**0**

votes

**0**answers

38 views

### Coproduct Slice category [on hold]

In category theory, how can I prove that, if the category $C$ has co-product, also the slice category $C / I$ admit it?
Thanks a lot!

**2**

votes

**2**answers

114 views

### a (pseudo)adjunction for the functor sending a category C to PSh(C) the category of presheaves

I've read about free cocompletion of categories discussing on the adjunction between Cat and cocompleteCat (Cat: category of small categories, cocompleteCat: category of small cocomplete categories ...

**1**

vote

**1**answer

80 views

### Induced topology on site + Reconstructing global sections of a scheme (Orlov)

Let $(C,T,O)$ be a ringed site. Let $X$ be a presheaf on C.
We get an induced ringed site $(C/X,T_X,O_X)$. C/X is the over category wrt the presheaf X. The topology $T_X$ is the biggest topology ...

**5**

votes

**1**answer

109 views

### Direct proof that the model category of cdgas is left proper

Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...

**10**

votes

**1**answer

221 views

### Is there something like “Noncommutative geometry internal to a category”?

I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual ...

**2**

votes

**0**answers

73 views

### Do copairings provide dualities in derived categories?

Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...

**-2**

votes

**0**answers

60 views

### categorical constructions surfaces [on hold]

Is there any literature where construction of the sphere realize know , the banda , bull, klein bottle , the projective plane etc... Using the language of category theory for example through pullback ...

**2**

votes

**2**answers

346 views

### Exact sequences of pointed sets - two definitions

It seems to me that there are (at least) two notions of exact sequences in a category:
1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite ...

**1**

vote

**1**answer

151 views

### Classifying space of a colimit of topological categories

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...

**20**

votes

**1**answer

900 views

### Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...

**1**

vote

**1**answer

75 views

### Is $\textbf{FHILB}$ locally regular?

Is the category, $\textbf{FHILB}$, of finite dimensional Hilbert spaces and linear maps locally regular, where `locally regular' is defined like this
...

**1**

vote

**2**answers

96 views

### Poset-enrichment of abelian categories

Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another ...

**-2**

votes

**0**answers

237 views

### When do boring objects exist? [closed]

Let's provisionally call an integer boring if it is not the root of a polynomial over $\mathbb{Z}$ with a small number of variables and with small coefficients and arguments (note that this requires ...

**8**

votes

**0**answers

245 views

### End of the Ext functors

Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...

**8**

votes

**2**answers

521 views

### random category theory

This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references ...

**-2**

votes

**0**answers

70 views

### Specific examples or applications of homotopy coherent diagrams [closed]

A homotopy coherent diagram is a special case of a functor between higher categories where the source category is an ordinary category. Homotopy coherence can be precise in a topological category. In ...

**2**

votes

**0**answers

31 views

### Comparing right and left quasi-representable bimodules

Let $\mathcal V$ be your favourite (closed, symmetric) monoidal model category. To fix ideas, set $\mathcal V = \mathrm{Ch}(k)$, the category of chain complexes over a fixed commutative ring. Given a ...

**2**

votes

**1**answer

105 views

### Are lax functor categories into a cartesian closed 2-category cartesian closed?

Suppose that $C$ is a complete closed monoidal category and $I$ is any small category. Then the functor category $Fun(I,C)$ is again a closed monoidal category with the pointwise tensor product $F ...

**1**

vote

**1**answer

54 views

### Strict comma objects implies comma objects

I'm condusion on a statement in this page comma object in $n$lab. It states:
any strict comma object is a comma object, but the converse is not in general true.
My confusion is: the strict comma ...

**1**

vote

**0**answers

141 views

### Monoidal Forgetful/Free Adjunction for $E_2$-algebras

Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms ...

**7**

votes

**1**answer

331 views

### Explicit counter example to Vopěnka's principle in the constructible universe?

Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle ...

**0**

votes

**0**answers

27 views

### Morphisms between lax wedges

In the paper
BOZAPALIDES, S., Les fins cartésiennes
the following definition of a lax wedge for a 2-functor $S\colon \mathcal{A}^{op}\times \mathcal{A}\to \mathcal B$ between 2-categories is ...

**3**

votes

**0**answers

118 views

### Categories in which an epimorphism applied to a non-monic epimorphism can be monic

Let $\mathcal{C}$ be a category, and let $A$, $B$, and $C$ be objects.
Given $A \xrightarrow{f} B \xrightarrow{g} C$ such that:
$f$ is both epic and monic
$g$ is epic but not monic
$gf$ is epic and ...

**2**

votes

**0**answers

123 views

### (co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...

**1**

vote

**1**answer

181 views

### Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?

Let $\mathcal{A}$, $\mathcal{B}$ be two triangulated categories and $F: \mathcal{A}\to \mathcal{B}$ be a triangulated functor between them. Then $F$ induces an homomorphism between their Grothendieck ...

**3**

votes

**1**answer

89 views

### Is every frame monomorphism regular?

Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?

**6**

votes

**0**answers

148 views

### Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a ...

**4**

votes

**1**answer

114 views

### Extremal, but not regular monomorphism

Is there an example of a category, and a monomorphism $m:X\to Y$ between two objects such that $m$ is extremal, but not regular? (A monomorphism $m:X\to Y$ is said to be extremal if whenever $m=g\circ ...

**3**

votes

**0**answers

107 views

### Are all monomorphisms in the category of bounded lattices regular?

Let $\mathbf{Lat}_{01}$ be the category of bounded lattices with lattice homomorphisms that respect the smallest and the largest element. Is there a monomorphism in $\mathbf{Lat}_{01}$ that is not ...

**0**

votes

**0**answers

45 views

### About cartesian closure of lax.functors categories

Let $\mathscr{A}$ a category and $F, G, H: \mathscr{A}^{op}\to CAT$ lax.functors. I wish find a possible "natural correspondence" between categories: $[F\times G, H]_O \leftrightarrow [F, H^G]_O$ ...

**5**

votes

**0**answers

111 views

### Is there a classification of 2d extended TQFTs with defects?

Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly ...

**2**

votes

**1**answer

181 views

### Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...

**5**

votes

**1**answer

170 views

### Difference between coherent nerve of simplical model category and simplicial category

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in ...

**7**

votes

**1**answer

296 views

### Constructing unnatural transformations

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere?
Let $C$ ...

**8**

votes

**1**answer

251 views

### When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian ...

**3**

votes

**1**answer

306 views

### What are the automorphisms of $BG$?

Setup: Let's work in the category of schemes over $\mathbb C$. Let $G$ be a finite group. Let $BG=[pt/G]$ be the classifying stack of principal $G$ bundles. This is a fiberd category over the big ...

**13**

votes

**1**answer

541 views

### What's the cardinality of a higher category?

The cardinality of a set is just the number of elements.
To make sense of the cardinality of a category, one has to account for the morphisms. The usual definition is the sum over the isomorphism ...

**1**

vote

**0**answers

92 views

### How does one compute a colimit of monoidal categories?

The question is in the title. I'm also happy to get answers about (your favorite adjective) monoidal categories.
Here's a guess:
In order to compute a colimit of monoids we can push everything down ...

**2**

votes

**0**answers

210 views

### category theoretic approach to Sylow theorems and finite group theory?

Is there a category theoretic approach to Sylow theorems?
More generally, is there an exposition of finite group theory in terms of category theory which would include Sylow theorems and little facts ...

**1**

vote

**1**answer

58 views

### Quotients of termwise split injections, for additive categories

In the stack exchange notes found in Section 10 of this file, it is claimed that the category $K(\mathcal{A})$ of complexes up to homotopy is a triangulated category, if $\mathcal{A}$ is additive. In ...

**3**

votes

**0**answers

118 views

### Finitely presented categories and limits

Suppose I have a finite graph $G$, and I then take the free category $\mathcal{C}(G)$ over such a finite graph. Now, I would like to "force" some objects to be limits. Is there a way to do that ...

**1**

vote

**1**answer

88 views

### Projective family of probability spaces

This is a crosspost of this question from MSE.
I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions
...

**4**

votes

**0**answers

241 views

### A construction with homotopy colimits and homotopy pullbacks for descent

EDIT: Following the lines of some suggestions in the comments below, I try to add something more to explain the problem better. A map $\text{hocolim}Y\rightarrow\bar{Y}$ in $\text{Ho}(\mathbf{M})$ is ...

**2**

votes

**0**answers

69 views

### What should one “do” to “strictify” a triangle of transformations coming from a lax commutative triangle of functors?

I would like to apologize for this rather stupid abstract nonsense question.
Let $h=f\circ g$ for composable functors $f,g$; assume that there exist left or right adjoints to $f$ and $g$. Then it ...

**5**

votes

**2**answers

386 views

### When is the category of small (pre)sheaves a(n elementary) topos?

When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos.
When $C$ is ...

**5**

votes

**0**answers

166 views

### Is there a higher, “orientalish” version of geometric realisation?

Geometric realisation of simplicial sets can be roughly thought of like this:
In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...

**7**

votes

**0**answers

167 views

### The bifunctoriality of co/limits

I recently noticed that there are two senses in which colimits are functorial, and I'm curious about their interplay.
Let $C$ be a cocomplete category. Then, on the one hand, for any diagram ...

**2**

votes

**1**answer

179 views

### Loop defects in Walker-Wang model

My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model.
Elementary excitations in the WW model can be point particles, loop defects and more ...

**1**

vote

**0**answers

80 views

### is sufficient cohesion equivalent to the connectedness of subobject classifier?

I'm following Lawvere article Axiomatic Cohesion.
He states (Proposition VI.4) that sufficient cohesion is equivalent to the connectedness of subject classifier, but I can't follow the proof. I can't ...

**2**

votes

**0**answers

129 views

### Relationship between coherent toposes/coherent logic and coherent sheaves

I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in ...