درود بر همراهان گرامی،
آقای دکتر امیر همایون نجاح لطف فرمودند در این درس با من همکاری خواهند داشت. برخی از مفاد درسی (جبر، هم جبر بطور کلی تریپل ها) و همچنین حل تمرین و مثال‌های بیشتر توسط ایشان مدیریت خواهد شد.

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Let C be a category, prove that Set^C is both finitely complete and cocomplete. What about completeness and cocompleteness in general?

Let D:I ×J -> C be a diagram. Prove that lim_i lim_j D is isomorphic to lim_j lim_i D.

This is the generalisation of associativity

برای نحوه‌ی ارائه سوالات عذرخواهی میکنم. سعی ام بر این خواهد بود که تایپ شده باشه.

The category Rel has sets as its objects and morphisms are relations

If R is a relations from X to Y and S is a relation from Y to Z then S○R is defined such that (x,y) is in S○R iff there exists z in Y such that (x,z) is in R and (z,y) is in S.

You can see that Set is a subcategory of this category

Find limit and colimit of Rel

Specifically the product in this category is very interesting

What is the dual of Rel

Let C be a Cartesian closed category (CCC for short) with an initial object 0

Prove that if there exists a morphism from some object X to 0 then X is itself an initial object.

Please do not forget to do the exercise mentioned in class by Prof. Alizadeh which was the category of groups does not have exponential objects