**3**

votes

**1**answer

348 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

82 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

446 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

70 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

341 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 $\mathcal{B}$, ...

**3**

votes

**0**answers

136 views

### Internal categories in an endofunctor category

Here we see the definition of an internal category in a monoidal category. We also know that endofunctor categories support a monoidal product which is actually functor composition. It is the case ...

**10**

votes

**1**answer

733 views

### Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...

**1**

vote

**3**answers

424 views

### Is there a general notion of semigroup action?

The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a ...

**2**

votes

**2**answers

384 views

### Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...

**5**

votes

**0**answers

150 views

### Compact objects and ind-objects in triangulated categories

question : let $A$ be triangulated category compactly generated by subcategory $A^c$ of compact objects. Consider category of ind-objects $Ind(A^c)$. Is there relation between $A$ and $Ind(A^c)$? ...

**5**

votes

**2**answers

266 views

### What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...

**2**

votes

**1**answer

153 views

### what is the stabilization of pointed sets?

Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab).
The famous example is ...

**4**

votes

**1**answer

231 views

### Higher Degree Data in a Cosimplicial Quasicategory and Delooping

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.
My question is regarding accessing data ...

**25**

votes

**9**answers

1k views

### Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful?
I have a vague idea of the possibility of ...

**8**

votes

**3**answers

631 views

### In the category of sets epimorphisms are surjective - Constructive Proof?

The statement that surjective maps are epimorphisms in the category of sets can be shown in a constructive way.
What about the inverse?
Is it possible to show that every epimorphism in the category ...

**8**

votes

**1**answer

430 views

### Notion of infinity in categories

Please excuse me if the question is too vague or uninteresting.
Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. Motivated by the equivalence of Dedekind-finiteness and finiteness ...

**2**

votes

**0**answers

104 views

### Pseudomodules, “general coherence theorem”

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...

**0**

votes

**2**answers

102 views

### a dcpo seen as a category: when does a dcpo map induce a functor with an adjoint?

Take two posets $A, B$ (partially ordered sets). Now consider these posets to be categories $Cat(A), Cat(B)$ respectively. Consider a map from $A$ to $B$, $f: A \rightarrow B$. This can be seen as ...

**3**

votes

**2**answers

174 views

### Directed subposet of a poset containing the minimal elements

The following appears naturally in a certain context:
Let $P$ be a graded partially ordered set. Let $M$ be the subset of minimal elements of $P$. Define subsets $E_i$ inductively as follows: First, ...

**8**

votes

**1**answer

512 views

### Reference request: Book of Linear algebra from categorical point of view

Is there any book of Linear algebra in the modern language of Category theory?
I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are ...

**7**

votes

**1**answer

596 views

### Learning roadmap to TQFT from a mathematics perspective

I had asked a question on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posting it below.
...

**0**

votes

**0**answers

72 views

### Sharp objects and fixed points

Given an adjunction $F\colon \mathcal C \leftrightarrows \mathcal D\colon G$ I would like to call "sharp" a pair of objects $(C,D)$ if the bijection
$$
\hom_{\cal D}(FC,D) \cong \hom_{\cal C}(C, GD)
...

**2**

votes

**1**answer

142 views

### Algebraic objects and lifts of their represented functors

I've seen the following theorem around in various forms:
To give an object $A \in \mathcal{C}$ the structure of a $\Omega$-algebra object in $\mathcal{C}$ is equivalent to giving a lift of the ...

**1**

vote

**1**answer

121 views

### Are the pullback functors of adjoint functors also adjoint?

Given adjoint functors $F: A \to B$, $G: B \to A$, if you then take their pullback functors $F^* : Set^B \to Set^A$ and $G^* : Set^A \to Set^B$ given by pre-composition, are these two also adjoint ...

**5**

votes

**0**answers

109 views

### Lifting commutative diagrams of functors from the homotopy level to the “higher” level

Let $\mathcal A$ and $\mathcal B$ be differential graded categories over a field. Let $F, G, K : \mathcal A \to \mathcal B$ be quasi-functors (see here for definitions). Assume you have morphisms of ...

**3**

votes

**2**answers

379 views

### Zigzags and contractibility of categories

Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:
If there is ...

**0**

votes

**0**answers

73 views

### Linkage between homotopy equivalence and identification of algorithms

I vaguely recall that someone says there is linkage between homotopy equivalence and identification of algorithms which may be isomorphic or morphism or something like that,the algorithm may be ...

**4**

votes

**2**answers

253 views

### What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?

In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
...

**3**

votes

**0**answers

132 views

### When do limits and colimits of infinity-categories commute?

This is a question for someone who read (or wrote) enough of Lurie's HTT to know a reference. Suppose $D,E$ are small "diagram" $(\infty,1)$-categories, and $\mathcal{C}$ is a stable infinity-category ...

**6**

votes

**0**answers

181 views

### Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category

Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$.
In HigherAlgebra, the derived category ...

**6**

votes

**0**answers

167 views

### Can we “complete” model categories to compute derived functors in the usual way?

Suppose we have a functor between two model categories $F\colon \mathcal{C}\to \mathcal{D}$. Suppose we don't know that it is a part of a Quillen pair. Nevertheless, it can still happen that the ...

**1**

vote

**0**answers

107 views

### right adjoint functor for closed immersion of topoi

Let $i\colon (X,A)\rightarrow (Y,B)$ be a closed immersion of ringed topoi. Does functor $i_*\colon Mod(A)\rightarrow Mod(B)$ have a right adjoint?

**1**

vote

**1**answer

289 views

### Category which has no non-trivial adjoint functors

Does there exist a category C which such that there is no functor $F:C \rightarrow D$ with $D\not\cong C$ which has a left (or right) adjoint?

**5**

votes

**2**answers

341 views

### A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.
Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...

**7**

votes

**1**answer

248 views

### The category of categories and adjunctions

What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are ...

**0**

votes

**1**answer

345 views

### (Homotopy) limits and colimits in a dg-category

It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of ...

**1**

vote

**1**answer

106 views

### Taking zeroth cohomology of a dg-module over a dg-category is “good” on representables, but probably “bad” on cones

This question is possibly related to this other one.
Let $\mathcal A$ be a dg-category over a commutative ring $k$. I denote by $\text{dgm-}\mathcal A$ the dg-category of right dg-$\mathcal ...

**1**

vote

**0**answers

45 views

### Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones

This question is strictly related to this other one.
Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let
be a diagram in $Z^0(\mathcal A)$, where the rows are ...

**3**

votes

**1**answer

108 views

### Reconstructing a morphism of exact triangles in the homotopy cat. of a dg-cat. using a “functorial cone map”

I set this problem in the framework of (pretriangulated) dg-categories; everything can probably be translated in the world of stable $(\infty,1)$-categories.
Let $\mathcal A$ be a pretriangulated ...

**4**

votes

**2**answers

200 views

### Serre functor of a subcategory (in particular parabolic category O)

For an additive category $\mathcal C$ there is the notion of a Serre functor on $\mathcal C$, i.e. a an autoequivalence $S$ of $C$ such that there exist isomorphisms
$$Hom(A, S(B)) \cong Hom(B, A)^*$$
...

**3**

votes

**1**answer

112 views

### weak version of a Baez-Crans 2-vector space?

Baez and Crans defined a 2-vector space to be a category internal to the category of vector spaces (say over the reals). I am interested in categories that are equivalent to Baez-Crans vector spaces ...

**2**

votes

**2**answers

204 views

### Is antipode unique for bialgebras in arbitrary monoidal categories?

If $B$ is a bialgebra in the category $\tt{Vect}$ of vector spaces (over $\mathbb C$, for example) then $B$ can't have two different antipodes.
Is this true for bialgebras in an arbitrary symmetric ...

**0**

votes

**0**answers

114 views

### Name of Property $t=st \text{ and } s=ts$

What is the name of the property shared by a pair of functions $s,t$ with $$t=st \text{ and } s=ts$$
( Main example: relation-valued domain and range operations on relations, via ...

**3**

votes

**0**answers

89 views

### symmetric monoidal dagger endofunctor categories

Take an arbitrary category $C$, and consider $End(C)$, any endofunctor category consisting of some or all endofunctors of $C$.
I have several related questions:
What restrictions must we impose on ...

**1**

vote

**0**answers

107 views

### Comonads and the category of Sets

In Vicary's paper, after eq 15, he talks about how the category of internal comonoids $C_\times$ has many properties of the category of sets. We know that a comonad on a category has the same axioms ...

**1**

vote

**0**answers

82 views

### Existence of Colimits in the Definition of Locally Presentable Categories

Basically, my question is simple: why does the definition of a locally presentable category require all colimits exist?
The motivation for this is that I was learning about algebraic posets, and had ...

**4**

votes

**2**answers

203 views

### An orthogonal factorization system on 1-Cob?

Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W_{dom}:A\to W$ and $i^W_{cod}:B\to ...

**2**

votes

**1**answer

152 views

### When is an exponential functor a bialgebra?

My question is about the notion of exponential functors as they are frequently defined in the literature on (strict) polynomial functors, e.g., the paper "General Linear and Functor Cohomology" by ...

**9**

votes

**1**answer

284 views

### Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?

I asked this question already on math.stackexchange, but maybe it is also useful to ask this here, since it was not answered there.
Suppose we have three directed sequences of $C^*$-algebras, say ...