4 votes

When is $f$ not the coequalizer of its kernel pair?

As Zhen Lin noted in the comments, coequalizers are always epimorphisms. So, when $f$ is not an epimorphism, it will never be the coequalizer of its kernel pair. For a concrete example, in the ...
user avatar
  • 4,902
3 votes

Formalizing the universal property of the tensor product

It's a good question. I would personally say that this definition of universal property is not the best one and is not what comes naturally in many situations. On the other hand, I can tell you one ...
user avatar
  • 21.8k
2 votes

How to construct a Segal category from a quasicategory?

The canonical reference is the paper Quasi-categories vs Segal spaces by André Joyal and Myles Tierney. It works in the setting of Segal spaces, but there are standard tools to move between Segal ...
user avatar
  • 1,608
2 votes
Accepted

On the cartesian closed T-spaces

Regarding the first question, your interpretation is correct. Regarding the second, one can interpret "covering" as sending a pullback to what is known as a weak pullback. One can then show ...
user avatar
2 votes

When is $f$ not the coequalizer of its kernel pair?

The simplest example is the category with two distinct objects $X$ and $Y$, and a one non-identity morphism $f\colon X\to Y$. The coequalizer of its kernel pair is the identity morphism on $X$. Note ...
user avatar
1 vote

Initial object, final object and unit and counit

First one has to define $\epsilon_d$ for every $d$. For this, use the initiality of $\eta_{Gd}$: there is a unique map $\epsilon_d: FGd \to d$ making the triangle $1_{Gd}: Gd \xrightarrow{\eta_{Gd}} ...
user avatar
1 vote

Symmetries of a non-associative mapping

The paper "Plots and Their Applications - Part 1: Foundations" by Salvatore Tringali, available at https://arxiv.org/abs/1311.3524v1, explores how characterizations of objects by universal ...
user avatar
1 vote
Accepted

Continuity of the fibre product projection

It is clear that the x coordinates of the fibered product form a subset of X. Project from the fibered product to that subset, and utilize the definition of the product topology. Then project from ...
user avatar
  • 221
1 vote

Does the simplicial-set internal hom of quasicategories give a quasicategory?

Reference: see Lurie, Higher Topos Theory, Proposition 1.2.7.3. Also see the Proposition following 2.2.5.7; it is a proof of 1.2.7.3.
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible