# Tagged Questions

**2**

votes

**1**answer

133 views

### Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T ...

**2**

votes

**1**answer

56 views

### Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...

**1**

vote

**0**answers

93 views

### Retracts of 2-categories

Let $\mathbf{C}$ be a strict $2$-category and let $M = \{(x_\alpha,y_\alpha)\}$ be a collection of object-pairs so that each hom-category $\mathbf{C}(x_\alpha,y_\alpha)$ has an initial element ...

**2**

votes

**1**answer

271 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

**7**

votes

**1**answer

175 views

### Localizing 2-categories about a single morphism

This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and ...

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vote

**0**answers

125 views

### Axioms for a symmetric monoidal bicategory

I start reading the axioms for a symmetric monoidal bicategory. The axioms include so many diagrams to be satisfied. I am wondering if people really use these axioms directly to check a given data is ...

**1**

vote

**1**answer

197 views

### Inducing a Monoidal Structure using an Equivalence of Categories [closed]

Given an equivalence of categories $C \equiv D$, such that $C$ has a monoidal structure, is it clear that we can use the equivalence to induce a monoidal structure on $D$. Is there a standard ...

**15**

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**4**answers

781 views

### Expressing the Lebesgue integral using categories + the difficulty of describing estimates in category theory

In this question of mine in a comment to the accepted answer, someone remarked:
There are ways to express even basic things in analysis, such as the
spectral theorem or the Lebesgue integral, ...

**6**

votes

**2**answers

231 views

### Weakening simplicial identities

The generators $d_i, s_i$ for morphisms of the simplicial category satisfy simplicial identities:
$d_jd_i = d_id_{j−1}$ for $i < j$
$s_jd_i = d_is_{j−1}$ for $i < j$
$s_jd_i = id$ for $i = ...

**4**

votes

**2**answers

139 views

### symmetric monoidal double categories?

Let me preface this by saying that I don't know much category theory.
I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing ...

**4**

votes

**1**answer

252 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**

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**0**answers

86 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

156 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

269 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**

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**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 ...

**1**

vote

**1**answer

497 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

153 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

231 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

112 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**

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**0**answers

145 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

326 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**

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**0**answers

66 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**

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**2**answers

232 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

238 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 ...

**12**

votes

**3**answers

571 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

130 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 ...

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votes

**2**answers

214 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**

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**0**answers

151 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 ...

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votes

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226 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 ...

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votes

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210 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 ...

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168 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

162 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

277 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 ...

**21**

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**0**answers

618 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

164 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

224 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

163 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

137 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

229 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

289 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

3k 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 ...

**2**

votes

**2**answers

139 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 ...

**6**

votes

**2**answers

199 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

566 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

93 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 ...

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vote

**2**answers

134 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

267 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 ...

**9**

votes

**1**answer

331 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 ...

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votes

**0**answers

237 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

537 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 ...