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

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4
votes
1answer
108 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
0answers
122 views

Commutation of simplicial homotopy colimits and homotopy products in spaces

Edit: The claim below is wrong, as explained in the comments, because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy ...
2
votes
2answers
312 views

Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...
5
votes
2answers
471 views

Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...
1
vote
0answers
180 views

Are these two “FUNCTORS” adjoint?

I am considering the following correspondence: Let $X$ be quasi compact quasi separated schemes.Consider a pseudo functor \begin{equation}Sch\rightarrow CAT :U\mapsto Qcoh(U),f:U\rightarrow V\mapsto ...
2
votes
0answers
110 views

If the direct image of f preserves coherent sheaves on notherian schemes,how to show f is proper?

The other direction is well known I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove.I am also wondering whether ...
-3
votes
0answers
65 views

How do I calculate adjoint functors to forgetful functors? [closed]

Repost of question from mathstackexchange, because I thought it would be more appropriate here: Suppose I have a forgetful functor $F:Ab\hookrightarrow Grp$ where we forget that we have ...
3
votes
1answer
118 views

Categorical characterization of closed imbeddings

Let $f\colon X\to Y$ be a morphism of schemes. Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. ...
2
votes
1answer
62 views

Pivotal functors of that are substantially different from finite group homomorphisms

Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a ...
6
votes
1answer
165 views

Infinite dimensional 2-Hilbert spaces

Is there a definition of an infinite dimensional 2-Hilbert space? Finite dimensional 2-Hilbert spaces have been discussed by Baez in http://arxiv.org/abs/q-alg/9609018 In the more recent paper by ...
22
votes
8answers
1k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
3
votes
0answers
143 views

Category of modules over commutative monoid in symmetric monoidal category

Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes: In many cases, ...
2
votes
1answer
267 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 ...
1
vote
0answers
41 views

Are regular epi of locale stably epic?

It is well know that the category of locale is not a regular category, that is the pullback of a regular epimorphism is not always a regular epimorphism: for example, the classical counterexample ...
5
votes
1answer
221 views

Ends as a “cotrace” operation on profunctors

As mentioned here, there is a trace operation on the monoidal category of profunctors given by taking coends: for any profunctor $F : A\times X \nrightarrow B \times X$, there is a profunctor $Tr^X(F) ...
27
votes
6answers
1k views

Clifford algebra as an adjunction?

Background For definiteness (even though this is a categorical question!) let's agree that a vector space is a finite-dimensional real vector space and that an associative algebra is a ...
16
votes
2answers
913 views

Colimits in the category of smooth manifolds

In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of ...
-2
votes
0answers
141 views

Bi-dual of vector spaces [closed]

Back in the early 1940s, Eilenberg and MacLane introduced category theory, starting with the attempt to clarify such a simple looking issue as the following. There is a most simple and natural linear ...
15
votes
0answers
390 views

Relative consistency of ETCS over the theory of a well-pointed topos with NNO

EDIT: I'm bumping this, because I'm still curious, and because I have a relative consistency result over the theory of a well-pointed topos with NNO, and I am wondering how much baggage I save by not ...
4
votes
1answer
221 views

A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea. Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
1
vote
2answers
155 views

Directed subposet of a poset containing the minimal elements

The following appears naturally in a certain context: Let $P$ be a graded partially ordered set. Let $M$ be the subset of minimal elements of $P$. Define subsets $E_i$ inductively as follows: First, ...
2
votes
1answer
260 views

Brutal truncation of indecomposable complexes

Let $P^{\bullet} = (P^i, d^i)_{i\leq 0}$ be an indecomposable object in category of complexes bounded above, where each $P_i$ is a finitely generated projective module over an finite dimensional ...
2
votes
2answers
418 views

What structure has been found for functions with this relationship.

Given $f$ and $g$ $\forall x y. f(x) = f(y) \Longrightarrow f(g(x)) = f(g(y))$ Or equivalently $ker\ f \subseteq ker\ (f \circ g)$. Note: if $f$ is injective then this holds for any $g$. ...
9
votes
1answer
522 views

Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory? I refer to the (systematic, formalist) study of the category of sheaves on a site or the study of topology in a ...
13
votes
2answers
604 views

What's an example of a locally presentable category “in nature” that's not $\aleph_0$-locally presentable?

Recall the notion of locally presentable category (nLab): $\DeclareMathOperator{\Hom}{Hom}$ Definition: Fix a regular cardinal $\kappa$; a set is $\kappa$-small if its cardinality is strictly less ...
33
votes
4answers
2k views

Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
1
vote
3answers
329 views

Is there a general notion of semigroup action?

The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a ...
2
votes
0answers
64 views

Why does $\pi_t$ preserve pullbacks in this special case?

Let $X$ be a fibrant pointed cosimplicial space. Following Bousfield-Kan, let $\text{lim}^{\partial \Delta_{n+1}} X = M^{n}X$ be the nth matching object of $X$. Onecan then show that there is a ...
2
votes
1answer
64 views

From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...
4
votes
2answers
310 views

Aspheric functors and Grothendieck fibrations

Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, ...
2
votes
0answers
103 views

Internal categories in an endofunctor category

Here we see the definition of an internal category in a monoidal category. We also know that endofunctor categories support a monoidal product which is actually functor composition. It is the case ...
10
votes
3answers
1k views

Is there a categorical proof of Gödel's incompleteness theorem?

A significant result in set theory was shown by Cohen when he showed that the continuum hypothesis was independent of ZFC using a new technique called forcing. In Topos theory, this result has a new ...
2
votes
2answers
137 views

Adjoint of Pushout as Modal Operators in Internal Logic

Regarding the internalization of mathematics to a particular category as in the nLab article: Internal Logic, there is a peculiar table mentioned in the section on Categorical Semantics in which there ...
49
votes
11answers
3k views

Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations. Why was there the necessity of singling ...
5
votes
2answers
225 views

What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
6
votes
2answers
293 views

A continuous notion of realizability

I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to ...
5
votes
0answers
116 views

Compact objects and ind-objects in triangulated categories

question : let $A$ be triangulated category compactly generated by subcategory $A^c$ of compact objects. Consider category of ind-objects $Ind(A^c)$. Is there relation between $A$ and $Ind(A^c)$? ...
8
votes
2answers
453 views

Modern versions of Verdier's hypercovering theorem?

Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a ...
2
votes
1answer
132 views

what is the stabilization of pointed sets?

Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab). The famous example is ...
21
votes
3answers
1k views

t-structures and higher categories?

I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer). Given a triangulated category, one ...
4
votes
1answer
216 views

Higher Degree Data in a Cosimplicial Quasicategory and Delooping

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference. My question is regarding accessing data ...
0
votes
2answers
84 views

a dcpo seen as a category: when does a dcpo map induce a functor with an adjoint?

Take two posets $A, B$ (partially ordered sets). Now consider these posets to be categories $Cat(A), Cat(B)$ respectively. Consider a map from $A$ to $B$, $f: A \rightarrow B$. This can be seen as ...
9
votes
6answers
3k views

why haven't certain well-researched classes of mathematical object been framed by category theory?

Category theory is doing/has done a stellar job on Set, FinSet, Grp, Cob, Vect, cartesian closed categories provide a setting for $\lambda$-calculus, and Baez wrote a paper (Physics, Topology, Logic ...
7
votes
3answers
307 views

Koszul duality for modular operads

Has anyone defined what it means for a modular operad to be Koszul, or what the Koszul dual of a modular operad is? In particular, is it meaningful to say that a modular operad is quadratic? Merkulov, ...
2
votes
3answers
140 views

Locally finite varieties which are not finitely generated

Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...
6
votes
3answers
391 views

In the category of sets epimorphisms are surjective - Constructive Proof?

The statement that surjective maps are epimorphisms in the category of sets can be shown in a constructive way. What about the inverse? Is it possible to show that every epimorphism in the category ...
6
votes
3answers
866 views

Do Disjoint Unions and Fiber Products Commute?

Do disjoint unions and fiber products commute? In other words, is the following statement true? Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Let {$U_{i}$} be a ...
7
votes
1answer
440 views

Learning roadmap to TQFT from a mathematics perspective

I had asked a question on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posting it below. ...
4
votes
2answers
198 views

What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?

In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in ...
8
votes
1answer
361 views

Notion of infinity in categories

Please excuse me if the question is too vague or uninteresting. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. Motivated by the equivalence of Dedekind-finiteness and finiteness ...