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

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80 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
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0answers
84 views

Is the category Cat complete? [on hold]

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
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1answer
697 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
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2answers
820 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
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1answer
607 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
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0answers
81 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
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0answers
44 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
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104 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
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0answers
66 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
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1answer
110 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
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1answer
689 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 ...
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0answers
109 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
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0answers
319 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
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0answers
141 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
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0answers
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
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1answer
418 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
1answer
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
3answers
126 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
1answer
150 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
1answer
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
2answers
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
1answer
212 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
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1answer
67 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
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0answers
255 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
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1answer
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
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0answers
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
3answers
460 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
1answer
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
1answer
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
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0answers
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.
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0answers
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
1answer
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
1answer
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
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0answers
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
1answer
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
1answer
255 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
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0answers
90 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
1answer
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
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0answers
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
2answers
870 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
1answer
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
2answers
410 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
3answers
167 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
1answer
116 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
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1answer
230 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
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3answers
723 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
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0answers
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 ...
4
votes
1answer
144 views

Most natural equivalence between $C^*$-algebras in NCG

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
6
votes
0answers
234 views

Coherence and rewriting

In category theory there are numerous coherence theorems (https://ncatlab.org/nlab/show/coherence+theorem). One example is the Mac Lane's coherence theorem for monoidal categories. This and probably ...
8
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1answer
230 views

Relative version of Quillen's theorem A

Quillen's Theorem A is formulated as follows: Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category $F/y$ is contractible. Then $F$ induces a weak equivalence ...