**4**

votes

**1**answer

166 views

### When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$?

Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a ...

**4**

votes

**0**answers

83 views

### Limits in Span(Vec)

Let Vec be the category of real vector spaces and linear maps. Let Span(Vec) be the bicategory of correspondences between real vector spaces. I am trying to understand lax limits in Span(Vec). What ...

**0**

votes

**0**answers

80 views

### About stable maps passing through fixed points

In "Notes On Stable Maps and Quantum Cohomology", Fulton and Pandharipande present some results, and their proofs, about the representability of the functor $\mathcal{M}_{g, n}(X, \beta)$, which maps ...

**3**

votes

**0**answers

69 views

### Cofree Lie Coalgebra

I have problems finding anything about the cofree Lie coalgebra functor
$\mathcal{L}ie^c$ out there.
Basically all I found was that it appears in Harrison cohomology and that,
given a ...

**5**

votes

**1**answer

277 views

### Framed version of braided monoidal category

The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a ...

**2**

votes

**0**answers

231 views

### What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?

I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...

**28**

votes

**6**answers

3k views

### Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...

**0**

votes

**0**answers

62 views

### Complex of “homotopy coherent dg-natural transformations” between dg-functors

Let $\mathcal B$ be a dg-category. I define the dg-category of morphisms $\underline{\mathrm{Mor}}(\mathcal B)$ as follows: objects are triples $(A,B,f)$, where $A,B \in \mathcal B$ and $f \in ...

**-1**

votes

**1**answer

221 views

### Kan extensions and special cases

Kan extensions specify the adjoint structures between $\mathbf{Sets^{C^{op}}}$ and $\mathbf{Sets^{D^{op}}}$, where there exists a functor $f:\mathbf{C} \to \mathbf{D}$ and $\mathbf{C}$ and ...

**4**

votes

**1**answer

188 views

### Does “simplicial” commute with “Bousfield localization”?

Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the left Bousfield localization $L_SM$ of $M$ with respect ...

**2**

votes

**0**answers

127 views

### Infinite Fubini rule for co/limits

It is a well known fact that given a functor $F\colon I\times J\to C$ then (when everybody exists)
$$
\varinjlim_I \varinjlim_J F\cong \varinjlim_J \varinjlim_I F\cong \varinjlim_{I\times J}F
$$
Now, ...

**2**

votes

**1**answer

339 views

### Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.
Questions: are there definitions of image and kernel for a ...

**39**

votes

**5**answers

4k views

### Why higher category theory?

This is a soft question.
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...

**8**

votes

**0**answers

107 views

### A “small” definition of sub-(∞,1)-topoi

Suppose I have an $(\infty,1)$-topos $\mathcal{X}$ and a (small) set of maps $S$ in $\mathcal{X}$, which therefore generates an accessible localization $S^{-1}\mathcal{X}$. Is there any "small" ...

**12**

votes

**1**answer

300 views

### The real numbers object in Sh(Top)

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...

**3**

votes

**2**answers

222 views

### $ \text{Lan}_KN(-/\mathcal C)\cong N(-/K) $

Let $N(-/\mathcal C)\colon \mathcal C\to \mathbf{sSet}$ be the functor sending $c\in\mathcal C$ to the nerve of the coslice category $c/\mathcal C$.
Given a functor $K\colon\mathcal{C}\to ...

**2**

votes

**1**answer

198 views

### Graphs of tensoring modules

Let $A$ be a ring, and $L,M,N$ an $A^\mathrm{op} \otimes A$-modules.
$L \otimes_A M$ is then an $A \otimes A^\mathrm{op}$ module so we can tensor it again with $N$ to get an abelian group $(L ...

**4**

votes

**2**answers

267 views

### Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ ...

**15**

votes

**3**answers

586 views

### Characterize the category of rings

(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples:
Sets and functions, due to Lawvere.
Modules over some ...

**10**

votes

**1**answer

246 views

### A linear category with objects of infinite length but which is otherwise finite?

Fix a ground field $k$. By a linear category I will mean an Abelian category which is compatibly enriched over $k$-vector spaces. A linear category is called finite if it satisfies the following four ...

**1**

vote

**0**answers

94 views

### Finitely co/continuous monad induced by an operad

It is well known that any operad on a nice monoidal category induces a monad.
I was wondering which conditions can be imposed on the operad $T$ (say, in a monoidal closed $\cal V$) so that the ...

**3**

votes

**1**answer

188 views

### Do the algebras for a $\infty$-monad form a stable $\infty$-category?

I'm wondering if a monad $T$ on a stable $\infty$-category $\cal C$ has a stable $\infty$-category of algebras, provided $T$ preserves finite limits/colimits.
Is this true?
Edit: Is something ...

**6**

votes

**0**answers

130 views

### Fibrations of the injective model structure on G-simplicial sets

Let $G$ be a discrete group. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which ...

**8**

votes

**1**answer

563 views

### Higher vector spaces

As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at ...

**2**

votes

**2**answers

124 views

### $(LLP(Epi), Epi)$ is a WFS on any variety of algebras

This entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this ...

**1**

vote

**1**answer

197 views

### Relations In Category Theory

Probably a silly question. Suppose that $C$ is a category that does not have finite Cartesian products. So we cannot define a relation on some objects to be a sub object of their Cartesian product (a ...

**2**

votes

**0**answers

61 views

### relations between derived categories of ind-A and A

Let $A$ be an abelian category and $indA$ be its ind category. I want to know the relations between $D^b(A)$ and $D^b(indA)$. For example, I find in another question that if $A$ is thick in $indA$, ...

**1**

vote

**1**answer

247 views

### Concise definition of subobjects

Higher category theory tells us that it is a bad idea to identify isomorphic things. Rather, the isomorphism should belong to some additional data. Also, categorification tells us that one should, ...

**1**

vote

**1**answer

90 views

### Is the extension of a quasi-functor again a quasi-functor?

Let $\mathcal A, \mathcal A', \mathcal B$ be dg-categories over a field $k$ (this assumption allows me not to derive the tensor product, I don't think it is really essential). Let $F : \mathcal A \to ...

**2**

votes

**0**answers

74 views

### Orthogonal FS become weak FS if you switch classes

I've always been surprised by the fact that $(Epi, Mono)$ is, when possible, a strong factorization system (unique solution to lifting problems, existence of unique factorization), whereas $(Mono, ...

**8**

votes

**0**answers

95 views

### What suffices to check completeness in an n-fold Segal space?

Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the
Segal condition: For each $j$, the map
$$ X_j \to ...

**2**

votes

**0**answers

116 views

### The Karoubi model structure on Cat

I am looking for any kind of informations about the Karoubi model structure on $\bf Cat$. I discovered the presence of this structure a few months ago on the Joyal Lab and now I would like to use it ...

**2**

votes

**2**answers

183 views

### Is there a general notion of a subobject generated by a subset in any concrete category?

Let $A$ be an object of a concrete category $C$ with a forgetful functor $F\rightarrow Set$ and let $S \subseteq F(A)$. Is there a general construction that gives us a subobject $\left<S\right>$ ...

**2**

votes

**0**answers

117 views

### Do all full exceptional sequences of a triangulated category have the same length?

Let $\mathcal{D}$ be a $k$-linear triangulated category and $\langle E_n,E_{n-1},\ldots, E_0\rangle$ be a sequence of objects of $\mathcal{D}$. We call $\langle E_n,E_{n-1},\ldots, E_0\rangle$ an ...

**8**

votes

**1**answer

367 views

### Deligne's exterior power

In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...

**7**

votes

**1**answer

418 views

### Why $( \infty , n)$-categories are useful for?

I know that mathematicians are trying to construct adequate models for $( \infty, n)$-categories. Although, it seems to be an interesting task, I would like to know some explicity examples where this ...

**5**

votes

**0**answers

154 views

### Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...

**8**

votes

**0**answers

150 views

### Reference for maps whose pushouts are also homotopy pushouts

Consider a category C with weak equivalences, e.g., a model category.
For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) ...

**3**

votes

**1**answer

86 views

### Existence of ind-right adjoint functor for semi-simple category?

I'm just reading a lemma in Yves ANDRE's seminar on finite dimensional motives.
Soit $Σ:Rep_F G→T$ un ⊗-foncteur vers une categorie F-tensorielle T ……
where $G$ is a pro-reductive group scheme, ...

**0**

votes

**1**answer

155 views

### Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R.
Is it possible to extend this function to a smooth function on R²?
...

**4**

votes

**0**answers

244 views

### Commutation of simplicial homotopy colimits and homotopy products in spaces

Edit: The claim below is wrong, as explained in the comments,
because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy ...

**1**

vote

**1**answer

331 views

### Are these two “FUNCTORS” adjoint?

I am considering the following correspondence:
Let $X$ be quasi compact quasi separated schemes.Consider a pseudo functor \begin{equation}Sch\rightarrow CAT :U\mapsto Qcoh(U),f:U\rightarrow V\mapsto ...

**10**

votes

**2**answers

652 views

### If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?

The other direction is well known.
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering ...

**3**

votes

**0**answers

141 views

### Categorical characterization of closed imbeddings

Let $f\colon X\to Y$ be a morphism of schemes.
Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. ...

**3**

votes

**1**answer

334 views

### Category of modules over commutative monoid in symmetric monoidal category

Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:
In many cases, ...

**4**

votes

**0**answers

65 views

### Are regular epi of locale stably epic?

It is well know that the category of locale is not a regular category, that is the pullback of a regular epimorphism is not always a regular epimorphism: for example, the classical counterexample ...

**13**

votes

**2**answers

435 views

### Infinite dimensional 2-Hilbert spaces

Is there a definition of an infinite dimensional 2-Hilbert space?
Finite dimensional 2-Hilbert spaces have been discussed by Baez in
http://arxiv.org/abs/q-alg/9609018
In the more recent paper by ...

**3**

votes

**1**answer

67 views

### Pivotal functors of that are substantially different from finite group homomorphisms

Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a ...

**2**

votes

**0**answers

67 views

### Why does $\pi_t$ preserve pullbacks in this special case?

Let $X$ be a fibrant pointed cosimplicial space.
Following Bousfield-Kan, let $\text{lim}^{\partial \Delta_{n+1}} X = M^{n}X$ be the nth matching object of $X$. Onecan then show that there is a ...

**4**

votes

**2**answers

339 views

### Aspheric functors and Grothendieck fibrations

Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, ...