**6**

votes

**0**answers

176 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

142 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

186 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

95 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

638 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

123 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

288 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

126 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

157 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 ...

**4**

votes

**0**answers

87 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

120 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

104 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

87 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

409 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

152 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

106 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

242 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 – ...

**5**

votes

**0**answers

227 views

### Sign problem for the shift functor on DG modules

Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...

**6**

votes

**1**answer

285 views

### Can the monoidal structure on manifolds be strictified?

I'm asking this question purely out of curiosity.
Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...

**1**

vote

**4**answers

269 views

### Graphs with dangling edges

In conventional Graph Theory the role of Nodes and Edges is skewed: nodes are perfectly ok being aloof, but poor edges are always drawn between existing nodes (that is, the two maps from EDGES to ...

**3**

votes

**0**answers

218 views

### Why does this setting imply that a category is Grothendieck?

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf.
Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...

**2**

votes

**0**answers

117 views

### Homotopy limits of weak diagrams

This question is somehow related to my question at http://math.stackexchange.com/questions/683915/derived-pseudo-functor
and to the question here: A homotopy commutative diagram that cannot be ...

**1**

vote

**1**answer

501 views

### On the foundations for large categories

There are some basic discussions on the motivations of large categories and small categories [here] (On the large cardinals foundations of categories) and [here] (Large cardinal axioms and ...

**12**

votes

**3**answers

429 views

### Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...

**24**

votes

**5**answers

1k views

### (Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...

**0**

votes

**1**answer

161 views

### Homotopy limit of a cosimplicial category

Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?
...

**2**

votes

**1**answer

174 views

### n-limits and the Descent Category

Probably, this question could be at http://math.stackexchange.com/, but, for some reason, I can't access that site right now. So, I apologize in advance.
At this article ...

**3**

votes

**0**answers

142 views

### Is the collage of two spatial toposes a spatial topos?

Consider the collage operation along a profunctor, defined between two categories ${\bf C}, \bf D$.
Suppose now that the two categories are toposes, say $Sh(X), Sh(Y)$ for two topological spaces ...

**6**

votes

**2**answers

359 views

### Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone.
In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group ...

**4**

votes

**0**answers

173 views

### Does there exist a terminal surjective discrete fibration out of $C$?

Let $DF$ denote the category whose objects are categories and whose morphisms $F\colon R\to S$ are the discrete fibrations. This category has applications to the real-world problem of structuring ...

**5**

votes

**1**answer

286 views

### Not quite adjoint functors

What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...

**1**

vote

**1**answer

157 views

### Expression for functor strengths?

It's well known that endofunctors on $Set$ have an unique strength.
A strength for a functor $T : Set \to Set$ is a natural transformation $t_{A,B} : A \times T B \to T (A \times B)$ such that certain ...

**10**

votes

**0**answers

179 views

### Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...

**1**

vote

**1**answer

193 views

### What is this name of this 2-category without very much structure?

I was wondering if there is a name for this 2-category which is like the 2-category of natural transformations, but does not actually require the 1-morphisms to be functors or the 2-morphisms to be ...

**4**

votes

**1**answer

180 views

### Reference request: “unoriented composition” in generalized categories

I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category ...

**2**

votes

**0**answers

56 views

### A categorical analogue of Debreu's independent factors theorem

Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...

**3**

votes

**3**answers

315 views

### When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?

Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...

**3**

votes

**0**answers

175 views

### Invariants of groups that are invariant under passage to finite index subgroups

This question is mostly idle curiosity.
Recall the following standard terminology: if $P$ is a property of groups, a group $G$ is said to be virtually $P$ if it has a subgroup of finite index which ...

**2**

votes

**1**answer

234 views

### Cartesian closed category

Let $\bf{C}$ be a category with finite products.
(1) An object $X$ of $\bf{C}$ is called cartesian if the functor $(-)\times X$ has a right
adjoint.
(2) A morphism ...

**9**

votes

**1**answer

399 views

### Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category?

Following Anton Deitmar, let $\mathcal B$ be an "$\mathbb F_1$-linear category" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is balanced, pointed, contains finite products, kernels, and ...

**1**

vote

**1**answer

141 views

### comparison between two monadic definitions for an operad

According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is ...

**24**

votes

**4**answers

761 views

### What other monoidal structures exist on the category of sets?

I know of the following monoidal structures over $\mathbf{Set}$ (taken from here):
The Cartesian product $\otimes=\times$ (categorical product)
The disjoint union $\otimes=+$ (categorial coproduct)
...

**2**

votes

**1**answer

187 views

### A Criterion for a morphism to be a counit of an Adjunction

Suppose we have two functors $F:C\leftrightarrow D:G$ and a morphism $\varepsilon:FG\rightarrow\operatorname{Id}_D$. I am looking for a way to check whether $\varepsilon$ is the counit of an ...

**2**

votes

**0**answers

211 views

### When is the realisation of a simplicial set a manifold?

It is known that a simplicial complex is homeomorphic to a manifold if the link of every vertex is a simplicial sphere, for which there exists a definition. (I know that for high dimension, the ...

**11**

votes

**1**answer

359 views

### Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...

**3**

votes

**2**answers

289 views

### Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field.
This is a follow-up to this question.
I want to study varieties of objects generalizing ordered monoids, in particular using an ...

**4**

votes

**1**answer

98 views

### Does trace handle composition in a traced symmetric monoidal category?

Suppose that $(C,\otimes,I)$ is a traced symmetric monoidal category (TSMC) with symmetrizor $\sigma$ and trace $Tr$. Given two morphisms $f\colon A\to B$ and $g\colon B\to C$, I can tensor them to ...

**9**

votes

**1**answer

418 views

### Pushouts in the category of adjunctions

Let $\mathcal A$ be the category whose objects are all categories, and whose morphisms from a category $\mathcal C$ to a category $\mathcal D$ are all adjunctions $F \dashv G$ with $F : \mathcal C \to ...

**4**

votes

**1**answer

156 views

### Push-outs of fully faithful (enriched) functors

I'd like to know a reference where the following property, that I believe to be true, is checked: given a diagram of categories and functors $B\leftarrow A\rightarrow C$ (I'm actually interested in ...