**2**

votes

**1**answer

99 views

### Simple technical adjunction question

Suppose $F,G$ are adjoint, and $\epsilon:F\circ G\rightarrow Id$ is the counit. Is it always true that$$
Id_{FG}\epsilon=\epsilon Id_{FG}
$$
as maps from $FGFG$ to $FG$?
It's true if you precompose ...

**1**

vote

**0**answers

63 views

### When does a cogenerator determine a variety?

Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they ...

**0**

votes

**2**answers

379 views

### An isomorphism of categories

(This question was originally asked http://math.stackexchange.com/questions/725421/an-isomorphism-of-categories, with no affirmative answer there.)
Let $C$ be an (finite) extensive category with ...

**4**

votes

**1**answer

160 views

### defining a bicategory of real-valued matrices

Let $\mathbf{Rel}$ be the bicategory of sets, relations, and inclusions between relations. The following fact is well-known:
Any ordinary function $f : X \to Y$ between sets induces a pair of ...

**6**

votes

**1**answer

182 views

### Multi-categorical left Kan extensions?

Let ${\bf Set}$ be the category of sets with cartesian product denoted $\times$, and let Sets be the corresponding multi-category of sets, where
$$Hom_{\bf Sets}(A_1,\ldots,A_n;B)=Hom_{\bf ...

**5**

votes

**0**answers

75 views

### Given a 2-category, is the hammock localization wrt the equivalences equivalent to taking the hom-wise nerve of the maximal subgroupoids?

Consider a category $C$ enriched in categories. Here we have a natural class $W$ of equivalences, namely those morphisms having inverses up to 2-isomorphisms. One can take the hammock localization ...

**2**

votes

**0**answers

73 views

### A question about braiding represented as pseudofunctors

I just asked a very similar question here (About symmetry, braids, and pseudo-functors.), still now I dont get a answere. I hope to express myself more clearly and correctly this time.
Let ...

**4**

votes

**2**answers

258 views

### Transporting algebraic structure along adjoint equivalences

I have two questions, one general and the other particular to the case I am interested in.
The 'homotopically correct' notion of equivalence of categories is an adjoint equivalence (from one point of ...

**0**

votes

**3**answers

272 views

### Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...

**6**

votes

**1**answer

194 views

### Does the “free category on a reflexive graph” monad preserve weak pullbacks, and “why”?

Consider the category of reflexive graphs, and the monad $M$ on it taking the free category: $M(G)$ has all vertices of $G$ as objects, and as edges $x \to x'$ all identity-free paths $x \to x'$ in ...

**2**

votes

**0**answers

582 views

### The link between the subfactors and the motives as enriched Galois theories? [closed]

On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...

**2**

votes

**1**answer

84 views

### About reflective full subcategories and small-orthogonality classes

Let $\mathcal{A}\subset \mathcal{B}$ be two categories with $\mathcal{A}$ full and reflective in $\mathcal{B}$. Let $R:\mathcal{B}\to\mathcal{A}$ be the reflection. That $R$ is the left adjoint to the ...

**1**

vote

**1**answer

198 views

### Pushout of categories along embeddings gives homotopy pushout?

Consider the pushout of a diagram $A\leftarrow B\rightarrow C$ of categories and assume that at least one of the arrows is an embedding, i.e. injective on objects and arrows. When applying the nerve ...

**2**

votes

**0**answers

115 views

### Preimages of accessible full subcategories

My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories ...

**5**

votes

**3**answers

170 views

### Locally finite varieties which are not finitely generated

Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...

**2**

votes

**1**answer

91 views

### From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...

**4**

votes

**2**answers

301 views

### What's an initial object in a poset-enriched category?

I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the ...

**8**

votes

**1**answer

308 views

### Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...

**3**

votes

**0**answers

104 views

### Recognition principle for 2-categories (2-groupoids)

Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...

**9**

votes

**2**answers

354 views

### Is the category of schemes wellpowered? regularly wellpowered?

Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...

**2**

votes

**0**answers

63 views

### Is the category of spherical fusion categories regular? (i.e. is image factorisation possible?)

Consider the category where objects are strict spherical fusion categories and morphisms are strict spherical functors (preserving cups and caps). I am wondering whether there is some kind of image ...

**2**

votes

**1**answer

88 views

### Induced adjunctions

Suppose $F: C \rightarrow D$ is the left adjoint to a functor $G$. Then is it true that the functor $F^{\star}:[C : Sets]$ defined by prescomposing a functor $P: C \rightarrow Sets$ is still left ...

**4**

votes

**3**answers

688 views

### The most unexpected and/or the least natural category theory theorem?

Theory of categories is all natural and abstract nonsense. Or is it? What would be the most unexpected and/or the least natural theorem of the theory of category? (It does not really have to be THE).
...

**2**

votes

**1**answer

168 views

### Showing a functorial isomorphism

I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory.
The exercise in question is from chapter IV.
So, let ...

**6**

votes

**3**answers

404 views

### Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...

**4**

votes

**1**answer

192 views

### Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...

**2**

votes

**1**answer

338 views

### A question about the proof of Quillen's Theorem A

(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.)
Theorem (Quillen) ...

**15**

votes

**5**answers

2k views

### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...

**0**

votes

**1**answer

137 views

### Sheaffication using a $\lambda$-transfinite colimit

I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway.
I was reading this article ...

**8**

votes

**1**answer

264 views

### Adjoining adjoints in a 2-category

For a given 2-category $C$, does there exist a faithful and locally faithful 2-functor $C \to C^*$, such that the image of every 1-morphism of $C$ has a right adjoint in $C^*$?
Below are some of my ...

**16**

votes

**2**answers

468 views

### Internal logic of the topos of simplicial sets

I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...

**11**

votes

**2**answers

459 views

### Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...

**6**

votes

**0**answers

181 views

### Counting categories with at most $n$ morphisms

There are a number of results which count the number of possible algebraic structures on a set of $n$ elements. Notable previous MO questions are for example here and here. The analogous question for ...

**3**

votes

**0**answers

144 views

### Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...

**6**

votes

**1**answer

197 views

### In what generality does Eilenberg-Watts hold?

In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...

**1**

vote

**0**answers

97 views

### Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects?
What are some applications? For ...

**12**

votes

**4**answers

646 views

### Functors and coverings

A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical ...

**2**

votes

**0**answers

133 views

### Neeman's homotopy limits in stable $\infty$-categories

Related question (I hoped to turn this into an answer for the user, but in the end it became a question on its own right!): this
In the book
Neeman, Amnon. Triangulated categories. No. 148. ...

**6**

votes

**1**answer

294 views

### Are left adjoints a left adjoint?

Let $\mathcal C$ be a strict, locally small 2-category.
Consider a subcategory $\mathcal L$ of $\mathcal C$ such that $\mathcal L$ has the same objects as $\mathcal C$, and the arrows of $\mathcal L$ ...

**7**

votes

**1**answer

128 views

### coends of stable infinity categories

Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote ...

**5**

votes

**1**answer

159 views

### What do algebraic theories with strictly terminal trivial models look like?

By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...

**5**

votes

**1**answer

184 views

### How to show twisted complexes over a DG category is again a DG category?

In Bondal and Kapranov's paper enhanced triangulated categories, a twisted complex over a DG category $A$ is a set $\{(E_i)_{i\in \mathbb Z}, q_{ij}: E_i\to E_j\}$, where $E_i$ are objects in $A$, ...

**4**

votes

**0**answers

125 views

### Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that:
$\mathcal C$ is ...

**2**

votes

**2**answers

126 views

### Integral over integrals with different measures

I was trying to construct a category of measurable spaces and 'nondeterministic functions' (not the usual category of measurable spaces); specifically a map $(\Omega, \Sigma)\rightarrow (\Omega', ...

**6**

votes

**0**answers

106 views

### Do coherent toposes descend along open surjection?

Let $f:\mathcal{T} \rightarrow \mathcal{S}$ be a geometric morphism between two toposes. Let $g:\mathcal{L}\rightarrow \mathcal{S}$ be an open surjection (of toposes) and assume that the map ...

**1**

vote

**0**answers

89 views

### The classifying space of the groupoid $\pmb\Delta^n$

Consider the groupoid generated by the category $\{0\to 1\to\cdots\to n\}$; let's call this category $\pmb\Delta^n$ opposed to the category $\triangle^n$, which is "thinner".
I'm trying to figure out ...

**5**

votes

**1**answer

421 views

### Noncommutative geometry and category theory

The point in which one starts to talk about noncommutative geometry is the Gelfand Najmark theorem. It establishes an equivalence of the catgeories of commutative (non)unital $C^*$-algebras and the ...

**3**

votes

**1**answer

154 views

### Cogroup objects are to groups what — are to $k$-modules

Let us place ourselves in a category $\mathcal C$ with finite coproducts $X\amalg Y$, even cocomplete if necessary. It is well known that the morphism set $\mathcal C(X,Y)$ carries an abelian group ...

**3**

votes

**0**answers

114 views

### Categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$

This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains ...

**2**

votes

**1**answer

245 views

### Expressive power of first-order category theory

Given the signature $\lbrace \mathsf{dom}, \mathsf{cod}, \mathsf{id},\circ \rbrace$ and the axioms of category theory – which are expressible in the signature's first-order (FO) language – ...