This thesis is an exposition of the author's contribution on effective descent morphisms in various categories generalized categorical structures. It consists of: Chapter 1, where elementary description theory and content each remaining chapter provided, supplemented with references; 2, consisting theoretical definitions results employed remainder this work; four chapters, corresponding to article written by author during period his PhD studies.

We study effective descent V-functors for cartesian monoidal categories V with finite limits. This is carried out via the properties enjoyed by 2-functor V↦Fam(V), results about of bilimits categories, and fact that enrichment preserves certain bilimits. Since these rely on an understanding (effective) morphisms in Fam(V), we carefully free coproduct completions. Finally, provide refined conditions when a regular category.

We investigate the properties of lax comma categories over a base category $X$, focusing on topologicity, extensivity, cartesian closedness, and descent. establish that forgetful functor from $\mathsf{Cat}//X$ to $\mathsf{Cat}$ is topological if only $X$ large-complete. Moreover, we provide conditions for be complete, cocomplete, extensive closed. analyze descent in identify necessary effective morphisms. Our findings contribute literature foundation further research 2-dimensional...

We consider the canonical pseudodistributive law between various free limit completion pseudomonads and coproduct pseudomonad. When class of limits includes pullbacks, we show that this consideration leads to notions extensive categories. More precisely, categories with pullbacks infinitary lextensive are pseudoalgebras for resulting from two these pseudo­distributive laws. Moreover, introduce notion doubly-infinitary category , establish freely generated ones cartesian closed. From result,...

Abstract We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study morphisms are pursued. The first one relies on establishing the category multicategories as an equalizer diagrams. second approach extends techniques developed by Ivan Le Creurer his essentially algebraic structures.

For any suitable monoidal category $\mathcal{V}$, we find that $\mathcal{V}$-fully faithful lax epimorphisms in $\mathcal{V} \dash \mathcal{V}$ are precisely those $\mathcal{V}$-functors $F: \mathcal{A} \to \mathcal{B}$ whose induced ${\mathfrak C} F: {\mathfrak between the Cauchy completions equivalences. case $\mathcal{V}= {\rm Set}$, this is equivalent to requiring functor $F^*:{\rm CAT}({\mathcal A},{\rm Cat}) B}, Cat})$ an equivalence. By reducing study of effective descent functors...

We consider the canonical pseudodistributive law between various free limit completion pseudomonads and coproduct pseudomonad. When class of limits includes pullbacks, we show that this consideration leads to notions extensive categories. More precisely, categories with pullbacks infinitary lextensive are pseudoalgebras for resulting from laws. Moreover, introduce notion doubly-infinitary category, establish freely generated such cartesian closed. From result, further deduce that, in...

We present a characterization of effective descent morphisms in the lax comma category $\mathsf{Ord}//X$ when $X$ is locally complete ordered set with bottom element.

Abstract Via the adjunction $$ - *\mathbbm {1} \dashv \mathcal V(\mathbbm {1},-) :\textsf {Span}({\mathcal {V}}) \rightarrow {\mathcal {V}} \text {-} \textsf {Mat} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mn>1</mml:mn> <mml:mo>⊣</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mo>,</mml:mo> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>Span</mml:mi> <mml:mo>→</mml:mo> <mml:mtext>-</mml:mtext>...

Effective descent morphisms, originally defined in Grothendieck theory, form a class of special morphisms within category. Essentially, an effective morphism enables bundles over its codomain to be fully described as domain endowed with additional algebraic structure, called data. Like the study epimorphisms, studying is interesting own right, providing deeper insights into category under consideration. Moreover, these part foundations several applications notably including Janelidze-Galois...

We study effective descent $ \mathcal V $-functors for cartesian monoidal categories with finite limits. This is carried out via the properties enjoyed by $2$-functor \mapsto \mathsf{Fam}(\mathcal V) $, results about of bilimits categories, and fact that enrichment preserves certain bilimits. Since these rely on an understanding (effective) morphisms in we briefly those epimorphisms when a regular category.

Via the adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Span}(\mathcal V) \to V \text{-} \mathsf{Mat} and a cartesian monad T on an extensive category with finite limits, we construct \mathsf{Cat}(T,\mathcal (\overline T, V)\text{-}\mathsf{Cat} between categories of generalized enriched multicategories internal multicategories, provided satisfies suitable condition, which is satisfied by several examples. We verify, moreover, left adjoint fully faithful, preserves...

For any suitable base category $\mathcal{V} $, we find that $-fully faithful lax epimorphisms in $-$\mathsf{Cat} $ are precisely those $\mathcal{V}$-functors $F \colon \mathcal{A} \to \mathcal{B}$ whose induced $-functors $\mathsf{Cauchy} F \mathsf{Cauchy} \mathcal{B} between the Cauchy completions equivalences. case = \mathsf{Set} this is equivalent to requiring functor $\mathsf{CAT} \left( F,\mathsf{Cat}\right) categories of split (op)fibrations an equivalence. By reducing study effective...

We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study morphisms are pursued. The first one relies on establishing the category multicategories as an equalizer diagrams. second approach extends techniques developed by Ivan Le Creurer his essentially algebraic structures.