Definition df-termc 50715

Description: Definition of the proper class (termcnex 50821) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 50728). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50757). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50762). See also dftermc3 50773 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50773, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 50733.

Followed directly from the definition, terminal categories are thin (termcthin 50719). The opposite category of a terminal category is "almost" itself (oppctermco 50747). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50785).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50743) and terminal objects (dftermo4 50744). The universal properties provide an alternate proof of initoeu1 18193, termoeu1 18200, initoeu2 18198, and termoeu2 50467. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50758).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17869, 0thincg 50700, func0g 50311, 0funcg 50307, and initc 50313.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 50714	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1568	. . . . . 6 class 𝑐
4		cbs 17394	. . . . . 6 class Base
5	3, 4	cfv 6541	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1568	. . . . . 6 class 𝑥
8	7	csn 4584	. . . . 5 class {𝑥}
9	5, 8	wceq 1569	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1811	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 50659	. . 3 class ThinCat
12	10, 2, 11	crab 3412	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1569	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is used by:  istermc  50716