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.
4
votes
1answer
57 views
Flattening categories
Motivating example:
Given a ring, we can construct its lattice of ideals functorially
(with morphisms mapping to the preimage maps on ideals).
Next, we may flatten this category of lattices, obtaining ...
3
votes
0answers
63 views
Motivation for using etale topology in representability of functors problems
I am reading a paper that proves the representability for certain functors whose domain is the category of superschemes. The paper claims that to prove representability of functors (or possibly just ...
5
votes
1answer
125 views
Sheaves over a sheaf
Everything I write I mean in the in the sense of Lurie's HTT.
Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C}...
8
votes
0answers
75 views
Vertex algebras and factorization algebras
It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "...
4
votes
1answer
129 views
Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$?
Given a morphism of Lie groups $ \theta:G\rightarrow H$ and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle.
See that the morphism of Lie ...
8
votes
0answers
116 views
Adjoints to forcing
Forcing over a partial order $P$ can be viewed in a category theoretic sense as constructing the presheaf topos ${\bf Set}^{P^{op}}$ over the partial order (viewed as a category) then passing through ...
1
vote
1answer
110 views
Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?
In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization---the quotient of $F(UC +_{obj W} W^{op})$, the free category on the ...
14
votes
1answer
492 views
Are fully general Frobenioids necessary?
Mochizuki's notion of a Frobenioid introduced in The geometry of Frobenioids I is rather elaborate. However, he also introduces a myriad of further properties that a Frobenioid may satisfy, and his ...
2
votes
1answer
171 views
A module associated to an endomorphism of a vector bundle
Let $E$ be a vector bundle over a compact connected Hausdorff space $X$. To an endomorphism $\alpha \in End(E)$, we associate a $C(X)-$ module $\Gamma(E,\alpha)$ consisting of all $\beta\in End(E)$ ...
6
votes
0answers
106 views
Kleisli category with equalizers?
I was reading Hosseini's paper on "Equalizers in Kleisli Categories", yet I couldn't figure it out an example of a Kleisli category with (all) equalizers with the extra requirement of not being ...
5
votes
2answers
249 views
Semantics-structure adjunction
In the discussion on the nLab article for monadic adjunctions, John Baez suggests and Mike Shulman confirms that the relationship between adjunctions and monads itself constitutes an adjunction called ...
16
votes
2answers
818 views
A multicategory is a … with one object?
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly ...
2
votes
1answer
181 views
Understanding the definition of $G$-gerbe
In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following.
Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\...
6
votes
1answer
161 views
Derived category of a quotient
Le $A$ be an abelian category and $B$ a Serre subcategory of $A$. Under which conditions do we have $$D^*(A/B) \simeq D^*(A)/D^*(B),$$ where $D^*$ stands for the derived category with $* = +,-,b$ or ...
7
votes
0answers
93 views
Stability of accessible $\infty$-categories under some operations
I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.
In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...
1
vote
1answer
498 views
Is there any work related to Hypertype Theory? [closed]
Update: Since the question was put on hold, work on this will continue on this notebook.
So long, and thanks for all the fish!
First, let me state that I am not a formally-trained ...
2
votes
0answers
79 views
The Kleisli Category of the Monad of Measures of Finite Support and its composition formula
In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad. Let's call this ...
2
votes
0answers
179 views
Are there enough curves (to connect 'points' of f.g. algebras)?
(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the ...
3
votes
0answers
125 views
final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?
In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...
7
votes
2answers
232 views
Infinite Krull-Schmidt categories?
In a Krull--Schmidt category, if
$$
X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s},
$$
where the $X_{i}$ and $Y_j$ are all indecomposable, then $r = s$, ...
3
votes
0answers
121 views
Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable
I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu.
Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open ...
0
votes
1answer
142 views
A question about combinatorial model categories
I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.)
He uses this result to prove Jeff ...
25
votes
3answers
583 views
What is the structure preserved by strong equivalence of metrics?
Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
4
votes
1answer
212 views
Does the Eilenberg Moore Construction Preserve fibrations?
Say we have a Grothendieck fibration $p : E \to B$ and a monad $T$ on $B$ and a lift $T'$ of $T$ to $E$, i.e. a monad on $E$ such that $pT' = Tp$ and $p$ preserves $\eta, \mu$.
Then because the ...
0
votes
0answers
79 views
Relation between the inverse of a natural isomorphism and counit of an adjunction
Let $\mathcal{C}$ and $\mathcal{D}$ be two categories. Suppose that
$F\colon\mathcal{C}\to\mathcal{D}$ is a functor from $\mathcal{C}$ to $\mathcal{D}$;
$U\colon\mathcal{D}\to\mathcal{C}$ is a ...
4
votes
0answers
89 views
Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?
Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
1
vote
0answers
66 views
Notions of connected components in a finite family fibration
Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...
4
votes
0answers
74 views
What minimal structure is required to define Clifford modules in a way as abstract as possible?
Start with a quadratic form $q$ on a vector space $V$. A module $M$ over the corresponding Clifford algebra is determined by a map $\cdot:V\otimes M\to M$ satisfying $v\cdot(v\cdot m)=-q(v)m$.
Now ...
14
votes
2answers
324 views
Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?
It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...
9
votes
1answer
514 views
Higher Topos Theory Theorem 2.2.5.3
The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.
We have a trivial Kan ...
14
votes
0answers
257 views
The “concreteness preorder” on categories
A category is called concrete if it is equipped with faithful functor to $\mathrm{Set}$. Let us say a category $C$ can be made concrete if there exists a faithful functor $F \colon C \to \mathrm{Set}$...
2
votes
1answer
178 views
Category with one isomorphism class
What one can say about a category in which, any two objects are isomorphic. I know that this is a strange question.
Thanks
6
votes
1answer
270 views
Is the embedding ${\sf cat}\subset {\sf Cat}$ dense?
A quick question.
Let $j : {\sf cat} \to {\sf Cat}$ be the inclusion of small categories in locally small categories; is $j$ a dense functor?
If it is not (as I expect, since it's hard for a ...
6
votes
1answer
225 views
Does there exist a full and faithful embedding of $\mathsf{Poset}$ in $\mathsf{Set}$?
Does there exist a full and faithful embedding of the category of posets into the category of sets? I suspect no, but I don't know how to prove or disprove this.
2
votes
1answer
58 views
Reflexive coequalizer and singular functor.
Does the singular functor $sing: Top\rightarrow SimpSet$ form the category of topological spaces to simplicial sets commutes with reflexive coequalizers?
Recall that the singular functor $sing$ is ...
6
votes
1answer
210 views
Does the Dwyer-Kan model structure make dgCat a model $2$-category?
Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
1
vote
0answers
236 views
Does this axiomatic system satisfy requirements for founding mathematics?
In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is ...
10
votes
0answers
576 views
Toposophy vs Set theoretical multiverse philosophy
Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...
3
votes
0answers
133 views
filetered colimit of fibrant-cofibrant objects
Suppose that we have a $\lambda$-combinatorial model category $M$ (for some cardinal $\lambda$) such that any $\lambda$-filtered colimit of fibrant-cofibrant objects is always fibrant. My question is ...
5
votes
1answer
271 views
Universal property of sheaf category
Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...
9
votes
1answer
234 views
Functoriality of (co)limits in $\infty$-categories
I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) ...
5
votes
1answer
189 views
Comonad for normalized pseudofunctors for strict higher categories
Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{...
2
votes
1answer
79 views
Definition of $\in_c$ for power objects
On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...
7
votes
1answer
269 views
Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?
Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...
10
votes
3answers
732 views
Are inclusions “canonical” injections?
[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]
Summary of question: the inclusions are a particularly ...
3
votes
0answers
58 views
Do we have criteria of strict localization of a Grothendieck category?
Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...
21
votes
5answers
1k views
are quotients by equivalence relations “better” than surjections?
This might be a load of old nonsense.
I have always had it in my head that if $f:X\to Y$ is an injection, then $f$ has some sort of "canonical factorization" as a bijection $X\to f(X)$ followed by an ...
3
votes
0answers
83 views
How to show mapping cones are homotopy cofibers
In a dg-category $\mathcal{C}$, the $n$-translation of an object $C$ is an object $C[n]$ representing the functor
$$
{\rm Hom}(-,C)[n].
$$
The cone of a closed morphism $f\colon C \to D$ of degree ...
4
votes
0answers
155 views
How to construct the espace étalé (space of sections) for an arbitrary category?
I want to consider the sheaf valued in an arbitrary category (not only of sets, groups, modules and so on) on a topological space, using the language of étalé space.
In all references I am reading (...
11
votes
3answers
656 views
Comparisons of convenient categories for algebraic topology
I've heard that there are many convenient categories for algebraic topology. Such categories often have many nice properties like being cartesian closed, complete, cocomplete, the forgetful functor ...
