# Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

**36**

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### What parts of the theory of quasicategories have been simplified since the publication of HTT?

It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression ...

**2**

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**0**answers

141 views

### Derived functors and derivatives

When looking at derived functors of a non-exact functor (e.g., ext, tor, sheaf cohomology groups) I am struck by their similarity to derivatives of a non-constant function in that they are both ...

**7**

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**1**answer

262 views

### What are the “smallest” topoi?

Yesterday I was talking to somebody from the Haskell community.
Late in the night we found ourselves discussing possible topoi.
Lets order topoi (up to equivalence, ...) by number of objects/...

**6**

votes

**1**answer

173 views

### About a canonical model structure on topologically enriched categories

Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying ...

**7**

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176 views

### When a localization of a category is (non-)reflective?

Let $S$ be a class of maps of a category $\mathcal{C}$. The localization of $\mathcal{C}$ at $S$ is reflective when the localization functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ has a fully faithful ...

**6**

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691 views

### What's the point of “created limits”?

Seeing as there's an nLab page about creation of limits, I take it that at least some people think this is an important notion. There's also a discussion here about what the "correct" definition of ...

**15**

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**1**answer

340 views

### When is the category of models of a limit theory a topos?

If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.
Is there a characterization of ...

**10**

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**0**answers

104 views

### Colimits of algebras for $\infty$-Monad

I would like to know in anyone has developed method for constructing colimits in the category of algebra for a monad in the $(\infty,1)$-categorical framework, using transfinite constructions.
I have ...

**7**

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**4**answers

395 views

### Localization of $\infty$-categories

In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...

**5**

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**1**answer

153 views

### Factorization of colimits through slices?

I could swear I remember a result of the following form:
Suppose we have a pair of functors $$C\xrightarrow{F}D\xrightarrow{G} X,$$ with $X$ cocomplete.
then we obtain a functor $$D\to X$$ sending $$...

**0**

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62 views

### Define a directed-complete partial order of lists

The List monad takes a set and produces the set of lists on that set. The elements of the set become the symbols or objects of the list. I would like to define a directed-complete partial order (...

**4**

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201 views

### Map from the Multiset Monad to the Giry Monad: From Data to Probabilities

The Mulitiset monad, aka the free commutative monoid monad or "Bag" monad, takes a set to the set of all Multisets for that set. A Multiset is like a set, but can have duplicates. It is used in ...

**4**

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**1**answer

182 views

### Conceptual and practical reasons and consequences of inverting weak equivalences

Although dealing with this in one or other form for many years, to my shame this question only struck me now.
One of the most radical differences between categories of "algebraic" and "topological" ...

**18**

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**2**answers

430 views

### Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$?

To a symmetric sequence $V_\bullet$ of vector spaces, associate the generating function $F_V(z) = \sum_n \frac{\dim(V_n)}{n!} z^n$. Then
$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$
...

**2**

votes

**1**answer

102 views

### Commutation of filtered colimits and finite limits in $\mathbb{CGWH}$

Do filtered colimits and finite limits (in particular pullbacks) commute in the category of compactly generated weak Hausdorff spaces?

**4**

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**1**answer

134 views

### DGA for a general abelian category

Differential graded algebras dga-s are fundamental objects of study in homological algebra and category theory. On the nlab webpage, they are defined as follows:
a dga is a monoid in the symmetric ...

**15**

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209 views

### What is the group completion of finite sets with respect to cartesian product?

Let $\Sigma_+$ be the groupoid of finite pointed sets. The Barratt-Priddy-Quillen theorem tells us that the group completion of $\Sigma_+$ with respect to the symmetric monoidal structure given by ...

**0**

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**1**answer

123 views

### The property of category of Semirings

I’m now thinking about the property of category of semirings Rig.
Is it complete or co-complete?
I think that Rig has projective and inductive limits, and finite products and co-products, so it ...

**6**

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154 views

### About a zig-zag of Quillen adjunctions

I have the following situation:
Three combinatorial model categories $\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$ such that all objects are fibrant (so no need to bother with the fibrant ...

**2**

votes

**1**answer

154 views

### Relaxing a natural isomorphism to a natural transformation to obtain a more general $2$-category

Many definitions of $2$-categories are given as categories equipped with some extra structure encoded by some functors and some natural (or extranatural) transformations between these functors (or ...

**2**

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**1**answer

83 views

### When does a 2-functor or 2-monad of Cat lift to a psuedofunctor or pseudomonad on Prof?

I'm currently reading Richard Garner's paper Polycategories via pseudo-distributive laws, and a central construction is the lifting of the symmetric strict monoidal category 2-monad to a pseudomonad ...

**7**

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104 views

### Explicit data for $E_n$-monoidal model and simplicial categories

The definition of a monoidal model category requires that the tensor product is biclosed (such is needed to ensure that the tensor product is derivable in both variables). Obviously, in the situation ...

**4**

votes

**1**answer

197 views

### Simplicial model categories and simplicial equivalence

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow ...

**6**

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94 views

### Spatiality of products of locally compact locales

In Johnstone´s Sketches of an Elephant Volume 2, page 716,
lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the locale product $X\times Y$ is spatial.
Is this ...

**25**

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2k views

### Categorifications of the real numbers

For the purposes of this question, a categorification of the real numbers is a pair $(\mathcal{C},r)$ consisting of:
a symmetric monoidal category $\mathcal{C}$
a function $r\colon \mathrm{ob}(\...

**7**

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**0**answers

289 views

### Why are commutative diagrams called “commutative”?

Does anyone know the rationale behind the name of "commutative diagrams"? To be precise, what is(are) the reason(s) for calling those diagrams "commutative" and in what sense?
I have previously asked ...

**9**

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**2**answers

220 views

### Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups

For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...

**7**

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**1**answer

270 views

### Stationarity and Fodor's lemma for a (nice) poset?

The notion of a stationary set is peculiar in that it applies to subsets of certain very particular posets -- ordinals or powersets. At least to a non-set-theorist, the situation seems to beg for the ...

**8**

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100 views

### Valuation with values in a semiring?

The notion of "valuation" on a ring $R$ is peculiar in that as typically presented, it is really two notions, neither of which subsumes the other.
A valuation can be a homomorphism $v: (R,\times) \to ...

**5**

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**1**answer

179 views

### When does $\text{Set}^{C^{\text{op}}}$ have split regular epi?

Is it possible to understand when in a presheaf category $\text{Set}^{C^{\text{op}}}$ every regular epimorphism splits?
Obviously I am looking for conditions on $C$ such that this is the case.
Since ...

**11**

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**3**answers

249 views

### Inequivalent compact closed symmetric monoidal structures on the same category

I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures.
We know how to construct disconnected toy ...

**6**

votes

**1**answer

124 views

### Analogue of Urysohn metrization for Lawvere metric spaces?

Urysohn proved that any regular, Hausdorff, second-countable space $X$ is metrizable, i.e. there exists a metric space whose underlying topological space is $X$. But what if we ask the same question ...

**55**

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3k views

### Why did Voevodsky consider categories “posets in the next dimension”, and groupoids the correct generalisation of sets?

Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one ...

**9**

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**1**answer

273 views

### What is the consistency strength of weak Vopenka's principle?

Weak Vopěnka's principle says that
the opposite of the category of ordinals cannot be fully embedded in any locally presentable category.
Recall that one form of Vopěnka's principle says that the ...

**8**

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212 views

### In what context can enriched category theory be done?

There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.
My question is what ...

**11**

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206 views

### Is $\mathrm{Hom}(P^i,P^j)$ a finite set? ($P=$ power set functor, $i\equiv j\bmod2$)

Let $P:\textbf{Set}\to\textbf{Set}$ be the contravariant power set functor, and put $P^n:=P\circ\cdots\circ P$ ($n$ factors), so that $P^n$ is a covariant (resp. contravariant) endofunctor of $\textbf{...

**3**

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**1**answer

92 views

### Internal equality for Eq-fibrations' morphisms

I have posted this question here on M.SE but since it received little attention and since it seems difficult to find helpfule references I reposting it here.
In Jacob's Categorical logic and Type ...

**7**

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**1**answer

180 views

### The different gradings of a graded ring, and their schemes

Let $(A,g)$ be a graded commutative ring, where $A$ denotes the commutative ring, and $g$ its grading. What can be said about the set $\mathcal{G}_A := \{ \mathrm{Proj}\Big((A,g)\Big)\ \vert\ g \}$? ...

**7**

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115 views

### Hochschild-Mitchell Homology

There is the notion of Hochschild-Mitchell homology for a $k$-linear category $\mathcal{C}$ (here $k$ can be a field). The definition is straightforward and so are some general properties, but somehow ...

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110 views

### Enriched homotopy colimit and space of paths

I work in the category of $\Delta$-generated spaces. Let $G$ be the group of strictly increasing continuous bijections from $[0,1]$ to itself (they necessarily take $0$ to $0$ and $1$ to $1$). I call ...

**7**

votes

**1**answer

220 views

### Proposition in HTT on cofibrations of categories

Proposition A.3.3.9. in Higher Topos Theory is as follows:
Let $S$ be an excellent model category and let $f:C\rightarrow C'$ be a cofibration of small $S$-enriched categories. Then (1) for every ...

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107 views

### Does the 2 category of Groupoids Admit the Vector Space Monad?

We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad.
3.2. Vector spaces. For a semiring S one can define the ...

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**1**answer

138 views

### What is the idempotent completion of the (2,1)-category of spans of finite sets?

I don't believe the $(2,1)$-category $FinSpan$ has split idempotents.
Question: Is there a simple description of the idempotent completion of $FinSpan$?
Foundationally, we may think of $FinSpan$ as ...

**5**

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**0**answers

97 views

### Internal hom as 2-Kan lift of pseudofunctor

Consider a situation where there is a pseudofunctor from some category $C$.
Due to it's geometrical properties, it might induce some sort of an internal hom on the original category (making it a ...

**2**

votes

**2**answers

116 views

### Example: Accessible category without colimits

I am looking for intuitive examples of the way(s) that colimits may fail to exist in the category of (Set-valued) models for a limit/colimit sketch.
Bonus points if the sketch and/or the colimit ...

**8**

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158 views

### What is a true invariant of $G$-crossed braided fusion categories?

Definition. An invariant of a (spherical) fusion category with extra structure is a number or a set or tuple of numbers preserved under (appropriate) equivalences.
(Spherical) fusion categories have ...

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193 views

### $MK+CC$ as a foundation for category theory

Has any work been done on what $MK+CC$ looks like as a foundation for category theory? Is it 'the same' as restricting to inaccessibles in some precise manner?
According to wikipedia, any category ...

**3**

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**1**answer

216 views

### Is Quillen's bracket a “universal enveloping” something?

$\newcommand{\G}{\mathcal{G}}$
In K-theory, there is a construction due to Quillen as follows. Let $(\G, \oplus, 0)$ be a monoidal groupoid. Then the bracket $\langle \G, \G \rangle$, sometimes also ...

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175 views

### Atlas of gerbe over stack

Suppose $F:\mathcal{X}\rightarrow\mathcal{Y}$ is gerbe over stack and $p:X\rightarrow \mathcal{X}$ is an atlas $\mathcal{X}$.
Does this imply $F\circ p:X\rightarrow \mathcal{Y}$ is an atlas for $\...

**6**

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124 views

### Product-preserving fibrant replacement functor for the Joyal model structure

There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\...