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2
votes
1answer
84 views

Tannaka reconstruction: reference request

What is a classical and perhaps even original reference for the following result, often called Tannaka reconstruction? Let $G$ be a group and $R$ be a commutative ring in which $0,1$ are the only ...
1
vote
1answer
30 views

Mac Lane‘s proof of coherence theorem for symmetric monoidal categories.

In [CWM, Ch. XI, §1], Mac Lane prove the coherence theorem for symmetric monoidal categories by assuming the strictness. Thus we have a $n-$ary tensor functor $T$, and the theorem states that any two ...
0
votes
1answer
57 views

References about “algebras over monoids”

Please, could someone point me any reference (with a bit of details) about "algebras over monoids" (in the sense of Schwede & Shipley, Algebras and modules in monoidal model categories)? Thank you ...
13
votes
0answers
135 views

Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
12
votes
0answers
130 views

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 ...
11
votes
0answers
107 views

How to name these “ideals”?

Background. Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of ...
6
votes
0answers
152 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 ...
5
votes
0answers
150 views

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 $F$) ...
4
votes
0answers
33 views

What's the difference between a cartesian monoidal category and a semicartesian monoidal category?

According to ncatlab: In a semicartesian monoidal category, any tensor product of objects $x \otimes y$ comes equipped with morphisms $$ p_x : x \otimes y \to x $$ $$ p_y : x \otimes y \to ...
3
votes
0answers
53 views

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 ...
3
votes
0answers
42 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}: ...
3
votes
0answers
73 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
votes
0answers
66 views

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 ...
2
votes
0answers
107 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 ...
2
votes
0answers
87 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 ...
1
vote
0answers
58 views

Monoidal Categories

For a monoidal category $\mathcal{C}$ with $\alpha_{a,b,c}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c$, $\rho_a : a \otimes 1 \rightarrow a$, and $\lambda_a: 1 \otimes a ...
1
vote
0answers
52 views

Taking the quotient of a free monoidal category modulo a relation

I have recently been advised, in a particular example which is described below, that taking quotients of a category can be complicated, but I don't see where exactly lies the difficulty. More ...
1
vote
0answers
20 views

Symmetries and internal hom in monoidal categories

I need some help in trying to define an object. Suppose we have a biclosed monoidal category (possibly non-symmetric) and two natural isomorphisms: $B \otimes - \simeq - \otimes B, \ \ C \otimes - ...
1
vote
0answers
33 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 ...
1
vote
0answers
69 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 ...
1
vote
0answers
41 views

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 ...
1
vote
0answers
103 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 ...
0
votes
0answers
34 views

Tensor structure on the category of algebras for a strong monad

Proposition 1.4 in "Monads on tensor categories" by I. Moerdijk says: "If $S$ is a Hopf monad on a tensor category $\mathcal{C}$, then the category of $S$-algebras is again a tensor category." ...
0
votes
0answers
52 views

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 ...
0
votes
0answers
57 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 ...
0
votes
0answers
41 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 $X$. ...
0
votes
0answers
118 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 ...
0
votes
0answers
116 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 ...