**6**

votes

**4**answers

891 views

### What is the intuition of connections for cubical sets?

I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, ...

**5**

votes

**0**answers

18 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 ...

**7**

votes

**1**answer

108 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 ...

**1**

vote

**1**answer

123 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 ...

**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, ...

**11**

votes

**6**answers

3k views

### Is there a relationship between model theory and category theory?

According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relation while in Universal Algebra we ...

**10**

votes

**1**answer

120 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 ...

**10**

votes

**2**answers

377 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.
...

**5**

votes

**0**answers

134 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 ...

**5**

votes

**0**answers

200 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 ...

**7**

votes

**4**answers

598 views

### What are natural examples of “bimorphism” classes?

Note: As pointed out in a comment by Peter and echoed by Andrew in his answer, the question as stated does not make sense because "being bimorphic to" is not an equivalence relation. Nevertheless, I ...

**9**

votes

**3**answers

396 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 ...

**23**

votes

**1**answer

394 views

### Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”)

Lemma D4.5.3 of Johnstone’s Sketches of an Elephant states:
Lemma. For an object $A$
of a topos $\newcommand{\E}{\mathcal{E}}\E$, the following are equivalent:
$A$ is internally ...

**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" ...

**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 ...

**3**

votes

**1**answer

157 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 ...

**30**

votes

**5**answers

3k views

### Several Topos theory questions

Hey. I have a few off the wall questions about topos theory and algebraic geometry.
Do the following few sentences make sense?
Every scheme X is pinned down by its Hom functor Hom(-,X) by the ...

**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 $F \colon \mathcal{C} \to \mathcal{D}$ ...

**6**

votes

**1**answer

117 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
$$
...

**24**

votes

**2**answers

833 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 ...

**22**

votes

**10**answers

1k views

### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...

**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 ...

**10**

votes

**1**answer

219 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 ...

**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 ...

**1**

vote

**0**answers

71 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 ...

**5**

votes

**3**answers

608 views

### Nerves of (braided or symmetric) monoidal categories

I'm looking for references on the structure which can be roughtly described as follows: given a (braided or symmetric) monoidal category $C$, I want to consider a simplicial set $N(\mathbf{B}C)$ with ...

**8**

votes

**0**answers

277 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 ...

**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 ...

**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 ...

**26**

votes

**3**answers

701 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 ...

**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?
...

**31**

votes

**35**answers

2k views

### Formalizations of category theory in proof assistants

What are the existing formalizations of category theory in proof assistants?
I'm primarily interested in public-domain code implementing category theory in a proof assistant (Coq, Agda, ...

**6**

votes

**2**answers

2k views

### Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...

**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 ...

**6**

votes

**0**answers

227 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 ...

**2**

votes

**1**answer

126 views

### Product and coproduct for bipartite graphs

Consider the category $BiGraph$ of bipartite graphs (vertices are named "places" and "transitions" like in Petri nets) and continous maps. A subgraph is named open iff it is place-bordered, and a map ...

**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 ...

**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 ...

**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 ...

**2**

votes

**1**answer

208 views

### Special objects in a category - terminology

For an object $A$ in a category $\mathfrak{C}$, consider the following property.
($*$) For every object $B$ in $\mathfrak{C}$, the set of morphisms $\text{Hom}(B,A)$ is either empty or consists ...

**14**

votes

**3**answers

815 views

### Set-theoretic forcing over sites?

All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation ...

**4**

votes

**1**answer

135 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

**1**answer

330 views

### Needless axiom for Grothendieck topologies?

Hi,
The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a covering family.
Why ...

**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 ...

**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, ...

**5**

votes

**1**answer

484 views

### Reference request: sheaves on closed sets

I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. Specifically, I say a ...

**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 ...

**1**

vote

**0**answers

42 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 ...