Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

learn more… | top users | synonyms

0
votes
1answer
132 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
1answer
72 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 ...
4
votes
0answers
173 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
1answer
170 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 ...
14
votes
0answers
123 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 ...
21
votes
2answers
1k 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: ...
7
votes
1answer
279 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
0answers
134 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
0answers
119 views

For f: X -> Y -> X, what is the name of the property whereby for all x in X and y1, y2 in Y, f(f x y1)y2 = f(f x y2)y1?

My PL group has been discussing this (so we are really professional reject mathematicians, but this is more about the math behind what we are doing). Some of us called it a "generalized ...
1
vote
1answer
179 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 ...
9
votes
1answer
217 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 ...
7
votes
1answer
255 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 ...
7
votes
1answer
456 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 ...
24
votes
3answers
553 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
0answers
90 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
0answers
181 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. ...
36
votes
5answers
5k 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
1answer
253 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
2answers
277 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
1answer
324 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 ...
6
votes
2answers
330 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
1answer
298 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
1answer
201 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 @>!>> ...
3
votes
1answer
164 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 ...
13
votes
1answer
218 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 ...
7
votes
1answer
97 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 ...
8
votes
2answers
301 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 ...
6
votes
1answer
364 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 ...
8
votes
0answers
69 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 ...
5
votes
1answer
124 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 ...
2
votes
0answers
109 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
0answers
93 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 ...
5
votes
1answer
194 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 $E=\mathbb{k}<\mu_2,\mu_3,\dots,\mu_n,\dots>$ is the ...
2
votes
0answers
50 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 ...
8
votes
1answer
219 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 ...
25
votes
2answers
976 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 algebras over $\mathbb{C}$. The equivalence associates to an affine variety its ...
4
votes
1answer
127 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] ...
3
votes
0answers
125 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 ...
7
votes
0answers
166 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 $6j$ ...
5
votes
0answers
64 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 $\{N(x): x\in D\}$ ...
2
votes
0answers
192 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 ...
8
votes
3answers
893 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
0answers
96 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
1answer
137 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 ...
16
votes
3answers
420 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), ...
18
votes
0answers
580 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 ...
4
votes
1answer
269 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 ...
12
votes
2answers
404 views

Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction. Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ ...
3
votes
0answers
76 views

Natural transformations of $A_\infty$-functors (between dg-categories) are “directed homotopies” (reference?)

Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to ...
3
votes
0answers
98 views

Does this notion of “$\mathcal{F}$-digraph” appear in the literature?

By a digraph, I mean a quiver with no multiple edges. So in particular: Loops are okay. An infinite set of vertexes is okay. Furthermore, I will tend to identify each digraph with its underlying ...