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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5
votes
2answers
409 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
174 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
178 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
207 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
86 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
146 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
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
votes
1answer
343 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
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
votes
1answer
392 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
101 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
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
votes
1answer
123 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
347 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
131 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
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} & ...
0
votes
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. ...
0
votes
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 ...
11
votes
4answers
727 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
375 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
45 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
99 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
86 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
67 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
738 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
156 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 ...
8
votes
4answers
763 views

Topological Grothendieck Construction

Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and ...
0
votes
0answers
51 views

Intersection and union of torsion classes

One of the main result in Cassidy, C., M. Hébert, and G. M. Kelly. "Reflective subcategories, localizations and factorization systems." Journal of the Australian Mathematical Society (Series A) ...
7
votes
3answers
489 views

Category of Gödel Codings? [Reference Request]

Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with ...
4
votes
1answer
204 views

Establishing Duality in Tannakian Categories

I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid. However, I find the definition of rigid categories somewhat difficult. I don't ...
0
votes
2answers
260 views

Serre's Theorem for Coherent Sheaves

I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...
4
votes
0answers
127 views

When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?

Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal ...
8
votes
2answers
297 views

Definition of internal field objects

Let $\mathcal{C}$ be a category with finite limits and a strictly initial object $\mathbf{0}$. The final object is denoted by $\mathbf{1}$. I propose the following definition of a field object ...
5
votes
0answers
119 views

How do you categorify the cycle index series?

Let $F$ be a combinatorial species. The exponential generating series of $F$ is defined to be $$ \sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!} $$ It was observed by Baez and Dolan in their paper ...
1
vote
0answers
131 views

Sum-epimorphisms and prod-monomorphisms

        Sum-epimorphisms A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition: ...
4
votes
1answer
126 views

Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?

In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory ...
2
votes
1answer
174 views

strictifying tricategories

Every tricategory is equivalent to a Gray-categories. However any Gray-category is not equivalent to a 3-category. As far as I know, this is similar to the fact that braided monoidal categories are ...
1
vote
0answers
69 views

Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$. Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...
1
vote
1answer
287 views

Are chain complexes over a field always injective?

Question: Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and ...
0
votes
0answers
66 views

Subdivision of a small category

I am reading about subdivision of a category from this paper: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Delgado.pdf At the page 4, the author of this paper gives the first example ...
10
votes
1answer
289 views

Tensor product of dendroidal sets: counter-examples

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
6
votes
2answers
443 views

Primitive Recursive Arithmetic via Universal Algebra

From the wikipedia article on Primitive Recursive Arithmetic (see http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic): "Primitive recursive arithmetic, or PRA, is a quantifier-free ...
0
votes
0answers
120 views

Reference Request: Category of explicit maps between primitive recursive sets?

[Edited] Let $\mathsf{PR}$ be the category defined as follows: Choose a specific presentation of Primitive Recursive Arithmetic, that is, with a specific set of terms for primitive recursive ...
1
vote
2answers
165 views

How is a MacNeille completion “universal” like a beta-compactification is “universal”?

The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping ...
5
votes
3answers
456 views

Why are pushouts the right tool in these setups

$\newcommand{\cat}[1]{\mathcal{#1}}$ $\newcommand{\cod}{\operatorname{cod}}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\colim}{colim}$ The question is about two pushout constructions ...
1
vote
1answer
89 views

Lax monoids where only the unit triangle is lax

I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper ...
6
votes
1answer
232 views

In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective? Recall that an object ...
5
votes
1answer
294 views

Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?

Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume ...
3
votes
0answers
91 views

Hilb as a Colimit in the Category of Scott Complete Categories (foundations)

Here is a paper I found by Adamek that generalizes Domain theory into categories of categories called Scott Complete Categories. The category of Scott Complete categories is denoted SCC. For years, ...
2
votes
2answers
161 views

Computable Categories in the most direct sense?

While there is a lot of work in category related to notions of realizability and computability, etc... I've failed to find work on categories that are computable in the sense of having object and ...