to prove that $[\mathbf C,\mathbf{Sets}]$, where $\bf C$ is symmetric monoidal, is itself symmetric monoidal, but I'm stuck at the very beginning. Is there a way to ...
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0answers
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Do we need braiding to define a monoid structure on product of monoids?

Let $(\mathcal{C}, \otimes, I)$ be a monoidal, symmetric category. It is well-known that in this case, if $M_{1}, M_{2}$ are monoids, there is a natural monoid structure on $M_{1} \otimes M_{1}$. The ...
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1answer
107 views

Reasons for coherence for bi/monoidal categories

Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this ...
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1answer
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Why can I choose to work in a strict monoidal category without loss of generality?

Let $\mathcal A$ be a monoidal category. We know that $\mathcal A$ is monoidally equivalent to a strict monoidal category $\mathcal A^{\mathrm{str}}$. In many books/papers it is assumed without loss ...
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0answers
138 views

Mnemonic device for relationships between Hom and Tensor

Probably this is a stupid question, but nevertheless... Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
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0answers
87 views

Left-modules over a bialgebra form a monoidal category

Let $B = (B, \nabla, \eta, \Delta, \epsilon )$ be a bialgebra over a commutative ring $k$. Let $M$ and $N$ be two left $B$-modules. Then the tensor product $M \otimes_k N$ becomes a left $B$-module ...
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1answer
68 views

Morphisms in a symmetric monoidal closed category.

Let $\mathcal C$ be a symmetric monoidal closed category. This means that every functor $- \otimes B$ has a right adjoint $[B, -]$. Let $I$ be the unit and let $\rho \colon - \otimes I \to 1_{\mathcal ...
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Cartesian monoidal functors

Let $\mathcal{C}$ and $\mathcal{D}$ be categories with finite products, and consider them as monoidal categories in the obvious way. Every functor $\mathcal{C} \to \mathcal{D}$ can be canonically ...
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1answer
73 views

Is the inverse to a monoidal equivalence also monoidal?

Let ${\cal C,D}$ be two categories, and let $$ F:{\cal C} \to {\cal D}, ~~~~~~~~~~~~~~~~~ G:{\cal D} \to {\cal C}, $$ be an equivalence of categories. Let us now further assume that ${\cal C}$ can ...
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2answers
166 views

The abstract definition of commutative monoids

In trying to begin to understand the idea of a $k$-tuply monoidal $n$-category, I'm already a bit stuck on the idea (Baez, nLab) that a commutative monoid can be defined as a monoid object in the ...
5
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1answer
79 views

Coherence Conditions and Strict Monoidal Equivalences

Consider the following: two monoidal categories $({\cal C},\otimes)$, and $({\cal D},\odot)$, and a functor $F:{\cal C} \to {\cal D}$, that gives an equivalence (of ordinary categories) between ${\cal ...
2
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0answers
76 views

Thompson's group F and monoidal categories

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category generated by one object $A$ such that there exists an isomorphism $\alpha: A \otimes A \to A$, then for every object $X ...
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1answer
107 views

Endomorphisms in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category generated by one element $X$ such that $End(X)=G$ where $G$ is a finite group. Is it true that, for any object $A \in \mathcal{C}$, $End(A)$ is ...
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1answer
210 views

What is the categorical perspective on representations of topological groups?

One categorical definition of a group $G$ is that it is a category $C$ with a single object $X$ such that every morphism in the set $C(X,X)$ is invertible, i.e. such that $C(X,X)$ is precisely the ...
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1answer
298 views

Coherence for symmetric monoidal categories

Let $\mathcal{C}$ be a monoidal category. By Mac Lane's coherence theorem for monoidal categories, there are strong monoidal functors $F : \mathcal{C} \to \mathcal{C}_s$ and $G : \mathcal{C}_s \to ...
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What is the decategorification of a triangulated category?

The decategorification of an essentially small category $\mathcal C$ is the set $\lvert\mathcal C\rvert$ of isomorphism classes of $\mathcal C$. If $\mathcal C$ carries additional structure, then so ...
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0answers
107 views

Is push-forward of coherent sheaves a tensor functor?

Given a finite map between two Noetherian schemes $f: X \rightarrow Y$, is $f_*: \operatorname{Coh}{(X)} \rightarrow \operatorname{Coh}{(Y)}$ a tensor functor? If this is not true in general, is it ...
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1answer
93 views

Is every functor monoidal between monoidal categories where monoidal product is interpreted as sum?

Namely, for every functor $F$, is $(F, (A_0, A_1)\mapsto[F(\iota_0(A_0, A_1)), F(\iota_1(A_0, A_1))])$ monoidal? (For reference, $\iota_i(A_0, A_1):A_i\to A_0+A_1$ is an injection of the categorical ...
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1answer
196 views

Reference request: Deligne's reconstruction theorem

I've heard this result referenced a few times on MO now. It is supposed to be a theorem of Deligne that gives some natural conditions under which an (abelian?) tensor category $C$ is the category of ...
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1answer
142 views

May left and right unitors be equal in a monoidal category?

Monoidal category. $\lambda_I = \rho_I : I\otimes I\to I$? If this equality can not be proved, in what categories it is false?