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
0answers
47 views

Should an object in the category always be a formal mathematical structure? [on hold]

Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So ...
3
votes
0answers
46 views

Categories in which an epimorphism applied to a non-monic epimorphism can be monic

Let $\mathcal{C}$ be a category, and let $A$, $B$, and $C$ be objects. Given $A \xrightarrow{f} B \xrightarrow{g} C$ such that: $f$ is both epic and monic $g$ is epic but not monic $gf$ is epic and ...
2
votes
0answers
113 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
1
vote
1answer
166 views

Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?

Let $\mathcal{A}$, $\mathcal{B}$ be two triangulated categories and $F: \mathcal{A}\to \mathcal{B}$ be a triangulated functor between them. Then $F$ induces an homomorphism between their Grothendieck ...
-3
votes
0answers
112 views

How to use the derived functor to write down a lot of natural morphisms or isomorphisms? [closed]

In the Kashiwara’s book:Sheaves on manifolds, the derived functor is defined in the abstract form, you can see some discussing at here: Derived functor Kashiwara’s book gives a lot of natural ...
3
votes
1answer
86 views

Is every frame monomorphism regular?

Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?
5
votes
0answers
128 views

Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
4
votes
1answer
111 views

Extremal, but not regular monomorphism

Is there an example of a category, and a monomorphism $m:X\to Y$ between two objects such that $m$ is extremal, but not regular? (A monomorphism $m:X\to Y$ is said to be extremal if whenever $m=g\circ ...
3
votes
0answers
104 views

Are all monomorphisms in the category of bounded lattices regular?

Let $\mathbf{Lat}_{01}$ be the category of bounded lattices with lattice homomorphisms that respect the smallest and the largest element. Is there a monomorphism in $\mathbf{Lat}_{01}$ that is not ...
0
votes
0answers
40 views

About cartesian closure of lax.functors categories

Let $\mathscr{A}$ a category and $F, G, H: \mathscr{A}^{op}\to CAT$ lax.functors. I wish find a possible "natural correspondence" between categories: $[F\times G, H]_O \leftrightarrow [F, H^G]_O$ ...
3
votes
0answers
67 views

Is there a classification of 2d extended TQFTs with defects?

Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly ...
2
votes
1answer
177 views

Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...
5
votes
1answer
164 views

Difference between coherent nerve of simplical model category and simplicial category

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in ...
7
votes
1answer
281 views

Constructing unnatural transformations

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere? Let $C$ ...
8
votes
1answer
243 views

When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian ...
3
votes
1answer
300 views

What are the automorphisms of $BG$?

Setup: Let's work in the category of schemes over $\mathbb C$. Let $G$ be a finite group. Let $BG=[pt/G]$ be the classifying stack of principal $G$ bundles. This is a fiberd category over the big ...
13
votes
1answer
527 views

What's the cardinality of a higher category?

The cardinality of a set is just the number of elements. To make sense of the cardinality of a category, one has to account for the morphisms. The usual definition is the sum over the isomorphism ...
1
vote
0answers
88 views

How does one compute a colimit of monoidal categories?

The question is in the title. I'm also happy to get answers about (your favorite adjective) monoidal categories. Here's a guess: In order to compute a colimit of monoids we can push everything down ...
2
votes
0answers
203 views

category theoretic approach to Sylow theorems and finite group theory?

Is there a category theoretic approach to Sylow theorems? More generally, is there an exposition of finite group theory in terms of category theory which would include Sylow theorems and little facts ...
1
vote
1answer
55 views

Quotients of termwise split injections, for additive categories

In the stack exchange notes found in Section 10 of this file, it is claimed that the category $K(\mathcal{A})$ of complexes up to homotopy is a triangulated category, if $\mathcal{A}$ is additive. In ...
2
votes
0answers
110 views

Finitely presented categories and limits

Suppose I have a finite graph $G$, and I then take the free category $\mathcal{C}(G)$ over such a finite graph. Now, I would like to "force" some objects to be limits. Is there a way to do that ...
1
vote
1answer
76 views

Projective family of probability spaces

This is a crosspost of this question from MSE. I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions ...
4
votes
0answers
231 views

A construction with homotopy colimits and homotopy pullbacks for descent

EDIT: Following the lines of some suggestions in the comments below, I try to add something more to explain the problem better. A map $\text{hocolim}Y\rightarrow\bar{Y}$ in $\text{Ho}(\mathbf{M})$ is ...
2
votes
0answers
68 views

What should one “do” to “strictify” a triangle of transformations coming from a lax commutative triangle of functors?

I would like to apologize for this rather stupid abstract nonsense question. Let $h=f\circ g$ for composable functors $f,g$; assume that there exist left or right adjoints to $f$ and $g$. Then it ...
5
votes
2answers
376 views

When is the category of small (pre)sheaves a(n elementary) topos?

When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos. When $C$ is ...
5
votes
0answers
160 views

Is there a higher, “orientalish” version of geometric realisation?

Geometric realisation of simplicial sets can be roughly thought of like this: In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...
7
votes
0answers
160 views

The bifunctoriality of co/limits

I recently noticed that there are two senses in which colimits are functorial, and I'm curious about their interplay. Let $C$ be a cocomplete category. Then, on the one hand, for any diagram ...
2
votes
1answer
173 views

Loop defects in Walker-Wang model

My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model. Elementary excitations in the WW model can be point particles, loop defects and more ...
1
vote
0answers
78 views

is sufficient cohesion equivalent to the connectedness of subobject classifier?

I'm following Lawvere article Axiomatic Cohesion. He states (Proposition VI.4) that sufficient cohesion is equivalent to the connectedness of subject classifier, but I can't follow the proof. I can't ...
2
votes
0answers
127 views

Relationship between coherent toposes/coherent logic and coherent sheaves

I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in ...
2
votes
0answers
65 views

Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...
3
votes
1answer
320 views

Notation: Categories of measur(abl)e spaces

Is there a common notation in the literature for the category of measurable spaces and measurable maps? the category of measure spaces and measure-preserving maps? The nlab suggests ...
0
votes
0answers
64 views

Is this quasi-coherent sheaf a subsheaf of $\ker f$?

Let $f: \mathcal{F}\to \mathcal{G}$ be a morphism of quasi-coherent sheaves over a scheme $X$. Let also $T_U$ be a submodule of $\ker f_U$ with $|T_U|\leq \kappa$ for each open subset $U$ of $X$ ...
2
votes
0answers
148 views

Continuous maps to fat geometric realizations of simplicial spaces

The nLab page on partitions of unity mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel ...
0
votes
1answer
92 views

tree derived from monad is itself a monad

I have constructed a functor from a monad that appears (based on computer experiments to test the monad laws) to also have monad properties but I am having trouble proving it. Here is the idea: M[A] ...
3
votes
0answers
101 views

Proofs in monoidal categories [closed]

I have to do some pretty ugly proofs in monoidal categories. Basically, I have some long identities that I would like to prove. A random example: $$(a\otimes b)\circ (c\otimes d) \circ q = q $$ Are ...
5
votes
1answer
109 views

What homomorphisms $G \to BrPic(\mathcal{C})$ correspond to group-theoretical $G$-extensions of $\mathcal{C}$?

For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$. ...
2
votes
1answer
272 views

Recollement of multiple $t$-structures

Given a recollement $$ \mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} ...
1
vote
0answers
126 views

Récollement of stable $t$-structures

Given a recollement $$ \mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} ...
-1
votes
1answer
79 views

Is this apushout diagram [closed]

Let $A, B, C, E$ and $F$ be some objects in an abeleian category $\mathcal{C}$. Let we have a commutative diagram \begin{array}{ccccccccc} 0 & \xrightarrow{} & A & \xrightarrow{f} & ...
0
votes
0answers
34 views

How is the monoidal product defined on the functor category between symmetric monoidal dagger cats

I have found a quote in a paper by Abramsky and Heunen If C and D are symmetric monoidal dagger categories, then so is the category [C, D] of functors F : C → D that preserve the dagger. ...
0
votes
1answer
73 views

Canonical colimit and cartesian product of simplicial sets

Let $K$ be a simplicial set and let $\Delta K$ be the category of simplices, i.e the category where the objects are simplicial maps $$ \Delta[n]\to K $$ and the maps $\phi\: : \: (\Delta[n]\to K)\to ...
11
votes
4answers
687 views

Brandt's definition of groupoids (1926)

The definition of a category is usually attributed to Mac Lane and Eilenberg (1945). What seems to be less known is that the german mathematician Heinrich Brandt has developed the notion of a groupoid ...
5
votes
1answer
365 views

Functor generalisation

In an article I am writing, I am led to the following generalization of the notion of functor. Let $C$ and $D$ and be two categories. A generalized functor $F : C \to D$ is given by: a function $f : ...
2
votes
0answers
40 views

Lattice of subobjects of a particular coproduct

I have the following situation: $\mathcal C$ is a (good enough, say Grothendieck) Abelian category and $F:\mathcal C\to \mathcal C$ is self-equivalence. Given an object $C$ in $\mathcal C$, what can I ...
3
votes
1answer
95 views

Is the image of a idempotent morphism in $\mathcal{K}(\mathcal{A})$ defined in the naive way?

Let $\mathcal{A}$ be an abelian category and $\mathcal{K}(\mathcal{A})$ be the homotopy category of chain complexes in $\mathcal{A}$. It is well-known that $\mathcal{K}(\mathcal{A})$ is idempotent ...
1
vote
0answers
72 views

When is the category of endofunctors braided? When is it ribbon? Fusion? Modular?

Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural ...
0
votes
0answers
63 views

The first lemma in Auslander's functors and morphisms determined by objects

[lemma 1.1] Let $\mathcal{C}$ be a preadditive category. Suppose G is in ($\mathcal{C^{op}}$, $\mathcal{Ab}$). If for each $X$ in $\mathcal{C}$ we are given a subgroup $A_x$ of $G(X)$ such that ...
21
votes
3answers
722 views

Possible categorical reformulation for the usual definition of compactness

Let $X$ be a compact topological space, $f_i:Y_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f_i^{-1}(U)$ is open for each $i$, for ...
5
votes
1answer
142 views

Relations between functors in a recollement

Consider a recollement situation like the following by the very definition of the various functors it follows that $i^* j_*=0$, and $j^! i_* = 0 = j^* i_!$. Also, $j^! i_! = 0 = j^* i_*$ by ...