**6**

votes

**1**answer

255 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 ...

**6**

votes

**0**answers

112 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

**0**answers

112 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

**1**answer

172 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

**1**answer

207 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

**1**answer

225 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 ...

**6**

votes

**1**answer

361 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

**3**answers

479 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 ...

**2**

votes

**0**answers

56 views

### Colimits in 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

165 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. ...

**34**

votes

**4**answers

4k 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

242 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 ...

**9**

votes

**2**answers

251 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

292 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

**2**answers

303 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

282 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

192 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

**1**answer

149 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

**1**answer

182 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

**1**answer

95 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

**2**answers

257 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

**1**answer

339 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

**0**answers

62 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

**1**answer

115 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

**0**answers

88 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 ...

**2**

votes

**0**answers

76 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 ...

**4**

votes

**1**answer

145 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

**0**answers

44 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

**1**answer

197 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 ...

**24**

votes

**2**answers

868 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

**1**answer

113 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

**0**answers

106 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

**0**answers

151 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 $6j$ ...

**4**

votes

**0**answers

57 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

**0**answers

150 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

**3**answers

749 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

89 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 ...

**4**

votes

**1**answer

128 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

**3**answers

397 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

**0**answers

559 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

**1**answer

234 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

**2**answers

388 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

**0**answers

68 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 ...

**0**

votes

**0**answers

71 views

### Generating a series representation for the inverse of the operator $f(f)$

I was considering the following problem:
Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...

**3**

votes

**0**answers

93 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 ...

**4**

votes

**2**answers

279 views

### Algebras for probability monad

What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by
$$
DX = \left\{ p \in [0,1]^X \ ...

**-1**

votes

**1**answer

91 views

### Categorical product of graphs and chromatic number

Let $(G_i)_{i\in I}$ denote a family of simple, undirected graphs (finite or infinite). Let $\prod_{i\in I}G_i$ denote their categorical product. Why do we have the inequality
$$\chi(\prod_{i\in ...

**2**

votes

**0**answers

99 views

### About a (new?) definition of transformation (anti.transformation) as a link between natural and dinatural transformations

This is not a hard topic, but I post here as "reference request" or because elementary aspects (but not previously vocalized) can be interesting too for researchers.
Given $F: \mathscr{A}\to ...

**2**

votes

**1**answer

86 views

### Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors
$F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the ...

**10**

votes

**2**answers

591 views

### What's a good introduction to category theory for someone doing analysis?

I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory.
Any recommendations?