Optimisation Abstraite
The basic concepts in category theory are representables, adjoints, limits, and monads. In this talk, we define the notion of a Kan extension and show that this notion encompasses these concepts.
math.CT↗
arXiv subjects
Fethi Kadhi
Publications and source records attributed to Fethi Kadhi.
Lemme de Yoneda pour les foncteurs \`a valeurs monoidales
We consider a closed symmetric monoidal category $\mathcal{M}$. We show that if $I$ is a small category then $\mathcal{M}^I$ is a closed $\mathcal{M}$-module. We rewrite the Yoneda Lemma in the case of monoidal valued functors. We derive an adjoint functor theorem and we show that $\mathcal{M}^I$ is a closed symmetric monoidal category
math.CT↗
Accessible bridge between category theory and functional programming
Monadic programming presents a significant challenge for many programmers. In light of category theory, we offer a new perspective on the use of monads in functional programming. This perspective is clarified through numerous examples coded in Haskell.
cs.PL↗
Reedy diagrams in symmetric monoidal model categories
Given a small category $I$ and a closed symmetric monoidal category $\mm$, we show that the diagram category $\mm^I$ with the objectwise product is a closed symmetric monoidal category. We then prove that if $I$ is a Reedy category and $\mm$ has a model structure compatible with its product, then so is the Reedy model structure on $\mm^I$ provided that $\mm$ is cofibrantly generated.
math.AT↗
Reedy diagrams in V-model categories
We study the category of Reedy diagrams in a $\mm$-model category. Explicitly, we show that if K is a small category, V is a closed symmetric monoidal category and C is a closed V-module, then the diagram category V^K is a closed symmetric monoidal category and the diagram category C^K is a closed V^K-module. We then prove that if further K is a Reedy category, V is a monoidal model category and C is a V-model category, then with the Reedy model category structures, V^K is a monoidal model category and C^K$ is a $\mm^K-model category provided that either the unit 1 of V is cofibrant or V is cofibrantly generated.
math.AT↗
Analytical Study of a Class of Rational Difference Equations
We obtain the solution of the fourth order difference equation $$ x_{n+1}=\frac{ αx_{n-3}}{A+B x_{n-1}x_{n-3}}$$ with the initial conditions; $x_{-3}=d,$ $x_{-2}=c,$ $x_{-1}=b,$ and $x_{0}=a$ are arbitrary nonzero real numbers, $α$, $A$ and $B$ are arbitrary constants. The result is used to study the convergence of solutions, the existence of unbounded solutions and the convergence to periodic solutions. We illustrate the results by several numerical examples.
math.DS↗