Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,364
questions
2
votes
1
answer
436
views
The definition of unitary fusion category
I just come across a definition of the unitary fusion category:
A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have:
We have a Hilbert space structure on each ...
0
votes
1
answer
914
views
Representable functors and direct limits
Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation $\mathcal{F}(-)...
2
votes
1
answer
273
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: $...
0
votes
1
answer
98
views
A confusion about covering flatness
I'm reading this entry on nLab. But I'm confusing with the notion of covering-flatness. More precisely, I meet some trouble when I try to show that the $Sets$-valued flatness is a special case of ...
5
votes
0
answers
256
views
On the Axiom of Choice for Conglomerates and Skeletons
Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and $\...
2
votes
1
answer
381
views
Set theoretic issue of localization of abelian categories
For a small abelian category in which every object is also a set, consider its localization with respect to a Serre subcategory (thus a quotient category), is it true that under this localization ...
16
votes
0
answers
339
views
Specific cases of the tangle hypothesis in terms of "classical" n-categories
As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal $n$-...
10
votes
2
answers
2k
views
Lemma 1 from Beilinson's "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", intuition?
Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: \mathcal{...
7
votes
1
answer
511
views
What are some examples of total derived functors that can't be computed from a functorial replacement?
(Or more generally, what are some examples of Kan extensions which are not pointwise?)
By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...
8
votes
0
answers
252
views
A completeness criterion for $\infty$-categories
We all know that for ordinary categories $\mathscr{C}, \mathscr{D}$ (with $\mathscr{C}$ small) the limit of a functor $F:\mathscr{C} \to \mathscr{D}$, if it exists, can be constructed by using ...
2
votes
1
answer
265
views
Name of rule involving elements of two sets
For sets $X,Y$ and a function $f: X\times Y\to X$, what is the name of the property whereby for all $x\in X$ and $y_1, y_2 \in Y,$
$$f(f(x,y_1), y_2) = f(f(x, y_2),y_1)\qquad?$$
Some of us called it ...
1
vote
1
answer
250
views
About the canonical morphism from $f^{*}f_{*}f^{*}F$ to $f^{*}F$
Assume $X$ and $Y$ are noetherian schemes over $\mathbb{C}$ and there is a proper and faithfully flat morphism $f: X\rightarrow Y$.
Assume the canonical morphism $F\xrightarrow{\sim} f_{*}f^{*}F$ is ...
10
votes
1
answer
431
views
Enriching categories and equivalences
Let $\mathcal{C}$ and $\mathcal{D}$ be two equivalent categories. Furthermore, assume $\mathcal{C}$ is enriched over a monoidal category $(\mathcal{M}, \otimes)$. Can one use the equivalence to ...
9
votes
1
answer
384
views
Conditions on the fusion data of symmetric fusion category
We know that every symmetric fusion category (SFC) gives rise to data
$N^{ij}_k$ that describe the fusion of simple objects:
$i\times j = N^{ij}_k k$, and the data $\theta_i =\pm 1$ that describe the ...
11
votes
1
answer
2k
views
Does pullback in the category of smooth manifolds always exists?
I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist.
Remarks:
1) A pullback in a certain category is defined as ...
37
votes
3
answers
3k
views
What is the point of pointwise Kan extensions?
Recall that a Kan extension is called pointwise if it can be computed by the usual (co)limit formula, or equivalently if it is preserved by (co)representable functors.
I have seen pointwise Kan ...
3
votes
0
answers
114
views
Colimits of n-fold categories
An $n$-fold category is an internal category in the category of $(n-1)$-fold categories (and a $0$-fold category is just a Set).
General results about internal categories assure that the category of $...
1
vote
0
answers
216
views
Family $(X_y,D_y)$ with trivial canonical bundles
Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds
such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...
64
votes
6
answers
9k
views
Are dagger categories truly evil?
Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the ...
11
votes
1
answer
327
views
Reference request: sheaves on the site of d-manifolds
I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...
10
votes
2
answers
467
views
Is this a functor on the category of $C^{*}$ algebras?
The category of $C^{*}$ algebras is denoted by $\mathcal{A}$.
Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, ...
7
votes
1
answer
502
views
How (if at all) does category theory deal with situations where the usual notion of isomorphism isn't right?
This question will potentially rub some people the wrong way; I can't do much about this, except state right here at the outset that this question is motivated by a genuine desire to understand, and ...
8
votes
2
answers
622
views
Is dgCat a category or a 2-category?
Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has ...
11
votes
1
answer
374
views
Elementary consequences of commuting limits and colimits over groups
This is a crosspost of this MSE question.
In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...
10
votes
1
answer
336
views
Relationship between two universal properties of the category of elements?
Let $G: \mathcal{A} \to \mathsf{Set}$ be a functor. Recall that the category of elements $\mathsf{el}G$ is defined by the comma square
$\require{AMScd}$
\begin{CD}
\mathsf{el}G @>!>> \...
6
votes
1
answer
454
views
Universal covering and double cover functors
Initially posted on MSE
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...
15
votes
1
answer
520
views
Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds
Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming ...
6
votes
1
answer
122
views
Characterisations of closed embeddings in $Top_1$?
Let $Top_1$ be the category of topological spaces which are $T_1.$
I am curious as to whether there is a categorical definition of what a closed embedding is in this environment. With a ...
13
votes
2
answers
1k
views
The category of elements, enrichment, and weighted limits
This is a crosspost of this MSE question.
Every so often, when reading notes online or skimming through books, the category of elements and the Grothendieck construction pop up. I don't know anything ...
6
votes
1
answer
663
views
Iterated Homotopy Quotient
If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can ...
9
votes
1
answer
164
views
W-types and inverse image functor
All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.
I would like to know whether the ...
7
votes
1
answer
396
views
For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?
I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at http://ncatlab.org/nlab/...
2
votes
0
answers
169
views
Sufficient criteria for a nerve of a topological category to be good
I know that the following statement is true and I am looking for a reference:
Given a topological category $\mathcal C$ (i.e. morphisms and objects form a space and all maps in the definition of a ...
3
votes
0
answers
193
views
About the definition of lax.functor between tricategories
SUMMARY: Observing that monoids in a monoidal category are identified with lax.functors (with domain 1), I tried to generalize this argument wanting to get a skew-Monoidal-category as (tri)lax....
6
votes
1
answer
321
views
Minimal model (resolution) for a specific colored operad
We know that for the operad $As:=\mathcal{F}(\mu)/(\mu\circ_1\mu-\mu\circ_2\mu)$, its minimal model is the free operad $\mathcal{F}(E)$ where $E=\mathbb{k}\langle\mu_2,\mu_3,\dots,\mu_n,\dots\rangle$ ...
2
votes
0
answers
83
views
Cancellation property of groupoidal cartesian fibrations
I have an issue concerning a property of "left cancellation" for groupoidal cartesian fibrations of ∞-cosmoi (but everything works fine in a 2-category as well).
A 1-cell $p: E \to B$ is called ...
9
votes
1
answer
479
views
Is every locally compactly generated space compactly generated?
[Parse it as (locally compact)ly generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...
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 ...
5
votes
1
answer
323
views
Enriched Cauchy completions and underlying categories
The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] \...
4
votes
0
answers
313
views
Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety
Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$.
I read ...
10
votes
0
answers
271
views
6j symbols with Majorana indices
The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ symbols:...
5
votes
0
answers
70
views
Does the $D$-property have universal objects?
A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...
3
votes
0
answers
300
views
Can such categorical notion of action be formalized?
I'm wondering if it's possible to find an universal construction for a general concept of action for (single-sorted?) finite product sketches, such that one of those is "acting" on the second in the ...
11
votes
3
answers
4k
views
Category theory for Algebraic Geometry
How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?
1
vote
0
answers
125
views
Category-theoretic characterization of zero-dimensional spaces
Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
5
votes
2
answers
407
views
Kan extensions of pseudofunctors
Can anyone suggest a reference for (left) Kan extensions of pseudofunctors?
In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr C$...
20
votes
3
answers
867
views
Brouwer's theorem for the Cauchy reals
Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...
19
votes
0
answers
765
views
A Linear Order from AP Calculus
In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...
19
votes
2
answers
1k
views
A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?
My question is :
Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ?
What I mean by $\infty$-connected ...
4
votes
1
answer
491
views
The "$\infty$"-column in the periodic table of n-categories
A monoid is the same as a category with a single object.
A monoidal category is the same as a bi-category with a single object.
A commutative monoid is the same as a bi-category with a single object ...