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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115 views

Poset-enrichment of abelian categories

Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another ...
8
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274 views

End of the Ext functors

Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...
8
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2answers
595 views

random category theory

This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references ...
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0answers
36 views

Comparing right and left quasi-representable bimodules

Let $\mathcal V$ be your favourite (closed, symmetric) monoidal model category. To fix ideas, set $\mathcal V = \mathrm{Ch}(k)$, the category of chain complexes over a fixed commutative ring. Given a ...
2
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1answer
119 views

Are lax functor categories into a cartesian closed 2-category cartesian closed?

Suppose that $C$ is a complete closed monoidal category and $I$ is any small category. Then the functor category $Fun(I,C)$ is again a closed monoidal category with the pointwise tensor product $F ...
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1answer
64 views

Strict comma objects implies comma objects

I'm condusion on a statement in this page comma object in $n$lab. It states: any strict comma object is a comma object, but the converse is not in general true. My confusion is: the strict comma ...
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0answers
162 views

Monoidal Forgetful/Free Adjunction for $E_2$-algebras

Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms ...
7
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1answer
364 views

Explicit counter example to Vopěnka's principle in the constructible universe?

Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle ...
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32 views

Morphisms between lax wedges

In the paper BOZAPALIDES, S., Les fins cartésiennes the following definition of a lax wedge for a 2-functor $S\colon \mathcal{A}^{op}\times \mathcal{A}\to \mathcal B$ between 2-categories is ...
3
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0answers
142 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 ...
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0answers
133 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 ...
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1answer
207 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 ...
4
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1answer
98 views

Is every frame monomorphism regular?

Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?
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0answers
164 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 ...
5
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1answer
123 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 ...
4
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0answers
114 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 ...
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0answers
46 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: $[F\times G, H]_O \leftrightarrow [F, H^G]_O$ ...
6
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139 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
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1answer
194 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
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1answer
194 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 ...
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1answer
318 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 $C$ ...
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1answer
273 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 ...
4
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1answer
324 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 ...
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1answer
575 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 ...
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0answers
102 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 ...
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0answers
221 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 ...
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1answer
60 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 ...
3
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0answers
136 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
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1answer
101 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
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0answers
254 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
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0answers
75 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 ...
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2answers
416 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
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0answers
179 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
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0answers
180 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
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1answer
214 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 ...
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0answers
88 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
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0answers
149 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
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0answers
70 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
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1answer
345 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 ...
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67 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 $X$ ...
4
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1answer
397 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 ...
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1answer
103 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
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0answers
107 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
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1answer
125 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 $\boxtimes_\mathcal{C}$. ...
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1answer
406 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} ...
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0answers
132 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} ...
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1answer
90 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} & ...
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0answers
38 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. ...
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1answer
76 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 ...
12
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4answers
741 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 ...