**2**

votes

**0**answers

37 views

### Can I combine the category of Drinfeld modules and the category of the base O_S

I am learning about Drinfeld modules,T-modules,...They are said to be analogues of elliptic curves, abelian varieties,...
Let K be a finite extension of k = Frac(A), and $O_K$ the integral closure of ...

**1**

vote

**0**answers

81 views

### Decomposable elements in cohomology ring

I feel this must be well known, but I don't have enough knowledge on the matter. So, if there is a literature on this, then please provide a reference.
Let $E$ be a ring spectrum, and $X$ a pointed ...

**4**

votes

**0**answers

134 views

### Categorification of covering morphisms

Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct ...

**0**

votes

**0**answers

86 views

### “Sheaf” on the nerve of a category

So I have the nerve of a category and want to communicate information about the sets at each level in the simplicial set (nerve). Is there a way to regard the nerve as a category itself? Then one ...

**0**

votes

**0**answers

87 views

### Is the category Cat complete? [closed]

Let $Cat$ denote the 2-category of small categories. Is $Cat$ complete?
That is, given a diagram $\phi:J\rightarrow Cat$, does the limit over the diagram exist in $Cat$?

**9**

votes

**1**answer

710 views

### What is a field [Körper] really?

The notion of a field (a commutative ring $R$ with $0\neq 1$ and $R^\times=R-\{0\}$) seems to fit uncomfortably into modern algebra. To see what I mean, consider the following statements:
The ...

**19**

votes

**2**answers

839 views

### The homotopy category is not complete nor cocomplete

I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What ...

**9**

votes

**1**answer

629 views

### Why do we denote (co)ends with integral notation (beyond Fubini's Theorem)?

I know that (co)ends (i.e. universal wedges) follow Fubini-like relation, i.e.
$$ \int_{\langle c,d\rangle} F(c,d,c,d) \cong \int_c\int_d F(c,c,d,d) \cong \int_d\int_c F(c,c,d,d) $$
where we regard ...

**1**

vote

**0**answers

83 views

### Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.

**1**

vote

**0**answers

45 views

### Dual equivalence for multioperators

This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...

**4**

votes

**0**answers

107 views

### Categories where every Mono Splits

When every epi splits a category is said to satisfy the Axiom of Choice.
When every idempotent splits a category is called Cauchy Complete or Idempotent complete.
These look to be well-studied ...

**3**

votes

**0**answers

67 views

### Reference for generalized ind-completions?

I am wondering whether any enriched versions of ind- and pro- completions have been studied? I can not find any literature on them, even though I believe people (most likely the Australian school) ...

**-5**

votes

**1**answer

111 views

### Equivalence of categeories-variants of definition [closed]

There is a notion of equivalence of categories which is the functor $F:\mathcal{C} \to \mathcal{D}$ such that there is a functor $G:\mathcal{D} \to \mathcal{C}$ such that $FG \cong id_{\mathcal{D}}$ ...

**28**

votes

**1**answer

691 views

### What is the meaning of this analogy between lattices and topological spaces?

Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...

**1**

vote

**0**answers

111 views

### $\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.
...

**6**

votes

**0**answers

322 views

### Competing notions of étaleness

I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate.
Here is a list of ...

**5**

votes

**0**answers

142 views

### If $A$ is an algebra, $Sym^n(A)$ is an algebra. Where can I learn more about this algebra structure?

$\newcommand{\Vect}{\mathsf{Vect}}
\newcommand{\nats}{\mathbb{N}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\Alg}{\mathsf{Alg}}
\newcommand{\CAlg}{\mathsf{CAlg}}
\newcommand{\Hom}{\mathrm{Hom}}$
Let ...

**1**

vote

**0**answers

59 views

### A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...

**8**

votes

**1**answer

420 views

### Cohomology theory “from” Grothendieck's six operations?

How, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos?
I would like some ...

**2**

votes

**1**answer

59 views

### How nontrivial can “central extensions of ribbon fusion categories” be?

In a sense, this is a follow up on this question, but one PhD programme later.
Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of ...

**5**

votes

**3**answers

127 views

### Set of functions is not a bifunctor on Rel

Let Rel be the category whose objects are sets and whose morphisms are binary relations, with composition defined by $x (S \circ R) z \Leftrightarrow (\exists y : x R y \wedge y S z)$, and identity ...

**10**

votes

**1**answer

153 views

### Which simplicial objects are Čech nerves?

In 1-categories, a regular epimorphism is a coequalizer of some parallel pair. An effective epimorphism is one which coequalizes its kernel pair. In the presence of kernel pairs, regular and ...

**8**

votes

**1**answer

153 views

### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...

**10**

votes

**2**answers

408 views

### From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...

**2**

votes

**1**answer

215 views

### List is a monad, but is it a comonad with these natural transformations?

List is known to be a monad. It takes a set and maps it to lists of elements of that set. The natural transformations are, singleton and flatten, whereby we map a set to a set of singleton lists ...

**7**

votes

**1**answer

70 views

### natural weak factorization systems

I am trying to understand the definition of natural weak factorization systems from this article by Tholen and Grandis, and these notes by Emily Riehl.
In Riehl's notes, the splitting $s,t$ are of ...

**5**

votes

**0**answers

258 views

### Deligne's theorem on the characterisation of Tannakian categories

I am trying to understand the proof of the Theorem 7.1 from Catégories Tannakiennes by Pierre Deligne https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf.
Essentially, it is ...

**2**

votes

**1**answer

134 views

### Appropriate morphisms and 2-morphisms in Ind(C)

As I was trying to understand the category $Ind(C)$ of diagrams of the form $I \to C$, where $I$ is a small filtered $(0,1)$-category, I wondered whether it is possible to define morphisms directly, ...

**5**

votes

**0**answers

142 views

### Hemi-Semi Direct Product

In the category of racks (similarly quandles), instead of well-known semi direct product, we have hemi-semi direct product construction as seen on Wagemann & Crans.
As far as I know, semi direct ...

**9**

votes

**3**answers

462 views

### “Spatial (geometrical)” realization of Elementary topos?

It is well known that (Grothendieck) Topos (in fact, Model topos too) has many good geometrical properties. In many senses reflects general forms of generic geometry.
Note: Grothendieck view of Topos ...

**2**

votes

**1**answer

142 views

### Small object argument for multiple factorization systems

Is there something similar to the small object argument, but related to a chain of factorization systems on a category $\cal C$?
It is easy to see that one can give a chain of "generating morphisms" ...

**3**

votes

**1**answer

162 views

### discrete Grothendieck construction

In "BASIC CONCEPTS OF ENRICHED CATEGORY THEORY", (version Reprints in Theory and Applications of Categories, No. 10, 2005), chapter 4.7 p.75-76, Kelly introduces the "discrete Grothendieck ...

**3**

votes

**0**answers

85 views

### Obtaining Lawvere's “State categories and response functors”

I'm looking to get my hands on a (.pdf) copy of Lawvere's 1986 preprint State categories and response functors. If someone can post it and answer this question by offering a link, I'd appreciate it.

**1**

vote

**0**answers

57 views

### Adjunction of Crossed Module Functors

I am wondering about the following two related questions and don't know if they have already clear answers or not.
1) Suppose that we already know the functor $F \colon \mathcal{C} \to \mathcal{D}$ ...

**4**

votes

**1**answer

77 views

### Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker ...

**6**

votes

**1**answer

120 views

### Factorization system “tilted” by $(L,R)$

Suppose you have a pair of orthogonal factorization systems, $(E_0, M_0), (E_1, M_1)$ in a category $\cal C$ such that $M_0\subseteq M_1$; this entails that there is a ternary factorization
$$
...

**9**

votes

**0**answers

148 views

### Is there any notion of “smoothification” from $\mathbb{R}$-schemes to generalized smooth spaces?

I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor ...

**7**

votes

**1**answer

89 views

### Can a weak fibration category be non saturated?

A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories
$$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$
containing all the isomorphisms, such that the following ...

**10**

votes

**1**answer

261 views

### Difficulties with descent data as homotopy limit of image of Čech nerve

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ...

**1**

vote

**0**answers

92 views

### Example of non-locally finite stability condition

I am trying to work out example 5.6 in Bridgeland's paper "Stability conditions on triangulated categories".
http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf
A standard ...

**1**

vote

**1**answer

167 views

### Real/complex addition, multiplication, and exponentiation from a categorical viewpoint? [closed]

Edit: As an attempt to make the question somewhat more precise, I'll describe my intuitions in more detail here. Yes, I'm changing the details of question slightly, but the spirit of it remains the ...

**8**

votes

**0**answers

283 views

### Are all formal schemes *really* Ind-schemes?

I'm trying to understand whether there's a fully faithful functor $LRS \supset FormalSch \to IndSch$ and in what sense. Here's my progress so far:
Let $\mathsf{A}$ be the category of adic rings. The ...

**23**

votes

**2**answers

872 views

### What is the relation between sphere spectrum and supersymmetry?

In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab).
Are there some ...

**2**

votes

**1**answer

137 views

### Differential equations → predicate logic mapping

I was watching Tom LaGatta's talk on category theory recently, and one of the audience brought up a statement that caught my attention (around the 56:15 mark):
I was gonna say, there was a book I ...

**7**

votes

**2**answers

411 views

### Reference for Manin's idea on algebraic geometry over the symmetric monoidal model category of Motives

Reference for Y. Manin's idea of "algebraic geometry over the symmetric monoidal model category of motives."
Has been sugested to me that this was made in a Manin's letter. There is an escaned copy?
...

**3**

votes

**3**answers

169 views

### Coproducts and “Error Conditions” in Math vs CS

First, some background: recently in learning more about functional programming I saw one use for coproducts that surprised me a little bit: A function $f: A \rightarrow B \coprod C$ may result when ...

**2**

votes

**1**answer

117 views

### What word can I use for a poset with equivalences

Often I want to define a structure on a set $S$ which is like a poset, but lacks the antisymmetry condition: i.e., one is allowed both $a\succeq b$ and $a \preceq b$ for $a, b$ different elements of ...

**12**

votes

**1**answer

231 views

### From relative categories to marked simplicial sets

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories.
Naturally, one could ask whether there is a reasonably direct way to pass between these two ...

**25**

votes

**3**answers

724 views

### Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?

Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between ...

**6**

votes

**0**answers

113 views

### Reference for supergroupoids in supersymmetry?

I would like to know some references on supergroupoids in supersymmetry.
A supersymmetry is invariance under a supergroup action (nLab, Supersymmetry).
It is know that groupoids provide a local ...