# Tagged Questions

**4**

votes

**1**answer

190 views

### Universal property of module categories over monads

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...

**3**

votes

**0**answers

81 views

### Bicategorical limits with parameters

(This question was asked in http://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.)
Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a ...

**5**

votes

**1**answer

143 views

### Making additive envelopes of monoidal categories monoidal

I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$.
(This is defined as the category with ...

**4**

votes

**2**answers

249 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 ...

**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

437 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 ...

**4**

votes

**1**answer

145 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 ...

**3**

votes

**2**answers

220 views

### Intersection of free objects

I am aware that the following question is a very basic one and therefore I would not be at all offended if it were to be closed. Moreover, I am not familiar at all with category theory.
Let ...

**3**

votes

**0**answers

108 views

### Reference request: colimits of locally presentable categories

Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small ...

**5**

votes

**0**answers

143 views

### Nice references for injective model structures and Quillen functors between motivic homotopy categories

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...

**6**

votes

**1**answer

316 views

### Morita theorem for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting?
I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...

**3**

votes

**0**answers

64 views

### When is the localic reflection of a topos discrete?

Recall that the inclusion of locales into topoi has a left adjoint, called the localic reflection. It sends a topos $\mathcal{E}$ to the poset of subobjects of the terminal object, which is a locale. ...

**3**

votes

**2**answers

225 views

### Analogues of 'cone' distinguished triangles for pointed model categories?

For an additive $A$ and any morphism $f:X\to Y$ in $C(A)$ one has the following distinguished triangle in the homotopy category $K(A)$: $X\to Y\to Cone(f)\to X[1]$.
What is the closest analogue of ...

**4**

votes

**1**answer

232 views

### Two definitions of modules in monoidal category

The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here ...

**11**

votes

**3**answers

554 views

### The interplay between certain aspects of interpretability, model theory and category theory

I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would ...

**4**

votes

**1**answer

127 views

### Yoneda embeddings of stable model categories; composition with Bousfield localizations

For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question ...

**2**

votes

**2**answers

209 views

### Is $\text{Ind-}\bf C$ the category of models for a sketch?

It seems to me that the "indization" process of a category can be formulated in the language of sketches (by sketch I mean what is defined in [LPAC].2.F, Def. 2.55); in particular, see this answer by ...

**3**

votes

**0**answers

139 views

### Homotopy theory of acyclic categories

Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the ...

**6**

votes

**0**answers

225 views

### Generalizing prime numbers to product-indecomposable objects in toposes

Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of ...

**6**

votes

**0**answers

203 views

### Extension to a right exact functor

Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...

**3**

votes

**0**answers

160 views

### Can one construct Freyd-Mitchell's embeddings that respect sheafifications?

For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...

**5**

votes

**1**answer

159 views

### Monoidal transformations are isomorphisms at dualizable objects

Here is a cute observation: Let $F,G : \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functors between symmetric monoidal categories, and let $\eta : F \to G$ be a monoidal transformation. Then ...

**8**

votes

**1**answer

260 views

### The state of the art in the rectification of homotopy-coherent structures

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of:
Cordier and Porter proved a ...

**17**

votes

**0**answers

466 views

### categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...

**3**

votes

**1**answer

155 views

### Limits in span categories

What are the limits in the span categories? and what is known about them in the literature?

**3**

votes

**1**answer

221 views

### Categories where morphisms are pairs of adjoint functors

Is there a particular name for those categories (of categories) where morphisms "come as adjoint pairs", as in the case of toposes and model categories?
Is there any attempt to study them as general ...

**5**

votes

**1**answer

160 views

### Reference request: (co)limits in Eilenberg--Moore (V-)categories

The following result seems to be well known:
If T is a (V-)monad on a (V-)category C, then the forgetful functor $U^T \colon C^T \to C$ creates
any limits that exist in C, and
any ...

**1**

vote

**1**answer

133 views

### “Uniformly (co)well-powered” categories?

Let $C$ be a category and for each object $X$ let us denote by ${\sf Mono}(X)$ the category of all monomorphisms in $C$ going to $X$ (i.e. monomorphisms which have $X$ as range -- I hope it is clear ...

**4**

votes

**1**answer

224 views

### Quillen equivalence of diagram categories

Let $C$ be a model category and $B$ a direct category. By Theorem 5.1.3 in Mark Hovey's book Model categories, there is a model category structure on the diagram category $C^B$ such that weak ...

**4**

votes

**1**answer

288 views

### What is the image of the intial object inside the final object called?

A recent project has forced me to consider a rather special object in a rather nasty category. Consider any category $\mathcal{C}$ which has
image objects, meaning for each morphism $f: x \to y$ ...

**35**

votes

**4**answers

2k views

### What is the source of this famous Grothendieck quote?

I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck.
It is better to have a good category with bad objects than a bad category ...

**1**

vote

**2**answers

131 views

### Equivalence relations in suplattices

I am wondering about generalisations of the concept of equivalence relations to suplattices.
Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. For suplattices ...

**5**

votes

**2**answers

195 views

### Is there a name for (pre)sheaves satisfying this transitivity condition?

Suppose that $F:\mathscr{C}^{op} \to Set$ is a presheaf. For an object $C$, denote by $Aut(C)$ the group of isomorphisms $f:C \to C.$ There is a canonical action $Aut(C)$ of on $F(C).$ Is there a ...

**6**

votes

**5**answers

549 views

### What is a good basic reference on model categories?

I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ...

**4**

votes

**0**answers

92 views

### Metrized categories

Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let ...

**1**

vote

**2**answers

127 views

### Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)?

Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain complexes over $\scr ...

**3**

votes

**1**answer

253 views

### Reference request: sheaves on closed sets

I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. Specifically, I say a ...

**8**

votes

**1**answer

301 views

### Reference for “lax monoidal functors” = “monoids under Day convolution”

Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves colimits separately in ...

**2**

votes

**0**answers

230 views

### Transformation between Left and Right Kan Extensions

The following object has turned up in my research recently, and it would be surprising (to put it mildly) if no one else has seen, used or studied it before. I am hoping for a name and some references ...

**5**

votes

**1**answer

526 views

### Internal Day convolution

Let me recall that any small category $\mathbb{A}$ enriched in a complete and cocomplete symmetric monoidal closed category $\mathbb{V}$ admits embedding (the Yoneda embedding):
$$y_\mathbb{A} \colon ...

**6**

votes

**1**answer

281 views

### Tannaka duality for C*-algebras?

Tannaka-Krein
duality shows
how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional
complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to
...

**2**

votes

**2**answers

284 views

### Is there any nontrivial monad on the category of graphs?

The question is in the title, but let me specify what I mean by the category of graphs.
In the context of this question, the category of graphs is the category of symmetric irreflexive relations. ...

**3**

votes

**5**answers

368 views

### Connected groupoids and action groupoids

It is written in Wikipedia http://en.wikipedia.org/wiki/Groupoid, that any connected groupoid $A\rightrightarrows X$ is isomorphic to an action groupoid $G\ltimes X$ coming from a transitive action ...

**1**

vote

**1**answer

150 views

### How would you say that a small category is embedded into functors from a large $C'$ to abelian groups?

How would you say that a small additive category $C$ embedds (contravariantly) into the category of exact functors from a 'large' abelian $C'$ into abelian groups (this is something like Yoneda's ...

**1**

vote

**0**answers

89 views

### The semicat of morphisms which are neither right nor left invertible

Given a cat $\bf C$, the class $\mathcal{S}$ of all $\mathbf{C}$-morphisms that are neither left nor right invertible, generates a "genuine" subsemicat $\bf S$ of $\bf C$ (if necessary, see here for ...

**7**

votes

**0**answers

202 views

### Colimits of quasi-coherent sheaves on a ringed space

Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is ...

**2**

votes

**1**answer

282 views

### The point of view of semicats in functional analysis

I'm completing a paper about (Mitchell's) semicats (well, not exactly, but let's say so for simplicity), and as a motivational example I'd like to mention at some point that the monic/epic morphisms ...

**9**

votes

**1**answer

390 views

### Reference for my monads?

I'm looking for a reference for a certain pair of monads on $Cat$. One problem is that I don't know the modern way of thinking about some basic things, so excuse me if my presentation is naive.
First ...

**8**

votes

**0**answers

239 views

### Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples?
Definition. Let $\mathcal{C}$ be a ...

**2**

votes

**1**answer

361 views

### Any abelian category as filtered colimit of categories of projective modules

Recently I have heard somewhere that any (edit: small) abelian category can be expressed as the colimit of categories of projective modules over some rings. The remark was that this is "basically just ...