All Questions
Tagged with duality ct.category-theory
48 questions
5
votes
1
answer
109
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Duals and direct summands in an abelian monoidal category
This question may be seen as a continuation of Duals and sub-objects in a monoidal category.
In an abelian monoidal category, i.e. an abelian category with biadditive monoidal product, if $X \oplus Y$ ...
- 1,484
3
votes
2
answers
129
views
Duals and sub-objects in a monoidal category
Consider $\mathcal{M}$ a monoidal category. Let $V$ be an object that admits a left/right dual. If $U$ is a subject of $V$ then does it also admit a left right dual?
2
votes
1
answer
178
views
In a monoidal category with duals is the coevaluation map determined by the evaluation?
For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...
4
votes
0
answers
149
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Examples of $\ast$-autonomous $(\infty,1)$-categories
A $\ast$-autonomous category is a biclosed monoidal category together with a dualizing object. An object $\bot$ in a biclosed monoidal category $(\mathcal{C},\otimes)$ with left internal hom $[-,-]$ ...
- 764
3
votes
0
answers
90
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Tensor product of functors, central Hopf monad and star-autonomy
Setting.
Let $\mathcal{C}$ be a category and $(\mathcal{V},\otimes,I, \multimap)$ a (symmetric) closed monoidal category. Let $F:\mathcal{C}\rightarrow \mathcal{V}$ be a functor and $X\in \mathcal{V}$ ...
- 764
10
votes
1
answer
603
views
Isbell Duality and Dualizing Scheme Objects
I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\...
- 223
10
votes
0
answers
534
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Isbell duality between algebras and sheaves
nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...
- 1,347
4
votes
1
answer
222
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$\ast$-autonomous categories with non-invertible dualizing object?
1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:
A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...
- 764
4
votes
0
answers
180
views
Spanier-Whitehead dual of space of natural transformations
Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).
...
1
vote
1
answer
286
views
Product-coproduct duality
Let $T$ be a set, $R$ be a ring with $1$ and $B, S_t$ be $R$-modules $\forall t \in T$
My task is to state and prove the dual to the following statement:
Given momomorphisms $j_t: S_t \rightarrow B$. ...
- 13
4
votes
0
answers
150
views
How does Gabriel–Ulmer duality extend to (limit, colimit) sketches?
$\newcommand\Sketch{\mathit{Sketch}}\newcommand\Set{\mathit{Set}}
\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\Colim{Colim}\DeclareMathOperator\Mod{Mod}\newcommand\mod{\operatorname{mod}}\...
- 64k
8
votes
1
answer
1k
views
Why aren‘t op and co switched?
When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^...
- 355
12
votes
2
answers
409
views
Reference for free symmetric monoidal categories with duals on symmetric monoidal categories
The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons.
In ...
- 37.8k
4
votes
1
answer
114
views
Order of factors for unit/counit of duals
While working through Categorical quantum mechanics by Abramsky and Coecke, I noticed that their definition of the order of factors for the unit and counit of a dualizable object disagree with the one ...
- 10.1k
13
votes
0
answers
810
views
Reference request for a complete and formal Duality Principle in category theory
Most textbooks on category theory only sketch the meaning of the Duality Principle. But even when they do it more formally, I have only seen a version so far which concerns the language of a (single) ...
- 63.1k
5
votes
1
answer
338
views
Does rigidity imply a unique dualizing functor?
Let $\mathcal{C}$ be a rigid, monoidal category. Can I talk about $\mathcal{C}$ as having a unique, well-defined, dualizing functor (i.e. one that maps objects and morphisms onto their respective ...
- 1,389
38
votes
2
answers
1k
views
What are all the natural maps between iterated duals of vector spaces, and equations between these?
Fix a field. Given a vector space $V$ it has a dual $V^\ast$, which has its own dual $V^{\ast \ast}$, which has its own dual $V^{\ast \ast \ast}$, and so on ad infinitum.
There are lots of natural ...
- 22.3k
5
votes
2
answers
193
views
Category with binary biproducts but no zero morphism
Is there a category with binary biproducts but no zero morphism?
I'm wondering if the definition of biproducts as objects that are simultaneously products and coproducts that obey some identities on ...
- 10.1k
3
votes
0
answers
162
views
Reference for duality inducing bijections between subobjects and quotients?
I'm not the most category-theoretic person but I have run into the following statement in my work. Suppose we have a category duality $\mathcal F:\mathcal C\to\mathcal D$. For my needs you can ...
- 3,513
35
votes
12
answers
3k
views
No canonical isomorphism [duplicate]
I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
9
votes
1
answer
306
views
Origin and context of adjunctions inducing equivalences between full subcategories
The following is well-known.
Theorem. Let $F\dashv U$ be a pair of adjoint functors
$$F\colon \mathcal C\to \mathcal D, \qquad U\colon \mathcal D\to\mathcal C$$
with unit $(\eta_A\colon A\to U(F(A)))_{...
- 111
16
votes
1
answer
1k
views
What is a module over a Boolean ring?
Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
- 64k
8
votes
0
answers
451
views
Does any 'logical' theory have a bounded ∞-pretopos as syntactic category?
Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order ...
- 5,094
12
votes
1
answer
266
views
Closed embeddings of monoidal categories in *-autonomous ones
It's often very convenient for objects in a monoidal category to have duals. Hence, it's natural to wonder whether an arbitrary monoidal category can be embedded in one where all objects have duals. ...
- 66.8k
9
votes
0
answers
217
views
Point-free topology, but with $\sigma$-algebras instead of spaces
I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...
user30211
7
votes
1
answer
422
views
Dualizable presheaves with respect to Day convolution
This question was posted on MSE and got very little attention, so I'm also posting it here.
Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...
- 3,019
11
votes
1
answer
314
views
What do you call $C$ if $[D,C] = D^\vee \otimes C$ for all $D$?
This is different from $C$ being dualizable ($[C,D] = C^\vee \otimes D$). (EDIT: It turns out to be the same -- see Mike Shulman's answer!) But for example, if $C$ is a locally free sheaf of finite ...
- 64k
13
votes
1
answer
624
views
Gabriel-Ulmer duality for $\infty$-categories
Gabriel-Ulmer duality states that 2-categories $\mathrm{Lex}$ (of small finitely complete categories and functors preserving finite limits) and $\mathrm{LFP}$ (of locally finitely presentable ...
- 4,459
12
votes
1
answer
591
views
Uniqueness of dualizing objects
One definition of (symmetric) star-autonomous category is as a closed symmetric monoidal category $(C,\otimes,I,\multimap)$ equipped with an object $\bot$ such that all double-dualization maps $A \to (...
- 66.8k
20
votes
5
answers
965
views
If a $\otimes$-idempotent object has a dual, must it be self-dual?
Let $C$ be a symmetric monoidal category.
Recall that a dual for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the ...
- 64k
1
vote
0
answers
190
views
Is the category of profunctors $Prof(A,B)$ equivalent to $Prof(B,A)^{op}$?
$\def\Prof{\mathsf{Prof}}\def\Set{\mathsf{Set}}\def\tobar{\mathrel{\mkern3mu \vcenter{\hbox{$\scriptscriptstyle+$}}\mkern-12mu{\to}}}$Let $A$ and $B$ be categories. Define a profunctor $A\tobar B$ to ...
- 474
8
votes
1
answer
631
views
Are there other dualities on finite vector spaces besides the canonical one?
Let $\text{FinVec}$ denote the category of finite dimensional vector spaces over some field $k$, and let $F:\text{FinVec}\to \text{FinVec}$ be a contravariant functor such that $F^2$ is naturally ...
- 5,343
2
votes
1
answer
144
views
Associator of the "dual" monoidal structure of a Grothendieck--Verdier Category
In A duality formalism in the spirit of Grothendieck and Verdier, Boyarchenko and Drinfeld consider a monoidal category $(\mathcal{M}, \otimes, \mathbf{1})$ together with an object $K \in \mathcal{M}$ ...
- 95
18
votes
2
answers
731
views
What categorical property of monoidal categories picks out the ones with duals?
Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...
- 54.6k
2
votes
1
answer
299
views
Creating Duals in A Category
Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: $...
- 5,405
49
votes
4
answers
4k
views
Why is there a duality between spaces and commutative algebras?
1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety ...
- 9,491
2
votes
0
answers
93
views
Do copairings provide dualities in derived categories?
Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...
- 54.6k
3
votes
1
answer
320
views
Profinite completion of a partial order
In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
- 161
6
votes
3
answers
1k
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opposite category
In the 2-category Cat of small categories, for each category $C$ (an object of Cat) there is also the dual category (I dare not write "dual object") $C^{op}$.
Is ${op}$ the instance in Cat of a more ...
- 476
1
vote
3
answers
450
views
Smooth affine algebras are Calabi-Yau
Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
- 27
8
votes
3
answers
962
views
Do any Stone-like dualities have some self-dualities hidden inside them?
This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
- 18.4k
12
votes
1
answer
311
views
Looking for concrete description of a category derived from abelian groups
The category of abelian groups $\mathsf{Ab}$ is the $\mathcal{Ind}$-completion of the full subcategory of finitely presentable abelian groups $\mathsf{Ab}_{fp}$. This is not so special, since the ...
- 1,291
16
votes
1
answer
976
views
Which categories are the categories of models of a Lawvere theory?
Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...
- 118k
9
votes
0
answers
322
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Twisted duality in a symmetric monoidal category
I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples?
Definition. Let $\mathcal{C}$ be a ...
- 63.1k
8
votes
4
answers
734
views
Why are possibility and necessity dual?
Hello,
Recently, I'm studying modal logic for my master's thesis, and my research background is category theory.
So, I naturally have a question that why it is said that necessity (box) and ...
3
votes
1
answer
845
views
A category being self-dual vs. it being a groupoid
What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference?
To go from a self-duality $F$ on a ...
- 291
4
votes
1
answer
197
views
Show that duality functor is anti-monoidal
Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{U,V}: U^*\otimes V^*\to (V\otimes U)^*$ be the canonical isomorphism for every objects $...
- 41
3
votes
1
answer
2k
views
a “self-dual” adjunction
Is there a name for $(U,\eta)$ such that $(\eta, \eta^{op}):U^{op}\dashv U$ (is an adjunction). To clarify — $C:category$, $(I,I^{op})$ is the contravariant isomorphism with $I:C^{op}\to C$, $U:C^{op}\...
- 530