**10**

votes

**1**answer

125 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 ...

**7**

votes

**1**answer

130 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

382 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.
...

**1**

vote

**1**answer

131 views

### List is a monad, but is it a comonad with these natural transformations? [on hold]

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 ...

**5**

votes

**1**answer

48 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

205 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

126 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

135 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

403 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

138 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

159 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

81 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

55 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

75 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

119 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

146 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

85 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

222 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

72 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

166 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

278 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 ...

**24**

votes

**2**answers

838 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

136 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

403 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

163 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

114 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

220 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 ...

**26**

votes

**3**answers

704 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

112 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

**1**answer

136 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

**0**answers

228 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**

votes

**1**answer

225 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 ...

**2**

votes

**2**answers

94 views

### Reedy model structure on sSet

According to this question, there is a model structure on $\mathrm{Set}$ in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, ...

**1**

vote

**0**answers

43 views

### Pseudopullback of dimension three

What is the name of the appropriate analogue of the pseudopullback for dimension three?
That is to say, a pseudonatural equivalence $fg\simeq hj $ which is universal in the obvious sense...
Thank ...

**7**

votes

**0**answers

223 views

### Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$
$$\require{AMScd} \begin{CD} M^D ...

**3**

votes

**1**answer

88 views

### Self-enrichment of reflective subcategories of self-enriched categories

I'll go straight to the point of my question:
Say $\mathscr{A}$ is a reflexive subcategory of $\mathscr{B}$, meaning the inclusion functor $i: \mathscr{A} \to \mathscr{B}$ is fully faithful and ...

**2**

votes

**1**answer

55 views

### Why does vertical multiplication in 2-groups not follow the same order as horizontal one when constructed from crossed modules?

Consider a 2-group (seen as a 2-category with only one object $\star$) constructed from a crossed module $(G,H,t,\rhd)$ ($\circ$ will denote 1-morphisms composition, $\circ_{h}$ 2-morphisms horizontal ...

**8**

votes

**0**answers

85 views

### Stability of adjunctions of infinity-categories by base change

Let $O \to O'$ be a functor between locally presentable symmetric monoidal $(\infty,1)$-categories (assume that the tensor product commutes in each argument with colimits, if necessary). Suppose that ...

**3**

votes

**1**answer

143 views

### Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?

Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form ...

**5**

votes

**0**answers

163 views

### What makes Reedy model categories useful?

I have been reading up a bit on the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$ in Goerss-Jardine. One thing I find a bit unclear is what the Reedy ...

**1**

vote

**1**answer

116 views

### pullback square in Goerss-Jardine

In proving the the existence of the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$, Goerss-Jardine prove there is a pullback square induced by a map of ...

**3**

votes

**0**answers

191 views

### N-periodic derived categories

I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places.
Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...

**4**

votes

**0**answers

45 views

### Temporal semantics for string diagrams

Suppose I have a string diagram $D$ which involves a set of strings $S$ and atomic processes $A$. Formally, we should think of this as a canonically chosen map in the free symmetric monoidal category ...

**8**

votes

**1**answer

227 views

### Can the groupoid completion of a topological category be recovered from its classifying space?

Let $C$ be a category. The groupoid completion of $C$ is the free groupoid on $C$, i.e. the category $C[C^{-1}]$ obtained by localizing at everything. Recall that the classifying space $\mathbf{B}C$ ...

**5**

votes

**1**answer

166 views

### Twisted Day convolution

Has anyone studied a version of Day convolution for an enriched presheaf category $V^{A^{\mathrm{op}}}$ where the monoidal structure of $V$ is "twisted" on one side by an action of $A$? I'm thinking ...

**2**

votes

**0**answers

48 views

### Factorization systems on tensor product of presentable categories

This question is motivated by the following particular problem.
I have two presentable categories $\cal A,B$ with orthogonal factorization systems $({\cal E}, {\cal M})$ (on A) and $({\cal U},{\cal ...

**5**

votes

**4**answers

190 views

### On the tensor product of presentable categories

I am trying to understand how the tensor product of presentable categories works: let $\otimes\colon {\cal A}\times {\cal B}\to {\cal A}\otimes{\cal B}$ the universal bilinear functor corresponding to ...

**2**

votes

**0**answers

42 views

### Branching behavior in string diagrams/monoidal categories?

I am currently working through Peter Selinger's paper "Towards a Quantum Programming Language", and trying to connect it with what I already know about monoidal categories and string diagrams.
...

**2**

votes

**0**answers

145 views

### Toward Axiomatic sheaf theory? (References)

It is known that one of Lawvere and Tierney's goals was to provide an axiomatic approach to sheaves. Their notion of elementary topos constituted a preliminary step in that respect.
Question: Has ...

**2**

votes

**0**answers

132 views

### Is the class of Heyting algebras originating from directed graphs a variety?

The category RefGph of reflexive directed graphs
is the functor category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is
the simplex category truncated at level 1.
Hence the poset Sub(X) of ...